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+{"text":"\\section{Introduction}\nIn his 1984 AMS Memoir, George Andrews \n\\cite{AndMem} \ndefined two families of combinatorial objects known as {\\it generalized Frobenius partitions}.  These are generalizations of the two--rowed arrays, often known as Frobenius symbols, which arise from considering the rows and columns of the Ferrers graph of an ordinary partition once the Durfee square has been ``removed''.  In the process, Andrews defined two families of functions, $\\phi_k(n)$ and $c\\phi_k(n),$ as the number of generalized Frobenius partitions of weight $n$ in these two families of objects, respectively.  In \n\\cite{AndMem}, Andrews studied these functions $\\phi_k(n)$ and $c\\phi_k(n)$ from several perspectives, including proving a number of Ramanujan--like congruences satisfied by these functions.  This, in turn, led a number of others to extend Andrews' congruence results.  \n\nWhile there exists an extensive literature on the subject of congruences satisfied by generalized Frobenius partition functions, our focus in this note will be on parity results.  We highlight here that a number of authors have proven congruence results with even moduli for these functions; see, for example, the work of \nAndrews \n\\cite[Theorem 10.2]{AndMem}, \nBaruah and Sarmah \n\\cite{BarSar1, BarSar2},   \nChan, Wang, and Yang \n\\cite{ChanWangYang}, \nCui and Gu, \n\\cite{CuiGu}, \nCui, Gu, and Huang \n\\cite{CuiGuHuang},  \nand \nJameson and Wieczorek \n\\cite{JamWie}\nwhere specific congruence results with even moduli are proved.  Several additional papers involving congruence results for generalized Frobenius partitions, but with odd moduli, also appear in the literature.  \n\nWhat is striking about many of the works cited above is that the authors focus specifically on a particular value of the parameter $k$ in order to manipulate the generating function in question to prove their results.  One exception to this rule is Andrews' Theorem 10.2 in \n\\cite[Theorem 10.2]{AndMem}:  \n\n\\begin{theorem}\nLet $p$ be prime and and let $r$ be an integer such that $0<r<p.$ For all $n\\geq 0,$ $$c\\phi_p(pn+r) \\equiv 0\\pmod{p^2}.$$ \n\\end{theorem}\n\nAnother exception to focusing on a particular value of the subscript $k$ appears in the work of Garvan and Sellers \n\\cite[Theorem 2.2]{GarSel}\nwhere the authors prove the following theorem:  \n\n\\begin{theorem}\n\\label{GarSelmainthm}\nLet $p$ be prime and let $r$ be an integer such that $0<r<p.$  If \n$$\nc\\phi_k(pn+r) \\equiv 0\\pmod{p}\n$$ \nfor all $n\\geq 0,$ then \n$$\nc\\phi_{pN+k}(pn+r) \\equiv 0\\pmod{p}\n$$ \nfor all $N\\geq 0$ and $n\\geq 0.$\n\\end{theorem}\nOur goal in this note is to follow a path similar to the above theorem of Garvan and Sellers, where an infinite family of values of $k$ is identified while the value of the modulus is fixed.  \n\nIt is clear, when one reviews the literature on the subject of congruences satisfied by generalized Frobenius partition functions, that the functions $c\\phi_k(n)$ satisfy many more congruences than their counterpart functions $\\phi_k(n).$  One might argue that this is true ``combinatorially'' given the structure of the objects being counted by these functions (and symmetries that are inherent in the two--rowed arrays counted by $c\\phi_k(n)$). It is also true that, although the generating functions for each of these two families of functions are extremely similar, the presence of certain powers of roots of unity in the generating function for $\\phi_k(n),$ and the corresponding absence of such roots of unity in the generating function for $c\\phi_k(n),$ may contribute to the relative lack of congruences satisfied by $\\phi_k(n).$  Whatever the case, our primary goal in this note is to alter this narrative by proving the following surprising result:  \n\n\\begin{theorem}\n\\label{mainresult} \nLet $\\ell$ be a positive integer,  $p\\geq 5$ be prime, and let  $r,$ $ 0< r < p,$ be an integer such that $24r+1$ is a quadratic nonresidue modulo $p.$ For all $n\\geq 0,$\n $$\n \\phi_{p\\ell-1}(pn+r) \\equiv 0 \\pmod{2}.\n $$\n\\end{theorem}\n\n\n\\section{Proof of Theorem \\ref{mainresult}}\n\nIn order to prove Theorem \\ref{mainresult}, we need a few preliminary facts.  First, we remind the reader of the $q$--Pochhammer symbol which is defined as follows:  \n$$(A;q)_\\infty = (1-A)(1-Aq)\\cdots(1-Aq^{n})\\cdots$$\n\nWe will also need the following two well--known results:  \n\\begin{theorem}\n\\label{eulerPNT}\n\\begin{equation*}\n    (q;q)_\\infty = \\sum\\limits_{k = -\\infty}^\\infty (-1)^k q^{\\frac{3k^2-k}{2}}\n\\end{equation*}\n\\end{theorem}\n\\begin{proof}\nSee Hirschhorn \n\\cite[(1.6.1)]{Hir}.  \n\\end{proof}\n\n\\begin{theorem}\n\\label{jacobi_cube} \n\\begin{equation*}\n    (q;q)_\\infty^3 = \\sum\\limits_{k=0}^\\infty (-1)^k(2k+1)q^{\\frac{k^2+k}{2}}.\n\\end{equation*}\n\\end{theorem}\n\\begin{proof}\nSee Hirschhorn \n\\cite[(1.7.1)]{Hir}.  \n\\end{proof}\n\nNext, we prove an extremely important fact about the generating function for $\\phi_k(n)$ for all $k\\geq 1$ using a key result that appears in Andrews \n\\cite{AndMem}.\n\n\\begin{theorem}\n\\label{Mod2Lemma} \nLet $$\n\\Phi_k(q) = \\sum_{n=0}^\\infty \\phi_k(n)q^n\n$$\nbe the generating function for the generalized Frobenius partition function $\\phi_k(n).$  Then \n$$\n\\Phi_k(q) \\equiv \\frac{(q;q)_\\infty}{(q^{k+1};q^{k+1})_\\infty} \\pmod{2}.  \n$$\n\\end{theorem}\n\\begin{proof}\nFrom Andrews' Memoir \n\\cite[Theorem 7.1]{AndMem},\nwe know \n\\begin{equation*}\n    \\Phi_k(q) = \\frac{1}{ (q;q)_\\infty^2(q^{k+1};q^{k+1})_\\infty}\\sum\\limits_{j,r = -\\infty \\atop r \\geq (k+1)|j|}^{\\infty}(-1)^{r+kj}q^{\\binom{r+1}{2}-\\binom{k+1}{2}j^2}.\n\\end{equation*}\nTherefore, we have \n\\begin{eqnarray*}\n\\Phi_k(q) \n&=& \n\\frac{1}{ (q;q)_\\infty^2(q^{k+1};q^{k+1})_\\infty}\\sum\\limits_{j,r = -\\infty \\atop r \\geq (k+1)|j|}^{\\infty}(-1)^{r+kj}q^{\\binom{r+1}{2}-\\binom{k+1}{2}j^2}\\\\\n&= & \n\\frac{(q;q)_\\infty}{ (q;q)_\\infty^3(q^{k+1};q^{k+1})_\\infty}\\sum\\limits_{r =0}^\\infty q^{\\binom{r+1}{2}} \\sum\\limits_{|j|\\leq r\/(k+1)}(-1)^{r+kj}q^{-\\binom{k+1}{2}j^2}  \\\\\n&\\equiv & \n\\frac{(q;q)_\\infty}{ (q;q)_\\infty^3(q^{k+1};q^{k+1})_\\infty}\\sum\\limits_{r =0}^\\infty q^{\\binom{r+1}{2}} \\sum\\limits_{|j|\\leq r\/(k+1)}q^{-\\binom{k+1}{2}j^2} \\pmod{2} \\\\\n&=& \n\\frac{(q;q)_\\infty}{ (q;q)_\\infty^3(q^{k+1};q^{k+1})_\\infty}\\sum\\limits_{r =0}^\\infty q^{\\binom{r+1}{2}}\\left(1+2 \\sum\\limits_{1\\leq j \\leq r\/(k+1)}q^{-\\binom{k+1}{2}j^2} \\right)   \\\\\n&\\equiv & \n\\frac{(q;q)_\\infty}{ (q;q)_\\infty^3(q^{k+1};q^{k+1})_\\infty}\\sum\\limits_{r =0}^\\infty q^{\\binom{r+1}{2}} \\pmod{2} \\\\\n&\\equiv & \n\\frac{(q;q)_\\infty(q;q)_\\infty^3}{ (q;q)_\\infty^3(q^{k+1};q^{k+1})_\\infty} \\pmod{2} \\\\\n\\end{eqnarray*}\nthanks to Theorem \\ref{jacobi_cube}.  The result immediately follows.  \n\\end{proof} \n\n\nWe are now in an excellent position to prove Theorem \\ref{mainresult}.  \n\n\\begin{proof}(of Theorem \\ref{mainresult}) \nThanks to Theorem \\ref{Mod2Lemma}, we know that the generating function for $\\phi_{p\\ell-1}$ satisfies \n\\begin{eqnarray*}\n\\sum_{n=0}^\\infty \\phi_{p\\ell-1}(n)q^n \n&\\equiv &\n\\frac{(q;q)_\\infty}{(q^{p\\ell};q^{p\\ell})_\\infty} \\pmod{2} \\\\\n&\\equiv & \n\\frac{1}{(q^{p\\ell};q^{p\\ell})_\\infty} \\sum\\limits_{k = -\\infty}^\\infty q^{\\frac{3k^2-k}{2}} \\pmod{2}  \n\\end{eqnarray*}\nwhere the last statement follows from Theorem \\ref{eulerPNT}.  \nSince $(q^{p\\ell};q^{p\\ell})_\\infty$ is a function of $q^p,$ and since we are interested in the parity of the values $\\phi_{p\\ell-1}(pn+r)$ where $0<r<p,$ we simply need to determine when $$pn+r = \\frac{3k^2-k}{2}$$ for some integer $k.$  After completing the square, this is equivalent to asking whether\n\\begin{align*}\n    24r+1 & \\equiv 36k^2 -12k + 1 \\pmod{p}\\\\\n    & = (6k-1)^2.\n\\end{align*}\nHowever, we assumed that $24r+1$ is a quadratic nonresidue modulo $p$ in the statement of this theorem.   Therefore, $pn+r$ can never be represented as $\\frac{3}{2}k^2-\\frac{1}{2}k$ for any integer $k$. This implies that \n$$\\phi_{p\\ell-1}(pn+r) \\equiv 0 \\pmod{p}.$$\n\\end{proof}\n\n\\section{Concluding Remarks}  \nIt should be noted that a companion result to Theorem \\ref{mainresult}  exists for the functions $c\\phi_k(n)$ as well (which is not all that surprising).  \n\n\\begin{theorem} \n\\label{cphi_mod2} \nFor all $k\\geq 1$ and all $n\\geq 0,$\n $$\n c\\phi_{2k}(2n+1) \\equiv 0 \\pmod{2}.\n $$\n\\end{theorem}\n\nIn a real sense, the proof of this result already appears in the literature; see, for example, Garvan and Sellers \n\\cite{GarSel}\nas well as Baruah and Sarmah \n\\cite{BarSar2}. \nFor completeness' sake, we provide a proof here.  \n\n\\begin{proof} \nThe generating function for $c\\phi_{2k}(n),$ as provided by Andrews \n\\cite{AndMem},\nis the constant term in \n$$\nCG_{2k}(z) = \\prod_{n=0}^\\infty(1+zq^{n+1})^{2k}(1+z^{-1}q^n)^{2k}.\n$$\nThanks to the Binomial Theorem, we have \n\\begin{align*}\n    CG_{2k}(z) \n    &= \\prod_{n=0}^\\infty(1+zq^{n+1})^{2k}(1+z^{-1}q^n)^{2k}\\\\\n    & \\equiv \\prod_{n=0}^\\infty(1+z^2q^{2n+2})^{k}(1+z^{-2}q^{2n})^{k} \\pmod{2}.\n\\end{align*}\nThe theorem immediately follows because the last product above is a function of $q^2.$\n\\end{proof}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe goal of this survey paper is to give an overview of results for the uniqueness of bounded solutions\nof the heat equation with continuous time parameter, aka stochastic completeness, on infinite weighted graphs.\nWe first discuss some equivalent formulations of stochastic completeness and how\nwe have to go beyond the realm of bounded operators on graphs\nin order for this property to be of interest.  \nFurthermore, we discuss how the combinatorial graph distance is not an appropriate choice\nof metric for the purpose of finding results analogous to those found in the setting of Riemannian manifolds.\nThis leads to the use of so-called intrinsic metrics which give an appropriate notion of\nvolume growth. Finally, we discuss how some recently developed\nversions of curvature on graphs can be used to give conditions for stochastic completeness.\n\n\n\\subsection{Stochastic completeness, uniqueness class and volume growth}\nIn 1986, Alexander Grigor$\\cprime$yan published an optimal volume growth condition\nfor the stochastic completeness of a geodesically complete Riemannian manifold. More specifically, if\na geodesically complete manifold $M$ satisfies\n$$\\int^\\infty \\frac{r}{\\log V(r)}dr =\\infty$$\nwhere $V(r)$ denotes the volume of a ball of radius $r$ with an arbitrary center, then \n$M$ is stochastically complete \\cite{Gri86}. In particular, we note\nthat any Riemannian manifold with volume growth satisfying $V(r) \\leq Ce^{r^2}$ for $C>0$\nwill be stochastically complete. This result improved all other known volume growth\ncriteria at the time \\cite{Gaf59, KLi} and is sharp in the sense that there exist stochastically incomplete manifolds\nwith volume growth of order $e^{r^{2+\\epsilon}}$ for any $\\epsilon>0$. In particular, the class of model manifolds\nalready provides such examples, see the survey article of Grigor$'$yan for this and \nmany other results \\cite{Gri99}.\n\nGrigor$'$yan's volume growth criterion was proven via a uniqueness class result for solutions\nof the heat equation. More specifically, if $u$ is a solution of the heat equation on $M \\times (0,T)$\nwith initial condition $0$ and for all $r$ large $u$ satisfies\n$$\\int_0^T \\int_{B_r} u^2(x,t)\\  d\\mu \\ dt \\leq e^{f(r)}$$\nwhere $f$ is a monotone increasing function on $(0,\\infty)$ which satisfies\n$$\\int^\\infty \\frac{r}{f(r)}dr=\\infty,$$\nthen $u=0$ on $M \\times (0,T)$.\nIt follows that bounded solutions of the heat equation are unique by taking the difference\nof two solutions and letting $f(r)=\\log(C^2TV(r))$ where $C$ is a bound on the solutions.\nSee \\cite{Gaf59, KLi, Dav92, Tak89} for other techniques for proving volume\ngrowth criteria for stochastic completeness in the manifold setting.\n\nIn the setting of graphs, an explicit study of geometric conditions for the \nuniqueness of bounded solutions of the heat equation\nwith continuous time parameter can be found in\nwork of J{\\'o}zef Dodziuk and Varghese Mathai \\cite{DM06} as well as that of Dodziuk \\cite{Dod06}\nand subsequently taken up in the author's Ph.D. thesis \\cite{Woj08} and independently\nin work of Andreas Weber \n\\cite{Web10}. In particular,\nDodziuk\/Mathai show that whenever the Laplacian on a graph with standard weights is a bounded\noperator, then the graph is stochastically complete. Dodziuk then extends this result\nto allow weights on edges. The technique used to establish these results is that \nof a minimum principle for the heat equation.\n\nIn the thesis \\cite{Woj08} and follow-up paper \\cite{Woj09}\nwe rather use an equivalent formulation of stochastic completeness in terms\nof bounded $\\lambda$-harmonic functions to derive criteria for stochastic completeness which allow for\nunbounded operators. Furthermore, we give a full characterization of stochastic completeness\nin the case of trees which enjoy a certain symmetry. This characterization\nalready shows a disparity between the graph and manifold\nsettings in that there  exist stochastically incomplete graphs with factorial volume growth, that is, \nif $V(r)$ denotes the  \nnumber of vertices within $r$ steps of a center vertex, then the tree is stochastically\nincomplete and $V(r)$ grows factorially with $r$.\nHowever, a more striking disparity appeared in a subsequent paper \nwhich introduced a class of graphs called anti-trees. \nThese graphs can be stochastically incomplete and have polynomial volume growth \\cite{Woj11}. \nIn particular, there exist stochastically incomplete anti-trees with volume growth like \n$$V(r)\\sim r^{3+\\epsilon}$$\nfor any $\\epsilon>0$. This provides a very strong contrast with the borderline for the manifold case\ngiven by Grigor$'$yan's result.\n\nThe volume growth in these examples involves taking balls via the usual combinatorial graph metric, that is,\ntaking the least number of edges in a path connecting two vertices. This notion reflects only the global connectedness properties of the graph. However, it is natural to expect that a metric should\nalso reflect the local geometry of a graph, i.e., the valence or degree of vertices. Furthermore, if the\ngraph has weights on both edges and a measure on vertices, then the metric should \ninteract with both the edge weights and the vertex measure.\nFor Riemannian manifolds, there exists the notion of an intrinsic metric which naturally arises\nfrom the energy form as well as from\nthe geometry of the manifold. This notion of an intrinsic metric for the energy form was then extended to strongly \nlocal Dirichlet forms by Karl-Theodor Sturm \\cite{Stu94}. Now, graphs which have both a weight on \nedges and a measure on vertices can be put into a one-to-one correspondence\nwith regular Dirichlet forms on discrete measure spaces as discussed in work of Matthias Keller and Daniel Lenz \\cite{KL12}. \nHowever, the Dirichlet forms that arise from graphs are not strongly local. Thus, the notion\nof an intrinsic metric from strongly local Dirichlet froms has to be extended to the non-local setting. \nThis was done systematically in a paper by Rupert Frank, Daniel Lenz and Daniel Wingert \\cite{FLW14}. \n\nThe notion of intrinsic metrics for non-local Dirichlet forms was quickly put to use by the graph theory community. \nA first example of a concrete intrinsic metric for weighted graphs already appers in the \nPh.D. thesis of Xueping Huang \\cite{Hua11b} and can also be found in the work of Matthew Folz \non heat kernel estimates around the same time \\cite{Fol11}.\nHowever, even given the tool of intrinsic metrics, there are still difficulties in proving an analogue\nto Grigor$'$yan's criterion for graphs. In particular, Huang gives an example of a graph for which there\nexists a non-zero bounded solution of the heat equation $u$ with zero initial condition which satisfies\n$$\\int_0^T \\int_{B_r} u^2(x,t)\\  d\\mu \\ dt \\leq e^{f(r)}$$\nfor $f(r)=C r \\log r$ for some constant $C$, see \\cite{Hua11b, Hua12}. \nHence, as $f$ in this case clearly satisfies $\\int^\\infty {r}\/{f(r)} \\ dr=\\infty$, \nwe see that even when using intrinsic metrics, a direct analogue\nto Grigor$'$yan's proof is not possible for all graphs. \n\nA recent breakthrough in resolving this issue can be found in the work of Xueping Huang, Matthias Keller\nand Marcel Schmidt \\cite{HKS}. In this paper, the authors first prove a uniqueness class result\nwhich is valid for a certain class of graphs called globally local. They can then\nreduce the study of stochastic completeness of general graphs to that of globally local graphs using\nthe technique of refinements first found in \\cite{HS14}. With these two results, they are able to establish\nan exact analogue to the volume growth criterion of Grigor$'$yan which is valid for all graphs. That is, letting\n$V_\\varrho(r)$ denote the measure of a ball with respect to an intrinsic metric and letting \n$\\log^{\\#}(x)=\\max\\{\\log(x), 1\\}$ if\n$$\\int^\\infty \\frac{r}{\\log^{\\#} V_\\varrho(r)} dr = \\infty,$$\nthen the graph is stochastically complete. We note that taking the minimum with 1 \nis only necessary to cover the case of when the entire graph has small measure;\nthe actual value of the constant 1 is not relevant.\n\nLet us mention that the volume growth criterion for stochastic completeness of graphs involving\nintrinsic metrics was first proven under \nsome additional assumptions by Folz \\cite{Fol14}. \nThe proof technique of Folz, however, is different from that of Grigor$'$yan.\nMore specifically, Folz bypasses Grigor$'$yan's uniqueness class technique via a\nprobabilistic approach involving synchronizing the random walk on the graph with a random walk\non an associated quantum graph and then applying a generalization of Grigor$'$yan's result for manifolds\nto strongly local Dirichlet forms found in work of Sturm \\cite{Stu94}. A similar proof involving quantum graphs\nbut using analytic techniques can also be found in a paper by Huang \\cite{Hua14b}.  \n\n\nWe would also like to highlight earlier work focused on a volume growth criterion \nby Alexander Grigor$'$yan, Xueping Huang and Jun Masamune \\cite{GHM12} \nusing a technique from \\cite{Dav92}. While this did not yield the optimal volume growth condition when using\nintrinsic metrics, it did yield an optimal volume growth condition for the combinatorial graph metric in that \n$$V(r) \\leq Cr^3$$\nimplies stochastic completeness where $V(r)$ is the volume defined with respect to the combinatorial graph metric.\nThus, we see that the anti-tree examples found in \\cite{Woj11} are the smallest stochastically incomplete\ngraphs in the combinatorial graph distance. \n\n\n\\subsection{Curvature and stochastic completeness}\nLet us now turn to curvature. For Riemannian manifolds, in a paper from 1974, \nRobert Azencott gave both a curvature criterion for stochastic completeness and the first examples of stochastically incomplete manifolds \\cite{Az74}.\nIn Azencott's example, the curvature decays to negative infinity rapidly, thus it is natural to expect that\nlower curvature bounds are necessary for stochastic completeness.\nAn optimal result in this direction involving Ricci curvature\nwas established by Nicholas Varopoulos \\cite{Var83} and Pei Hsu\n\\cite{Hsu89}. It can be formulated as follows: let $M$ be a geodesically complete Riemannian manifold\nand suppose that $\\kappa$ is a positive increasing continuous function on $(0,\\infty)$ such that for all points\naway from the cut locus on the sphere of radius $r$ we have $\\mathrm{Ric}(x)\\geq -C \\kappa^2(r)$ for all $r$ large\nand $C>0$. \nIf\n$$\\int^\\infty \\frac{1}{\\kappa(r)}dr=\\infty,$$\nthen $M$ is stochastically complete. This improved the previously known results which gave that Ricci curvature\nuniformly bounded from below implied stochastic completeness as proven by Shing-Tung Yau \\cite{Yau78}, \nsee also the work of Dodziuk \\cite{Dod83}. However, due to the connection between\nRicci curvature and volume growth, this result is already implied by Grigor$'$yan's volume growth result.\nThere is also a number of comparison results for stochastic completeness involving curvature, see\n\\cite{Ichi82} or Section 15 in the survey of Grigor$'$yan \\cite{Gri99}.\n\nIn recent years, there has been a tremendous interest in notions of curvature\non graphs. We focus here on two formulation. One definition of curvature \noriginates in work of Dominique Bakry\nand Michele \\'{E}mery on hypercontractive semigroups \\cite{BE85}. Thus, we refer to it as Bakry--\\'{E}mery curvature.\nA second formulation comes from the work of Yann Ollivier on Markov chains on metric spaces in \\cite{Oll07, Oll09}.\nThis was later modified to give an infinitesimal version by Yong Lin, Linyuan Lu and Shing-Tung Yau \\cite{LLY11}\nand then extended to the case of possibly unbounded operators on graphs by Florentin M{\\\"u}nch and the author \\cite{MW19}. In any case, we refer to this as Ollivier Ricci curvature.\nFor Bakry--\\'{E}mery curvature, Bobo Hua and Yong Lin proved that a uniform lower bound implies stochastic completeness\nin \\cite{HL17}.\nOn the other hand, in \\cite{MW19} we prove that for Ollivier Ricci curvature, if\n$$\\kappa(r) \\geq -C\\log r$$\nfor $C>0$ and all large $r$ where $\\kappa(r)$ denotes the spherical curvature on a sphere\nof radius $r$, then the graph is stochastically complete.\nThis is optimal in the sense that for any $\\epsilon>0$ there exist stochastically incomplete\ngraphs with $\\kappa(r)$ decaying like $-(\\log r)^{1+\\epsilon}$. Thus, there is still a disparity in this condition for graphs and\nfor the Ricci curvature condition for manifolds as presented above. However, this disparity cannot\nbe resolved by using the notion of intrinsic metrics as was the case for stochastic completeness and volume growth.\n \n\\subsection{The structure of this paper}\nWe now briefly discuss the structure of this paper. Although we do not give full proofs of results,\nwe also do not assume any particular background of the reader and thus try to make the presentation\nas self-contained as possible in terms of concepts and definitions. We also give specific references\nfor all results that we do not prove completely.\n\nIn Section~\\ref{s:heat} we introduce the setting of weighted graphs and discuss the heat\nequation. In particular, we outline an elementary construction of bounded solutions of the \nheat equation using exhaustion techniques. In Section~\\ref{s:sc} we present some equivalent formulations for \nstochastic completeness. In particular, stochastic completeness is equivalent to the uniqueness of this bounded solution\nof the heat equation. In Section~\\ref{s:bounded} we discuss boundedness of the Laplacian and\nhow boundedness is related to stochastic completeness. In Section~\\ref{s:symmetry} we then introduce\nthe class of weakly spherically symmetric graphs and present the examples of anti-trees which show the\ndisparity between the continuous and discrete settings in the case of the combinatorial graph metric.\nIn Section~\\ref{s:intrinsic} we introduce intrinsic metrics and discuss how they can differ from the\ncombinatorial graph metric and how this is related to stochastic completeness. \nFinally, in Sections~\\ref{s:volume}~and~\\ref{s:curvature} we present the criteria\nfor stochastic completeness in terms of volume growth and curvature mentioned above.\n\nWe also mention here a recent survey article by Bobo Hua and Xueping Huang which has some contact\npoints with our article but also discusses heat kernel estimates, ancient solutions of the heat\nequation and upper escape rates \\cite{HH}.\n\n\n\\section{The heat equation on graphs}\\label{s:heat}\n\\subsection{Weighted graphs}\nWe start by introducing our setting following \\cite{KL12}. We note that this setting is very general\nin that we allow for arbitrary weights on both edges and vertices. We also do not assume local finiteness,\ni.e., that every vertex has only finitely many edges coming out of the vertex.\n\n\n\\begin{definition}[Weighted graphs] \nWe let $X$ be a countably infinite set whose elements we refer to as \n\\emph{vertices}. We then let $m:X \\longrightarrow (0,\\infty)$ denote a \\emph{measure} on the vertex set which can\nbe extended to all subsets by additivity. Finally, we let $b:X \\times X \\longrightarrow [0,\\infty)$ denote a function\ncalled the \\emph{edge weight} which satisfies\n\\begin{enumerate}\n\\item[(b1)] \\quad $b(x,x)=0$ for all $x \\in X$\n\\item[(b2)] \\quad $b(x,y)=b(y,x)$ for all $x, y \\in X$\n\\item[(b3)] \\quad $\\sum_{y \\in X} b(x,y) <\\infty$ for all $x \\in X$.\n\\end{enumerate}\nWhenever $b(x,y)>0$, we think of the vertices $x$ and $y$ as being \\emph{connected} by an edge with \nweight $b(x,y)$, call $x$ and $y$ \\emph{neighbors} and write $x \\sim y$. \nThus, (b1) gives that there are no loops,\n(b2) that edge weights are symmetric and (b3) that the total sum of the edge weights is finite.\nWe call the triple $G=(X, b, m)$ a \\emph{weighted graph} or just \\emph{graph} for short.\n\\end{definition}\n\nWe note, in particular, that condition (b3) above allows for a vertex to have infinitely many neighbors.\nWhenever, each vertex has only finitely neighbors, we call the graph \\emph{locally finite}. We call the quantity\n$${\\mathrm{Deg}}(x)= \\frac{1}{m(x)}\\sum_{y \\in X} b(x,y)$$\nthe \\emph{weighted vertex degree} of $x \\in X$ or just \\emph{degree} for short. We will see\nthat this function plays a significant role in what follows.\n\n\\begin{example}\nWe now present some standard choices for $b$ and $m$ to help orient the reader.\nIn particular, we discuss the case of standard edge weights, counting and degree measures.\n\\begin{enumerate}\n\\item Whenever $b(x,y) \\in \\{0,1\\}$ for all $x, y \\in X$, we say that the graph has \\emph{standard edge weights}.\nIn this case, it is clear that condition (b3) in the definition of the edge weights \nimplies that the graph must be locally finite.\n\\item One choice of vertex measure is the counting measure, that is, $m(x)=1$\nfor all $x \\in X$. In this case, \n$m(K)=\\# K$ is just the cardinality of any finite subset $K$. In the case of standard edge weights\nand counting measure, we then\nobtain\n$${\\mathrm{Deg}}(x) =\\# \\{y \\ | \\ y \\sim x\\}$$\nso that the weighted vertex degree is just the number of neighbors of $x$, that is, the valence\nor degree of a vertex.\n\\item Another choice for the vertex measure is \n$$m(x) = \\sum_{y \\in X}b(x,y)$$\nfor $x \\in X$. \nIn the case of standard edge weights, \nit then follows that $m(x) = \\# \\{ y \\ | \\ y \\sim x \\}$ is the number of neighbors of $x$.\nIn any case, with this choice of measure, it is clear that\n$${\\mathrm{Deg}}(x)=1$$ \nfor all $x \\in X$. \n\\end{enumerate}\n\\end{example}\n\nOften we will assume that graphs are \\emph{connected} in the usual geometric sense, namely,\nfor any two vertices $x, y \\in X$, there exists a sequence of vertices $(x_k)_{k=0}^n$ with $x_0=x$,\n$x_n =y$ and $x_k \\sim x_{k+1}$ for $k=0, 1, \\ldots n-1$. We note that we include the case of $x=y$\nwhen a vertex can be connected to itself via a path consisting of a single vertex and thus no edges.\nSuch a sequence is called a \\emph{path}\nconnecting $x$ and $y$. We then let \n$$d:X \\times X \\longrightarrow [0,\\infty)$$\ndenote the \\emph{combinatorial graph distance} on $X$,\nthat is, $d(x,y)$ equals the least number of edges in a path connecting $x$ and $y$.\nWe note that this metric only considers the combinatorial properties of the graph encoded in $b$ but not\nthe actual value of $b(x,y)$ nor the vertex measure $m$. We will have more to say about this later.\n\n\\subsection{Laplacians and forms}\nWe now denote the set of all functions on $X$ by $C(X)$, that is,\n$$C(X) = \\{ f: X \\longrightarrow {\\mathbb R}\\}$$\nand the subset of finitely supported functions by $C_c(X)$.  The Hilbert space\nthat we will be interested in at various points is $\\ell^2(X,m)$, the space of square summable functions\non $X$ with respect to the measure $m$. That is,\n$$\\ell^2(X,m) = \\{ f \\in C(X) \\ | \\ \\sum_{x \\in X}f^2(x)m(x)<\\infty\\}$$\nwith inner product\n$\\langle f, g \\rangle = \\sum_{x \\in X}f(x)g(x)m(x)$. \n\nIn order to introduce a formal Laplacian, we first have to restrict to a certain class of functions\nas we do not assume local finiteness so that summability becomes an issue. \n\\begin{definition}[Formal Laplacian and energy form]\nWe let\n$$\\mathcal{F}=\\{ f\\in C(X) \\ | \\ \\sum_{y \\in X} b(x,y) |f(y)| <\\infty \\textup{ for all } x \\in X \\}$$\nand for $f \\in \\mathcal{F}$, we let \n$$\\mathcal{L} f(x) = \\frac{1}{m(x)} \\sum_{y \\in X} b(x,y) (f(x)-f(y))$$\nfor $x \\in X$. The operator $\\mathcal{L}$ is then called the \\emph{formal Laplacian} associated to $G$. We furthermore let\n$$\\mathcal{D}=\\{ f\\in C(X) \\ | \\  \\sum_{x, y \\in X} b(x,y) (f(x)-f(y))^2 <\\infty\\}$$\ndenote the space of \\emph{functions of finite energy}. For $f, g \\in \\mathcal{D}$, we let\n$$\\mathcal{Q}(f,g) = \\frac{1}{2}\\sum_{x,y \\in X} b(x,y) (f(x)-f(y))(g(x)-g(y)) $$\ndenote the \\emph{energy form} associated to $G$.\n\\end{definition}\n\nWe denote the restriction of $\\mathcal{Q}$ to $C_c(X) \\times C_c(X)$ by $Q_c$. It then follows that a \nversion of Green's formula holds for $Q_c$:\n$$Q_c(\\varphi,\\psi) = \\sum_{x \\in X} \\mathcal{L} \\varphi(x)\\psi(x) m(x) = \\sum_{x \\in X} \\varphi(x) \\mathcal{L} \\psi(x) m(x) $$\nfor all $\\varphi, \\psi \\in C_c(X)\\subseteq \\ell^2(X,m)$, see, for example \\cite{HK11}.\nThe form $Q_c$ is closable and thus there exists a \nunique self-adjoint operator $L$ with domain \n$D(L) \\subseteq \\ell^2(X,m)$ associated to the closure of $Q_c$ denoted by $Q$. \nFor a discussion of the closure of a form and the construction of the \nassociated operator in a general Hilbert space, see Theorem~5.37 in \\cite{Weid80}.\nWe refer to $L$ as the \\emph{Laplacian} associated to the graph $G$. We note that with our sign\nconvention, we have\n$$\\langle Lf, f \\rangle = Q(f,f) \\geq 0$$\nfor all $f \\in D(L)$ so that $L$ is a positive operator.\n\n\n\n\\subsection{A word about essential self-adjointness and $\\ell^2$ theory}\nAlthough not a main concern of this article as we mostly deal with bounded solutions, \nwe want to mention another approach to the construction of the Laplacian $L$.\nIn this viewpoint, one starts by restricting $\\mathcal{L}$ to $C_c(X)$ and denoting the resulting operator by $L_c$,\nthat is, $D(L_c)=C_c(X)$ and $L_c$ acts as $\\mathcal{L}$. \n\nHowever, due to the lack of local finiteness,\n$\\mathcal{L}$ does not necessarily map $C_c(X)$ into $\\ell^2(X,m)$. Thus, whenever we want to consider $L_c$ as an operator on $\\ell^2(X,m)$, we have to assume\nthat $\\mathcal{L}$ maps $C_c(X)$ into $\\ell^2(X,m)$.\nUnder this additional assumption, it is easy to see that $L_c$ is a symmetric operator on $\\ell^2(X,m)$, i.e.,\n$$\\langle L_c \\varphi, \\psi \\rangle = \\langle \\varphi, L_c \\psi \\rangle$$\nfor all $\\varphi, \\psi \\in C_c(X)$ and that the Green's formula reads as\n$$Q_c(\\varphi,\\psi) = \\langle L_c \\varphi, \\psi \\rangle$$\nfor $\\varphi, \\psi \\in D(L_c)$. In this case, the self-adjoint operator associated to the closure of $Q_c$, which is just \nthe Laplacian $L$,\nis called the \\emph{Friedrichs extension} of $L_c$, see\nTheorem~5.38 in \\cite{Weid80} for further details on the construction of this extension for general Hilbert\nspaces. \n\nWe note that $\\mathcal{L}$ maps $C_c(X)$ into $\\ell^2(X,m)$ whenever \n$\\mathcal{L}1_x \\in \\ell^2(X,m)$ for all $x \\in X$ where $1_x$\ndenotes the characteristic function of the singleton set $\\{x\\}$. It is a direct calculation that \n$\\mathcal{L}1_x \\in \\ell^2(X,m)$ for all $x \\in X$ if and only if \n$$\\sum_{y \\in X} \\frac{b^2(x,y)}{m(y)} <\\infty$$\nfor all $x \\in X$. In particular, all locally finite graphs or, more generally, all graphs\nwith $\\inf_{y \\sim x} m(y)>0$ automatically satisfy this assumption.\nFurthermore, the condition $\\mathcal{L}1_x \\in \\ell^2(X,m)$ for all $x \\in X$ is equivalent\nto a variety of other conditions, for example, that $C_c(X) \\subseteq D(L)$, for more details,\nsee \\cite{Kel15}.\n\n\nWe further note that, in general, $L_c$ may have many self-adjoint extensions and\nthat processes associated to these different extensions may have different stochastic properties.\nWhen $L_c$ has a unique self-adjoint extension, $L_c$ is called \\emph{essentially self-adjoint}.\nIt was first shown as Theorem~1.3.1 in \\cite{Woj08} that $L_c$ is essentially self-adjoint in the case \nof standard edge weights and counting measure. This was then extended to allow for general edge \nweights and any measure such that the measure of infinite paths is infinite as Theorem~6 in \\cite{KL12}. \nThis criterion was further improved and generalized in \\cite{Gol14, Schm20} \nwhich consider more general operators on graphs.\nFor further discussion of essential self-adjointness, see \\cite{HKLW12, HKMW13, Schm20} and reference therein.\nWe will also discuss the connection between essential self-adjointness and metric completeness in Subsection~\\ref{ss:metric_esa} below.\n\n\\subsection{The heat equation: existence of solutions}\nWe now introduce a continuous time heat equation on $G$. We let $\\ell^\\infty(X)$ denote the set of bounded\nfunctions on $X$, that is,\n$$\\ell^\\infty(X) = \\{ f \\in C(X) \\ | \\ \\sup_{x \\in X} |f(x)| <\\infty \\}.$$\nWe now make precise the requirements\nfor summability, differentiability and boundedness of a solution.\n\n\\begin{definition}[Bounded solution of the heat equation]\nLet $u_0 \\in \\ell^\\infty(X).$\nBy a \\emph{bounded solution\nof the heat equation with initial condition} $u_0$ we mean a bounded function\n$$u:X \\times [0,\\infty) \\longrightarrow {\\mathbb R}$$\nsuch that $u(x,\\cdot)$ is continuous for every $t\\geq0$, differentiable\nfor $t>0$ and all $x \\in X$ and\n$$(\\mathcal{L} + \\partial_t)u(x,t) = 0$$\nfor all $x \\in X$ and $t>0$ with $u(x,0)=u_0(x)$.\n\\end{definition}\n\nWe note, in particular, that as $u(\\cdot, t) \\in \\ell^\\infty(X)$ for every $t \\geq 0$, we obtain\nthat $u(\\cdot, t) \\in \\mathcal{F}$. Thus, we may apply the formal Laplacian to $u$ at every time $t \\geq 0$.\n\nThe definition of bounded solutions raises two immediate questions: the existence and uniqueness of solutions. \nWe will first address existence by showing that there always\nexists a bounded solution which is minimal in a certain sense. On the other hand, uniqueness \nis one of the various formulations of stochastic completeness as we will discuss in the next section.\n\nWe note that although we have a self-adjoint operator $L$ on $\\ell^2(X,m)$ so that we may apply the spectral\ntheorem and \nfunctional calculus to obtain a heat semigroup $e^{-tL}$ for $t \\geq 0$, this semigroup acts on $\\ell^2(X,m)$ and we\nare actually interested in bounded solutions, i.e., solutions in $\\ell^\\infty(X)$. There is a number\nof ways around this. One approach taken in \\cite{KL12} is to extend the heat semigroup on $\\ell^2(X,m)$\nto all $\\ell^p(X,m)$ spaces for $p \\in [1,\\infty]$ via monotone limits. Another approach is via the general\ntheory of Dirichlet forms and interpolation between $\\ell^p(X,m)$ spaces, see \\cite{Dav90, FOT94}.\nWe highlight a slightly different approach\nin that we rather exhaust the graph via finite subgraphs, apply the spectral theorem to each operator\non the finite subgraph in order to get a solution and then take the limit. This rather elementary approach\nhas its roots in \\cite{Dod83} which gave the first construction of the heat kernel\non a general Riemannian manifold without any geodesic completeness assumptions.\n\nA basic tool behind the construction is the following minimum principle.\nWe call a subset $K \\subseteq X$ \\emph{connected} if any two vertices in $K$ can be connected via a\npath that remains within $K$.\n\\begin{lemma}[Minimum principle for the heat equation]\\label{l:minimum_principle}\nLet $G$ be a connected weighted graph and let $K \\subset X$ be a finite connected subset.  \nLet $ T\\ge0 $ and let $ u:X\\times[0,T] \\longrightarrow {\\mathbb R} $ be such that $t \\mapsto u(x,t) $ is continuously \ndifferentiable on $(0,T)$ for every $ x\\in K $ and $u(\\cdot, t) \\in \\mathcal{F}$ for all $t\\in(0,T]$. \nAssume that $u$ satisfies\n\\begin{itemize}\n\\item[(A1)] \\quad $ (\\mathcal{L}+\\partial_t) u\\ge0 $ on $ K\\times (0, T) $\n\\item[(A2)] \\quad $ u\\ge0 $ on $ \\left(X\\setminus K \\times (0,T] \\right) \\cup (K\\times \\{0\\}). $\n\\end{itemize}\nThen, $ u\\ge0 $ on $ K\\times [0,T] $.\n\\end{lemma}\n\\begin{proof}\nSuppose to the contrary that there exists $(x_0,t_0) \\in K \\times [0,T]$ such that $u(x_0,t_0) <0$.\nBy continuity, we can assume that $(x_0,t_0)$ is a minimum for $u$ on $K \\times [0,T]$. \nBy assumption (A2), it follows\nthat $t_0 >0$ so that $\\partial_t u(x_0,t_0) \\leq 0$. Furthermore, by the definition of $\\mathcal{L}$, at a minimum we have\n$\\mathcal{L} u(x_0,t_0) \\leq 0$. Therefore, $(\\mathcal{L} + \\partial_t)u(x_0,t_0) \\leq 0$ and assumption (A1) \ngives $(\\mathcal{L}+\\partial_t) u(x_0,t_0)=0 $ from\nwhich \n$$\\mathcal{L} u(x_0,t_0)= \\frac{1}{m(x_0)}\\sum_{y \\in X} b(x_0,y)(u(x_0,t_0)-u(y,t_0))=0$$ \nfollows. Therefore, since we are at a minimum, we now obtain that\n$$u(y,t_0)=u(x_0,t_0)<0$$\nfor all $y \\sim x_0$. Iterating this argument and using the connectedness of $K$ now gives a contradiction\nto (A2) as $K \\neq X$ and we assume that $G$ is connected.\n\\end{proof}\n\n\\begin{remark}\nWe note that the finiteness of $K$ is not necessary. It suffices to assume that there is at least\none vertex outside of $K$ and that the negative part of $u$\nattains a minimum on $K \\times [0,T]$. The minimum principle then follows with basically the same proof,\nsee, for example, Lemma~3.5 in \\cite{KLW13}. However, assuming the finiteness of $K$ is sufficient\nfor our purposes. For a much more elaborate discrete integrated minimum principle for solutions of the heat \nequation, see Lemma~1.1 in \\cite{Hua12}.\n\\end{remark}\n\nWe now sketch the construction of the minimal bounded solution of the heat equation. \nWe note that if $G$ is not connected, we work on each connected component of $G$ separately.\nThus, for the construction, we can assume without loss of generality that $G$ is connected.\nWe let $(K_n)_{n=0}^\\infty$\nbe an \\emph{exhaustion sequence} of the graph $G$ by which we mean that each $K_n$ is finite and connected,\n$K_n \\subseteq K_{n+1}$ and $X=\\bigcup_n K_n$. For each $n$, we let $L_n$ denote the restriction of $\\mathcal{L}$ to\n$C(K_n)=\\ell^2(K_n,m)$. More precisely, for a function $f \\in C(K_n)$, we extend $f$ by 0 to be defined on all of $X$ and let\n$L_n f(x) = \\mathcal{L} f(x)$ for $x \\in K_n$. Then, $L_n$ is an operator on a finite dimensional Hilbert space and we can define\n$$e^{-tL_n}=\\sum_{k=0}^\\infty \\frac{(-t)^k}{k!}L_n^k$$\nfor $t \\geq 0$. We then define the \\emph{restricted heat kernels} $p_t^n(x,y)$ for $t \\geq 0$ and $x, y \\in K_n$ \nvia\n$$p_t^n(x,y)=e^{-tL_n}\\hat{1}_y(x)$$ \nwhere $\\hat{1}_y = 1_y\/m(y)$.\nIt is immediate that  \n$$u_n(x,t)=e^{-tL_n}u_0(x)=\\sum_{y\\in K_n}p_t^n(x,y)u_0(y)m(y)$$\nsatisfies the heat equation\non $K_n \\times [0,\\infty)$ with initial condition $u_0$. \n\nFurthermore, applying Lemma~\\ref{l:minimum_principle}, gives $0 \\leq p_t^n(x,y)m(y) \\leq 1$ and\n$p_t^n(x,y) \\leq p_t^{n+1}(x,y)$ for all $x,y \\in K_n, t \\geq 0$ and $n \\in {\\mathbb N}$. \nThus, we may take the limit\n$$p_t^n(x,y) \\to p_t(x,y)$$ \nas $n \\to \\infty$  to define $p_t(x,y)$ which is called the \n\\emph{heat kernel} on $G$. \nThen, by applying Dini's theorem and monotone convergence, we can show that\n$$u(x,t)=\\sum_{y \\in X}p_t(x,y)u_0(y)m(y)$$ \nis a bounded solution of the heat equation with initial condition $u_0$ on $G$. \nFor further details and proofs, see Section~2 in \\cite{Woj08} for the case of standard edge weights\nand counting measure. An alternative approach for general graphs involving resolvents is given\nin Section 2 of \\cite{KL12}, in particular, Proposition 2.7. \n\n\\begin{remark}\nThe approach via resolvents in \\cite{KL12}\nis equivalent to the heat kernel approach above via the Laplace transform formulas, that is,\n$$e^{-tL_n}= \\lim_{k\\to \\infty}\\left(\\frac{k}{t} \\left(L_n+\\frac{k}{t}\\right)^{-1}\\right)^k$$\nfor all $t>0$ and\n$$(L_n + \\alpha)^{-1} = \\int_0^\\infty e^{-t\\alpha}{e^{-tL_n}} dt$$\nfor all $\\alpha>0$. Both of these formulas also hold for the Laplacian $L$ defined \non the entire $\\ell^2(X,m)$ space.\nWe further note that $L_n$ is a positive definite operator on $\\ell^2(K_n,m)$\nas can be seen by direct calculation which gives\n$$\\langle L_n f, f \\rangle = \\frac{1}{2}\\sum_{x, y \\in K_n} b(x,y)(f(x)-f(y))^2 \n+ \\sum_{x \\in K_n}f^2(x) \\sum_{y \\not \\in K_n} b(x,y)$$\nfor all $f \\in \\ell^2(K_n,m)$.\n\\end{remark}\n\nWe mention two further properties that follow from the construction and Lemma~\\ref{l:minimum_principle}\nabove. First, the solution $u$ is minimal in the following sense: if $u_0\\geq0$ and $w \\geq 0$ is any\nsolution of the heat equation with initial condition $u_0$, then $u \\leq w$. Secondly, as $0 \\leq \\sum_{y \\in K_n} p_t^n(x,y)m(y) \\leq 1$ for all $n$, we\nget\n$$0 \\leq \\sum_{y \\in X} p_t(x,y)m(y) \\leq 1$$\nby taking the limit $n \\to \\infty$. \nWe will return to the second inequality in the following section.\n\nWe note that the approach above also gives that\nif $f \\in \\ell^\\infty(X)$ with $0 \\leq f \\leq 1$, then\n$$0 \\leq \\sum_{y \\in X} p_t(x,y)f(y)m(y) \\leq 1.$$ \nThis property is referred to by saying that the heat semigroup is \\emph{Markov}. The fact\nthat the semigroup is Markov will\nbe used later in our discussion of curvature on graphs.\n\n\n\n\n\\section{Formulations of stochastic completeness}\\label{s:sc}\n\\subsection{Stochastic completeness and uniqueness of solutions}\nWe have seen that given any bounded function, we can construct a bounded\nsolution of the heat equation with the given function as an initial condition. \nWe now address the uniqueness of this solution.\nIn fact, we will see that the uniqueness is equivalent to the following property.\n\n\\begin{definition}[Stochastic completness]\nLet $G$ be a weighted graph. If for all $x \\in X$ and all $t \\geq 0$, \n$$\\sum_{y \\in X} p_t(x,y)m(y)=1,$$ then \n$G$ is called \\emph{stochastically complete.} Otherwise, $G$ is called \\emph{stochastically incomplete}.\n\\end{definition}\n\n\\begin{remark}\nThere is a short way to state the definition above which we will have recourse to later in our\ndiscussion of curvature. Namely, letting $P_t = e^{-tL}$ denote the heat semigroup\non $\\ell^2(X,m)$ for $t \\geq 0$ defined via the spectral theorem, it follows that $P_t$ can be extended\nto $\\ell^\\infty(X)$, the space of bounded functions, via monotone limits, see Section~6 in \\cite{KL12} where\nthis is actually shown for all $\\ell^p(X,m)$ spaces with $p \\in [1,\\infty]$. In particular, letting $1 \\in \\ell^\\infty(X)$\ndenote the function which is constantly 1 on all vertices, we then have\n$$P_t 1(x) = \\sum_{y \\in X}p_t(x,y)m(y)$$\nfor $x \\in X$. Thus, stochastic completeness can also be written as $P_t 1 =1$ for all $t \\geq 0$.\n\\end{remark}\n\nThe goal of this section is to give a variety of characterizations for this property. We do not aim\nto be exhaustive but rather highlight the characterizations that will be useful later in our presentation.\nIf $v \\in \\mathcal{F}$ satisfies $\\mathcal{L} v = \\lambda v$ for $\\lambda \\in {\\mathbb R}$, then $v$ is called a $\\lambda$-\\emph{harmonic} function.\nIn particular, the theorem below characterizes stochastic completeness in terms of non-existence\nof $\\lambda$-harmonic bounded functions for $\\lambda<0$. This will be used in several places in what follows.\n\n\\begin{theorem}[Characterizations of stochastic completeness]\\label{t:sc_characterizations}\nLet $G$ be a weighted graph. The following statements are equivalent:\n\\begin{itemize}\n\\item[(i)] $G$ is stochastically complete.\n\\item[(ii)] Bounded solutions of the heat equations are uniquely determined by initial conditions.\n\\item[(ii$'$)] The only bounded solution of the heat equation with initial condition $u_0=0$ is $u=0$.\n\\item[(iii)] The only bounded solution to $\\mathcal{L} v = \\lambda v$ for some\/all $\\lambda <0$\nis $v=0$.\n\\item[(iii$'$)] The only non-negative bounded solution to $\\mathcal{L} v \\leq \\lambda v$ \nfor some\/all $\\lambda<0$ is $v=0$\n\\item[(iv)] Every bounded function $v$ with $v^* = \\sup v>0$ satisfies $\\sup_{\\Omega_\\alpha} \\mathcal{L} v \\geq 0$\nfor every $\\alpha < v^*$ where $\\Omega_\\alpha = \\{ x\\in X \\ | \\ v(x) > v^*- \\alpha\\}$.\n\\end{itemize}\n\\end{theorem}\n\\begin{remark}[History and intuition]\nWe give a partial history with references for the equivalences above in various settings. The equivalence of (i),\n(ii), and (iii) for diffusion processes on Euclidean spaces goes back to \\cite{Fel54, Has60}.\nFor manifolds, see Theorem~6.2~and~Corollary~6.3 in \\cite{Gri99} which also gives further\nhistorical references. For Markov processes on discrete spaces, these equivalence go back to \\cite{Fel57, Reu57}. \nFor a proof in the case of graphs with\nstandard edge weights and counting measure, see Theorem~3.1.3 in \\cite{Woj08}. For an extension\nto weighted graphs see Theorem~1~in~\\cite{KL12} which deals also with a more general phenomenon\ncalled stochastic completeness at infinity. This allows for a discussion of these properties\nin the case of operators of the type Laplacian plus a positive potential, see also \\cite{MS} for a \ndiscussion of this property in the case of manifolds.\nCondition (iv) is referred to as a weak Omori-Yau maximum principle after the original\nwork in \\cite{Om67, Yau75}. The equivalence\nof (iv) and stochastic completeness was shown for manifolds in \\cite{PRS03} and for graphs as\nTheorem 2.2 in \\cite{Hua11a}.\n\nWe would also like to mention some of the intuition behind the equivalences. Roughly speaking, as mentioned\nin the introduction and discussed further below,\na large volume growth or curvature decay is required for\nstochastic completeness to fail. Let us discuss how this large volume growth can cause the failure of the\nother properties listed in the theorem above. First, failure of (i) means that\nthe total probability of the process determined by the Laplacian to remain in the graph when\nstarting at a vertex $x$ is less than 1 at some time.\nHence, under a large enough volume growth (or curvature decay) the process can be swept off the graph\nto infinity in a finite time. \nSecond, failure of (ii) means that there exists a non-zero bounded solution of the heat equation with zero\ninitial condition. In other words, a large volume growth can create something out of nothing.\nThird, by looking at the equation $\\mathcal{L} v = \\lambda v$ for $v>0$ and $\\lambda<0$,\nwe see that $v$ must increase at some neighbor of each vertex. Hence, stochastic incompleteness\nmeans that there is a sufficient amount of space in the graph to accommodate this growth while keeping $v$ bounded.\nThis gives the intuition for the failure of condition (iii).\nFinally, $\\mathcal{L} v \\leq -C< 0$ on a set of vertices where $v$ is near its supremum means that there is always more \nroom for $v$ to grow in the graph. This gives the intuition behind the failure of (iv). \n\nTo summarize, in order for any of the four conditions above to fail requires a large amount of\nspace in the graph. Conversely, if there is no large growth, then the process remains in the graph\nand the graph cannot accommodate non-zero bounded solutions to various equations.\nThus, stochastic completeness is also referred to as \\emph{conservativeness}\nor \\emph{non-explosion}.\n\\end{remark}\n\n\\begin{proof}[Sketch of the proof of Theorem~\\ref{t:sc_characterizations}] \nWe now sketch a proof. For full details, please see the references\ngiven in the remark directly above.\nTo show the equivalence between (i) and (ii), observe that both the constant function \n1 and $u(x,t) = \\sum_{y \\in X}p_t(x,y)m(y)$ are bounded solutions of the heat equation with initial condition 1.\nThe equivalence between (ii) and (ii$'$) is shown by taking the difference of two solutions of the heat equation\nwith initial conditions $u_0$. \nThe equivalence between (ii$'$) and (iii) can be established via the fact that if $u$ is a\nbounded solution of the heat equation with initial condition 0, then\n$v(x) = \\int_0^\\infty e^{t\\lambda}u(x,t) dt$ is a bounded $\\lambda$-harmonic function for $\\lambda<0$.\nTo show the equivalence between (iii) and (iii$'$) one can use exhaustion and minimum principle arguments.\nFinally, if (iii) fails and $v$ is a non-trivial bounded $\\lambda$-harmonic function for $\\lambda<0$, then\nletting $\\alpha = \\sup v\/2$, it can be shown that $\\mathcal{L} v \\leq -C < 0$ on $\\Omega_\\alpha$ so that (iv)\nfails.\nConversely, if (iv) fails, then there exists a bounded function $v$ such that $\\mathcal{L}v \\leq -C$ for $C>0$\non $\\Omega_\\alpha$ for some $0< \\alpha<v^*$. \nThen, $w= (v+\\alpha -v^*)_+$ is a non-negative non-trivial bounded function with\n$\\mathcal{L} w \\leq \\lambda w$ for $\\lambda = -C\/\\alpha$. Thus, (iii$'$) fails.\n\\end{proof}\n\n\\begin{remark}\nWe note that connectedness of the graph and the\nsemigroup property can be used to show that if $\\sum_{y \\in X}p_t(x,y)m(y)=1$\nholds for one $x \\in X$ and one $t>0$, then it holds for all $x \\in X$ and all $t > 0$. However, we do not\nrequire connectedness for the equivalence of the properties above as later we will need to consider\na possibly unconnected scenario. Thus, in the definition, we assume that the sum is 1\nfor all $x \\in X$ and all $t \\geq 0$.\n\\end{remark}\n\n\\subsection{A Khas$'$minskii criterion}\nWe will also need another property which implies stochastic completeness. This is sometimes referred to \nas a Khas$'$minskii-type criterion after \\cite{Has60}. The formulation below is Theorem~3.3~in~\\cite{Hua11a}, the\nproof given there uses the weak Omori-Yau maximum principle, that is, condition (iv) in \nTheorem~\\ref{t:sc_characterizations}.\n\\begin{theorem}\\label{t:Khas}\nLet $G$ be a weighted graph. If there exists $v \\in \\mathcal{F}$ which satisfies $v \\geq 0$, $v(x_n) \\to \\infty$\nfor all sequences of vertices with ${\\mathrm{Deg}}(x_n) \\to \\infty$ and\n$$\\mathcal{L} v + f(v) \\geq 0$$\non $X \\setminus K$ where $K\\subseteq X$ is a set such that  ${\\mathrm{Deg}}$ is a bounded function on $K$ and \n$f:[0,\\infty) \\longrightarrow (0,\\infty)$ is an increasing continuously differentiable function with \n$$\\int^\\infty \\frac{1}{f(r)}dr = \\infty,$$\nthen $G$ is stochastically complete.\n\\end{theorem}\n\n\\begin{remark}\nThis formulation of a Khas$'$minskii-type criterion is very precise\nas it involves the weighted degree function\nas well as the function $f$. \nA more general formulation is that the existence of\na function $v$ which satisfies $\\mathcal{L} v \\geq \\lambda v$ outside of a compact set and which goes to infinity \nin all directions implies stochastic\ncompleteness, see Corollary~6.6 in \\cite{Gri99} for the manifold case and Proposition~5.5 in \\cite{KLW13} for \nweighted graphs.\nFurthermore, we note that, in the manifold setting, the equivalence of this formulation and stochastic completeness\nwas shown as Theorem~1.2 in \\cite{MV13}.\n\\end{remark}\n\n\n\n\\section{Boundedness of geometry and of Laplacians}\\label{s:bounded}\n\\subsection{Boundedness of the Laplacian}\nWe now discuss the boundedness of the Laplacian which turns out to be equivalent to boundedness of the\nweighted vertex degree. Furthermore, it turns out that boundedness always implies stochastic completeness.\n\nWe start by a simple observation. We recall that $L$ is the self-adjoint operator on $\\ell^2(X,m)$\nwhich is obtained from the closure of the form $Q_c$ acting on $C_c(X) \\times C_c(X)$ and that \n${\\mathrm{Deg}}(x) = 1\/m(x) \\sum_{y \\in X}b(x,y)$ for $x \\in X$ is the weighted degree of a vertex $x$. \nWe first characterize\nthe boundedness of this operator in terms of the boundedness of the weighted degree function. This\nfact is certainly well-known, see, for example Theorem~11 in \\cite{KL10}.\n\n\\begin{theorem}[Boundedness of $L$]\\label{t:boundedness_l2}\nLet $G$ be a weighted graph. The Laplacian $L$ is a bounded operator on $\\ell^2(X,m)$\nif and only if ${\\mathrm{Deg}}$ is bounded on $X$.\n\\end{theorem}\n\\begin{proof}\nA direct calculation gives \n$$\\langle L1_x, 1_x \\rangle = {\\mathrm{Deg}}(x)m(x)$$\nwhere $1_x$ is the characteristic function of the set $\\{x\\}$ for $x \\in X$.\nNow, the result follows from the general theory of self-adjoint operators on Hilbert space, see for \nexample Theorem~4.4 in \\cite{Weid80}, by noting that $\\{1_x\/\\sqrt{m(x)} \\ | \\ x\\in X\\}$ forms an\northonormal basis for $\\ell^2(X,m)$.\n\\end{proof}\n\nWeighted graphs which satisfy the condition that ${\\mathrm{Deg}}$ is a bounded function are sometimes referred\nto as having \\emph{bounded geometry}. We now have a look at this in the cases most commonly appearing\nin the graph theory literature.\n\n\\begin{example} Let $G$ be a weighted graph.\n\n\\begin{enumerate}\n\\item If $m(x)=\\sum_{y \\in X} b(x,y)$ is the sum of the edge weights, then ${\\mathrm{Deg}}(x)=1$\nfor all $x \\in X$. Thus, in this case, $L$ is always a bounded operator.\n\\item If $G$ has standard edge weights, i.e., $b(x,y)\\in \\{0,1\\}$ for all $x, y \\in X$ and $m$ is the counting\nmeasure, i.e.,  $m(x)=1$ for all $x \\in X$, then\n$${\\mathrm{Deg}}(x) = \\# \\{y \\ | \\ y \\sim x \\}$$\nfor all $x \\in X$. Thus, ${\\mathrm{Deg}}$ is just the usual vertex degree which counts the number of\nneighbors of $x$. We see that $L$ is bounded in this case if and only if there is a uniform upper \nbound on this quantity.\n\\end{enumerate}\n\\end{example}\n\n\\subsection{Boundedness and stochastic completeness}\nWe now discuss the connection between boundedness and stochastic completeness.\nIn particular, we show that if ${\\mathrm{Deg}}$ is bounded on $X$, then the graph is stochastically complete.\nThis follows from a more general result which allows for some growth of the weighted \nvertex degree which we state below.\n\n\\begin{theorem}[Boundedness implies stochastic completeness]\\label{t:bounded_SC}\nLet $G$ be a weighted graph. If for every infinite path $(x_n)_{n=0}^\\infty$ \n$$\\sum_{n=0}^\\infty \\frac{1}{{\\mathrm{Deg}}(x_n)} = \\infty,$$ \nthen $G$ is stochastically complete.\nIn particular, if\n${\\mathrm{Deg}}$ is bounded on $X$, then $G$ is stochastically complete.\n\\end{theorem}\n\\begin{proof}\nBy Theorem~\\ref{t:sc_characterizations}~(iii), it suffices to show that any non-trivial $v \\in \\mathcal{F}$ with \n$\\mathcal{L} v = \\lambda v$ for $\\lambda <0$ is not bounded. Suppose that $v(x_0)>0$ for some $x_0 \\in X$. The equation\n$\\mathcal{L} v(x_0) = \\lambda v(x_0)$ can be rewritten as\n$$\\frac{1}{m(x_0)} \\sum_{y \\in X} b(x_0,y)v(y) = \\left({\\mathrm{Deg}}(x_0)-\\lambda \\right)v(x_0).$$\nHence, there exists $x_1 \\sim x_0$ such that\n$$v(x_1) \\geq \\left( 1 - \\frac{\\lambda}{{\\mathrm{Deg}}(x_0)} \\right) v(x_0).$$\nNow, we iterate this argument to get a sequence of vertices $x_0 \\sim x_1 \\sim x_2 \\ldots$\nsuch that\n\\begin{align*}\nv(x_{n+1}) &\\geq \\left(1 - \\frac{\\lambda}{{\\mathrm{Deg}}(x_n)} \\right)v(x_n) \\\\\n&\\geq \\prod_{k=0}^n \\left(1 - \\frac{\\lambda}{{\\mathrm{Deg}}(x_k)}\\right) v(x_0).\n\\end{align*}\nAs $\\sum_{k=0}^\\infty 1\/{\\mathrm{Deg}}(x_k) = \\infty$ if and only if $\\prod_{k=0}^\\infty (1 - \\lambda\/{\\mathrm{Deg}}(x_k))=\\infty$,\nit follows that $v$ cannot be bounded.\n\\end{proof}\n\n\\begin{remark} \nFor an even shorter proof of the boundedness portion using the Omori-Yau maximum principle, see\nLemma~2.3 in \\cite{Hua11a}. This is then extended to a boundendess of a notion of a global weighted degree\nin Theorem~2.9 in \\cite{Hua11a}. For a more precise result which only considers the maximal outward degree\non spheres in the case of standard edge weights and counting measure, see Theorem~4.2 in \\cite{Woj11}\nand Theorem~5.5 in \\cite{Hua11a}.\n\nWhen $b$ is the standard edge weight and $m$ is the counting measure, the boundendess result\nwas first shown via a minimum principle in \\cite{DM06}. This proof was then extended to the case\nof arbitrary edge weights and counting measure in \\cite{Dod06}.\n\nA more structural proof of the boundedness portion of Theorem~\\ref{t:bounded_SC} can be \nfound as Corollary~27 in \\cite{KL10} and can be described as follows.  It turns out that \nthe boundedness of $L$ on $\\ell^2(X,m)$ also implies\nboundedness of $\\mathcal{L}$ acting on $\\ell^\\infty(X)$. In fact, the \nboundendess of $\\mathcal{L}$ restricted to $\\ell^p(X,m)$ for one $p \\in [1,\\infty]$ implies the boundedness of the restriction\nof $\\mathcal{L}$ to $\\ell^p(X,m)$ for all $p \\in [1,\\infty]$. This was shown via the Riesz-Thorin interpolation\ntheorem as Theorem~9.3 in \\cite{HKLW12} following earlier work presented as Theorem~11 in \\cite{KL10}. Now, \nif $\\mathcal{L}$ gives a bounded operator on $\\ell^\\infty(X)$,\nthen it is clear that the equation $\\mathcal{L} v = \\lambda v$ cannot have a non-zero bounded solution for all $\\lambda <0$. \nThus, by Theorem~\\ref{t:sc_characterizations}~(iii), $G$ is stochastically complete.\n\nWe also note that the lower bound for the $\\lambda$-harmonic function appearing in the proof above \ncan also be used to establish the essential self-adjointness\nof the restriction of $\\mathcal{L}$ to $C_c(X)$. See Proposition~2.2 in \\cite{Gol14} \nor, more generally, Theorem~11.5.2 in \\cite{Schm20}.\n\\end{remark}\n\n\\section{Weakly spherically symmetric graphs, trees and anti-trees}\\label{s:symmetry}\n\\subsection{Weakly spherically symmetric graphs}\nWe now discuss a class of graphs for which we will give a full characterization of stochastic completeness.\nThese are weakly spherically symmetric graphs. They are an analogue to model manifolds which are\nextensively discussed in \\cite{Gri99}. The definition we give here was first presented in \\cite{KLW13}\nand later generalized in \\cite{BG15}.\n\nWe start with some definitions. \nWe assume that $G$ is connected and\nrecall that $d(x,y)$ denotes the combinatorial graph distance between\nvertices $x$ and $y$, that is, the least number of edges in a path connecting $x$ and $y$. For a vertex\n$x_0 \\in X$ and $r \\in {\\mathbb N}_0$, we let $S_r(x_0)$ and $B_r(x_0)$ denote the sphere and ball of radius $r$\nabout $x_0$, that is,\n$$S_r(x_0) = \\{ x \\in X \\ | \\ d(x,x_0)=r \\}$$\nand $ B_r(x_0) = \\bigcup_{k=0}^r S_k(x_0) = \\{ x \\in X \\ | \\ d(x,x_0) \\leq r \\}.$\nTo ensure that these are finite sets, we now assume that $G$ is locally finite.\n\nWe will generally suppress the dependence on $x_0$ and just write $S_r$ and $B_r$.\nWe can then define the \\emph{outer} and \\emph{inner degrees} of a vertex $x \\in S_r$ as\n$${\\mathrm{Deg}}_{\\pm}(x)  =\\frac{1}{m(x)} \\sum_{y \\in S_{r \\pm 1}} b(x,y).$$\nThat is, ${\\mathrm{Deg}}_+(x)$ gives the total edge weight of edges going ``away'' from $x_0$ divided by the vertex\nmeasure while ${\\mathrm{Deg}}_-(x)$ of those going ``back'' towards $x_0$.\n\n\\begin{definition}[Weakly spherically symmetric graphs]\nA locally finite connected weighted graph $G$ is called \\emph{weakly spherically symmetric} \nif there exists a vertex $x_0 \\in X$ such\nthat the functions ${\\mathrm{Deg}}_\\pm$ depend only on the distance to $x_0$. In this case, we will write\n${\\mathrm{Deg}}_{\\pm}(r)$ for ${\\mathrm{Deg}}_{\\pm}(x)$ when $x \\in S_r(x_0)$.\n\\end{definition}\nAgain, although all of the concepts above depend on the choice of $x_0$, we will suppress this dependence\nin our notation.  We note that this notion of symmetry is weak in the sense that we do not assume\nanything about the edge weights between vertices on the same sphere nor do we assume anything about\nthe structure of the connections between vertices on successive spheres.\n\nWe will now state a full characterization for the stochastic completeness of such graphs. In order to do so, we introduce\nthe notion of \\emph{boundary growth} of a ball as\n$$\\partial B(r) = \\sum_{x \\in S_r}\\sum_{y \\in S_{r+1}} b(x,y).$$\nWe note that $\\partial B(r)$ reflects the total edge weight of edges leaving the ball $B_r$. Furthermore, \nfor weakly spherically symmetric graphs, this can be written as\n$$\\partial B(r) = {\\mathrm{Deg}}_+(r)m(S_r)= {\\mathrm{Deg}}_{-}(r+1)m(S_{r+1})$$\nas follows directly from the definitions. \nIn particular, we note that\n$$\\partial B(r)= \\frac{\\partial B(r-1) {\\mathrm{Deg}}_+(r)}{{\\mathrm{Deg}}_-(r)}.$$\nIn what follows, we also let\n$$V(r) = m(B_r)=\\sum_{k=0}^r m(S_k)$$\ndenote the measure of a combinatorial ball of radius $r$.\n\n\\begin{theorem}[Stochastic completeness of weakly spherically symmetric graphs]\\label{t:sc_symmetry}\nIf $G$ is a weakly spherically symmetric graph, then $G$ is stochastically complete if and only if\n$$\\sum_{r=0}^\\infty \\frac{V(r)}{\\partial B(r)} =  \\infty.$$\n\\end{theorem}\nWe give a sketch of the proof. For further details, see the proof of Theorem~5 in \\cite{KLW13}.\nFor standard edge weights and counting measure, this was first shown as Theorem~4.8 in \\cite{Woj11},\nsee also Theorem~5.10 in \\cite{Hua11a} for an alternative proof in this case using the weak Omori-Yau maximum\nprinciple.\n\\begin{proof}\nBy Theorem~\\ref{t:sc_characterizations}~(iii)\nit suffices to show that any bounded solution to $\\mathcal{L} v = \\lambda v$ for $\\lambda<0$ is zero if and only if \n$\\sum_{r=0}^\\infty \\frac{V(r)}{\\partial B(r)} = \\infty.$ By applying the characterization in \nterms of non-negative subsolutions in Theorem~\\ref{t:sc_characterizations}~(iii$'$) and \nthe Khas$'$minskii criterion from Theorem~\\ref{t:Khas}, it suffices to consider only non-negative solutions $v$.\nFinally, by averaging a solution over spheres, it suffices to consider only solutions depending on the\ndistance to $x_0$. \n\nThus, we may write $v(r)$ for $v(x)$ for all $x \\in S_r$ and note that stochastic completeness\nis equivalent to the triviality of $v$ if $v$ is bounded. Now, by induction on $r \\in {\\mathbb N}_0$, it can\nbe shown by using the formulas above that  \n$\\mathcal{L} v(r) = \\lambda v(r)$ if and only if\n$$v(r+1)-v(r) = \\frac{-\\lambda}{\\partial B(r)} \\sum_{k=0}^r v(k)m(S_k).$$\nIn particular, if $v(0)>0$, then $v$ is strictly increasing with respect to $r$. Therefore, we estimate\n$$ -\\frac{\\lambda V(r)}{\\partial B(r)} v(0) \\leq v(r+1)-v(r) \\leq -\\frac{\\lambda V(r)}{\\partial B(r)} v(r)$$\nso that \n$$ v(r) - \\frac{\\lambda V(r)}{\\partial B(r)} v(0) \\leq v(r+1) \\leq \\left(1 - \\frac{\\lambda V(r)}{\\partial B(r)} \\right) v(r).$$\nIterating this down to $r=0$, gives \n$$ -\\lambda \\sum_{k=0}^r \\frac{V(k)}{\\partial B(k)}  v(0) \\leq v(r+1) \\leq \n\\prod_{k=0}^r \\left(1 - \\frac{\\lambda V(k)}{\\partial B(k)} \\right) v(0).$$\nHence, if $v$ is bounded, then $\\sum_{k=0}^\\infty \\frac{V(k)}{\\partial B(k)} <\\infty$.\nOn the other hand, if $v$ is not bounded, then\n$ \\prod_{k=0}^\\infty \\left(1 - \\frac{\\lambda V(k)}{\\partial B(k)} \\right) =\\infty$ which is\nequivalent to\n$\\sum_{k=0}^\\infty \\frac{V(k)}{\\partial B(k)} =\\infty$. This completes the proof.\n\\end{proof}\n\n\\subsection{Trees and anti-trees} \nWe now illustrate the theorem above linking stochastic completeness of weakly spherically symmetric graphs \nand the ratio of the growth of the ball and the boundary of the ball with several examples. \nIn particular, we introduce the class of spherically symmetric trees and anti-trees.\n\nWe start with spherically symmetric trees. \nFor this, we first take standard edge weights and counting measure.\nSuch a graph $G$ is then called a \\emph{spherically symmetric tree} if $G$ contains no cycles\nand there exists a vertex $x_0 \\in X$ such for all $x \\in S_r$\n$${\\mathrm{Deg}}_+(x) = \\# \\{ y \\ | \\  y \\sim x, y \\in S_{r+1} \\}$$ \nonly depends on $r$. Thus, we may write ${\\mathrm{Deg}}_+(x)={\\mathrm{Deg}}_+(r)$ for\nall $x \\in S_r$.  Note that the lack of cycles implies that ${\\mathrm{Deg}}_-(r)=1$ for all\n$r \\in {\\mathbb N}$ so that the number of edges leading back to $x_0$ is minimal in order to \nhave a connected graph.\n\nWe note that for spherically symmetric trees, we have\n$$m(S_r) = \\prod_{k=0}^{r-1} {\\mathrm{Deg}}_+(k)$$ \nand $\\partial B(r) = m(S_{r+1})$ as follows by direct calculations.\nWe now apply our characterization of stochastic completeness of weakly spherically\nsymmetric graphs to the case of such trees.\n\n\\begin{corollary}[Stochastic completeness and spherically symmetric trees]\nIf $G$ is a spherically symmetric tree, then $G$ is stochastically complete\nif and only if\n$$\\sum_{r=0}^\\infty \\frac{1}{{\\mathrm{Deg}}_+(r)} = \\infty.$$\n\\end{corollary}\n\\begin{proof}\nFrom the remarks directly above we obtain\n$$\\sum_{r=0}^\\infty  \\frac{V(r)}{\\partial B(r)} = \\sum_{r=0}^\\infty \\frac{1+\\sum_{k=1}^r \\prod_{j=0}^{k-1} {\\mathrm{Deg}}_+(j)}{ \\prod_{k=0}^{r} {\\mathrm{Deg}}_+(k)}.$$\nBy the limit comparison test, it then follows that the divergence of the series above is equivalent to divergence\nof the series $\\sum_{r=0}^\\infty 1\/{\\mathrm{Deg}}_+(r)$. Thus, the conclusions follows by Theorem~\\ref{t:sc_symmetry}.\n\\end{proof}\n\nThe result above was first presented as Theorem~3.2.1 in \\cite{Woj08}.   \nIt establishes the sharpness of the condition given for stochastic completeness in terms of the weighted\nvertex degree on paths presented in Theorem~\\ref{t:bounded_SC} in the previous section.\n\nWe note that the case of\nspherically symmetric trees already provides a contrast with the manifold case as if we take\n$V(r) = m(B_r)$ to be the counterpart of the volume growth in the Riemannian setting, then there\nexist stochastically incomplete trees with factorial volume growth.\nHowever, a much more striking example is that of anti-trees which we define next.\nThe basic idea is that we choose an arbitrary sequence of natural numbers for the number of \nvertices on the sphere and then connect all vertices between successive spheres. Thus, these\nare the antithesis of trees in the sense that for trees the removal of a single edge between spheres\ncreates a disconnected graphs while for anti-trees one must remove all of the edges between spheres.\n\n\\begin{definition}[Anti-trees]\nLet $(a_r)$ be a sequence with $a_r \\in {\\mathbb N}$ for $r \\in {\\mathbb N}$ and $a_0=1$.\nA graph $G$ is called an \\emph{anti-tree} with sphere growth $(a_r)$ \nif $G$ has standard edge weights and counting measure and the vertex set $X$ can be written\nas a disjoint union $X= \\bigcup_r A_r$ where $m(A_r)=a_r$ and $b(x,y)= b(y,x) = 1$ for all $x \\in A_r, y \\in A_{r+1}$\nfor $r \\in {\\mathbb N}_0$ and zero otherwise.\n\\end{definition}\n\nThus, by the definition of the edge weight, an anti-tree with sphere growth $(a_r)$  satisfies\n$m(S_r)=a_r$ and is weakly spherically symmetric with ${\\mathrm{Deg}}_{\\pm}(r)=a_{r \\pm 1}$. \nFurthermore,\n$\\partial B(r) = a_r a_{r+1}$ as each vertex in the sphere $S_r$ is connected to\nall vertices in the sphere $S_{r+1}$. Therefore, we obtain the following characterization of stochastic completeness\nin the case of anti-trees.\n\\begin{corollary}[Stochastic completeness and anti-trees]\\label{c:antitree_sc}\nIf $G$ is an anti-tree with sphere growth $(a_r)$, then $G$ is stochastically complete\nif and only if\n$$\\sum_{r=0}^\\infty \\frac{\\sum_{k=0}^r a_k}{a_r a_{r+1}} = \\infty.$$\n\\end{corollary}\n\\begin{proof}\nThis follows directly from the definition of an anti-tree and Theorem~\\ref{t:sc_symmetry}.\n\\end{proof}\n\nWe note that if $a_r$ grows like $r^{2+\\epsilon}$ for any $\\epsilon>0$, then the corresponding anti-tree is\nstochastically incomplete. Furthermore, $V(r)$ grows like $r^{3+\\epsilon}$ in this case. Thus, unlike in the case\nof manifolds, for the combinatorial graph metric, there exists stochastically incomplete graphs\nwith polynomial volume growth. We will also see later that these are the smallest such examples.\nThis motivates the move to different graph metrics which take into\naccount not only the combinatorial graph structure but also the vertex degree. \nThese are the so-called intrinsic metrics which we introduce in the next section.\n\nThe result on stochastic incompleteness of anti-trees presented above originally appeared as \nExample~4.11 in \\cite{Woj11}. To the best of our knowledge, the first example of \nan anti-tree in the special case of sphere growth $a_r=r+1$ appears as Example~2.5 in \\cite{DK88}.\nThis anti-tree is a transient graph with the bottom of the spectrum at 0. The same graph\nappears in \\cite{Web10} as an example of a stochastically complete graph with unbounded\nvertex degree.\n\n\n\\section{Intrinsic metrics}\\label{s:intrinsic}\n\\subsection{A brief historical overview}\nAs we have seen, in order to hope for a counterpart for Grigor$'$yan's volume growth result\nfor graphs, we must go beyond the combinatorial graph distance when defining volume growth. In this section we introduce\nthe notion of an intrinsic metric for a weighted graph. This concept arises from Dirichlet form theory.\nAlthough beyond the scope of this article, we mention that the form associated to the Laplacian,\nwhich is a restriction of the graph energy form,\nis a regular Dirichlet form which is not strongly local. For background on Dirichlet forms see \\cite{FOT94},\nfor the connection between graphs and non-local regular Dirichlet forms see \\cite{KL12}.\nFurthermore, let us caution that the notion of an intrinsic metric for a Dirichlet form is distinct from the notion\nof an intrinsic metric in the sense of length spaces as discussed, for example, in \\cite{BBI01}.\n\n\nThe concept of an intrinsic metric for strongly local Dirichlet forms was brought into full fruition\nin \\cite{Stu94}. This allowed for the extension of a variety of results for Riemannian manifolds,\nincluding Grigor$'$yan's volume growth result, to the setting of strongly local Dirichlet forms.\nIn particular, this covers the Riemannian setting as \nthe Riemannian geodesic distance is an intrinsic metric for the strongly local\nDirichlet form arising in the manifold setting.\nHowever, as mentioned above, the energy form of a graph is not strongly local so that the notions of \\cite{Stu94}\ndo not cover the graph setting.\n\nFor non-local Dirichlet forms, such as particular restrictions of the energy form of a graph, the concept of an intrinsic\nmetric was discussed in full generality in \\cite{FLW14}, see also \\cite{MU11} as well as\n\\cite{Fol11, Fol14, GHM12} for the related notion of an adapted metric. \nHowever, as noted in \\cite{FLW14}, the concept\nof an intrinsic metric for a non-local form is more complicated than in the local setting\nas the maximum of two intrinsic metrics is not necessarily an intrinsic metric. This can already be seen\nin an easy example of a graph with three vertices, see Example~6.3 in \\cite{FLW14}. The fact that\nthe maximum of two intrinsic metrics is an intrinsic metric for strongly local Dirichlet forms is essential\nto establish the existence of a maximal intrinsic metric.\n\nThus, there does not exist a maximal intrinsic metric for graphs.\nHowever, for proving statements in graph theory which are analogous to the strongly local setting, \nthe tool of an intrinsic metric is \nquite useful, see the survey article \\cite{Kel15} for an overview of results in this direction and further\nhistorical notes and also \\cite{HH} for some further recent applications.\n\n\\subsection{Intrinsic metrics, combinatorial graph distance and boundedness}\nAfter this brief discussion of the history of intrinsic metrics, we now present the definition for graphs.\nWe call a function mapping pairs of vertices to non-negative real numbers \na \\emph{pseudo metric} if the map is symmetric, vanishes\non the diagonal and satisfies the triangle inequality. In other words, a pseudo metric is a metric\nexcept for the fact that it might be zero for pairs of distinct vertices.\nIn general, intrinsic metrics are only assumed to be pseudo metrics. However, we will follow\nconvention and refer to them as metrics in any case.\n\n\\begin{definition}[Intrinsic metrics, jump size]\nA pseudo metric $\\varrho:X \\times X \\longrightarrow [0,\\infty)$ is called an \\emph{intrinsic metric}\nif\n$$\\sum_{y \\in X} b(x,y) \\varrho^2(x,y) \\leq m(x)$$\nfor all $x \\in X$.\nThe quantity $j = \\sup_{x \\sim y} \\varrho(x,y)$ is called the \\emph{jump size} of $\\varrho$. \nWhen $j < \\infty$, we say that $\\varrho$ has \\emph{finite jump size}.\n\\end{definition}\n\nThe use of intrinsic metrics often lies in a scenario when we want to estimate the energy\nof a cut-off function defined with respect to an intrinsic metric via the measure. In the easiest example,\nlet $\\varrho$ be an intrinsic metric and let $\\rho(x) = \\varrho(x,x_0)$ be the distance with respect to $\\varrho$\nto a fixed vertex $x_0$. If $K \\subseteq X$, then\n\\begin{align*}\n\\sum_{x,y \\in K} b(x,y)(\\rho(x)-\\rho(y))^2 &= \\sum_{x,y \\in K} b(x,y)(\\varrho(x,x_0)-\\varrho(y,x_0))^2\\\\\n&\\leq \\sum_{x \\in K} \\sum_{y \\in K} b(x,y)\\varrho^2(x,y)\\\\\n&\\leq \\sum_{x \\in K} m(x) = m(K).\n\\end{align*}\nIn particular, we see that if $K$ is a set with finite measure, then the energy of $\\rho$ on $K$ is finite.\nThe jump size becomes relevant whenever we have a cut-off function which is supported on $K$\nand we need to control how far outside of the set $K$ the sum above reaches.\n\nAfter this brief discussion, let us mention some examples. We recall that $d$ denotes the combinatorial graph \ndistance, that is, the least number of edges in a path connecting two vertices. The case of when the\ncombinatorial graph distance is equivalent to an intrinsic metric can be characterized in terms\nof the boundedness of the weighted vertex degree.\n\\begin{proposition}\\label{p:intrinsic}\nLet $G$ be a connected weighted graph. The combinatorial graph distance $d$ is equivalent to an \nintrinsic metric if and only if ${\\mathrm{Deg}}$ is a bounded function on $X$.\n\\end{proposition}\n\\begin{proof}\nWe note that $d(x,y)=1$ for all $x \\sim y$. Thus,\n$$\\sum_{y \\in X} b(x,y)d^2(x,y) = \\sum_{y \\in X} b(x,y) = {\\mathrm{Deg}}(x)m(x).$$\nThe conclusion now follows directly.\n\\end{proof}\nRecall that by Theorem~\\ref{t:boundedness_l2} this is the case exactly when the Laplacian is a bounded\noperator and by Theorem~\\ref{t:sc_symmetry}, the graph is stochastically complete in this case. \n\nThus, we see that the combinatorial graph distance may or may not be intrinsic. We now look\nat some further examples. In particular, the first example below gives a case when the combinatorial\ngraph distance is in fact intrinsic and the second gives a pseudo metric which is intrinsic for any given graph.\n\\begin{example}[Intrinsic metrics]\\label{ex:intrinsic}\nWe now give two examples.\n\\begin{enumerate}\n\\item If $m(x)=\\sum_{y\\in X}b(x,y)$, then ${\\mathrm{Deg}}(x)=1$ for all $x \\in X$ and thus the combinatorial\ngraph distance $d$ is equivalent to an intrinsic metric by Proposition~\\ref{p:intrinsic} directly above.\n\\item For a pair of neighboring vertices $x \\sim y$ we let \n$$\\sigma(x,y) = \\left( \\max \\{ {\\mathrm{Deg}}(x), {\\mathrm{Deg}}(y) \\} \\right)^{-1\/2}$$\ndenote the length of the edge connecting $x$ and $y$. Now, we can extend from the length of an edge to the\nlength of a path in a natural way, that is, if $(x_k)=(x_k)_{k=0}^n$ is a path, we let\n$$l_\\sigma((x_k)) = \\sum_{k=0}^{n-1} \\sigma(x_k, x_{k+1})$$\ndenote the length of the path.\nFinally, we define a pseudo metric via\n$$\\varrho_\\sigma(x,y) = \\inf \\{ l_\\sigma((x_k)) \\ | \\ (x_k) \\textup{ is a path connecting } x \\textup{ and } y\\}.$$\nAs $\\varrho_\\sigma(x,y) \\leq \\sigma(x,y)$ for $x \\sim y$ it is then clear that $\\varrho_\\sigma$ is an intrinsic metric as\n$$\\sum_{y \\in X} b(x,y) \\varrho_\\sigma^2(x,y) \\leq \\frac{1}{{\\mathrm{Deg}}(x)} \\sum_{y \\in X} b(x,y) =m(x).$$\nThis metric was first introduced in \\cite{Hua11b}, see also \\cite{Fol11}. \n\nWe note that if the graph is not\nlocally finite, then this intrinsic metric may be only a pseudo metric; however, in the locally finite case,\npath metrics are metrics and give the discrete topology, see for example Lemma~A.3 in \\cite{HKMW13}.\n\\end{enumerate}\n\\end{example}\n\nThe metric $\\varrho_\\sigma$ introduced in the second example above shows that there always exists an intrinsic metric on a weighted\ngraph. This metric, which utilizes the weighted vertex degree function, makes sense in the context\nof the process determined by the heat kernel. Namely, if the Markov process with transition probabilities\ngiven by the heat kernel is a vertex $x$, then at the next jump time it moves with probability $b(x,y)\/\\sum_{z}b(x,z)$\nto a neighbor $y$ of $x$. Furthermore, the wait time at the vertex $x$ is an exponentially distributed\nrandom variable with parameter given by the weighted vertex degree, that is, the probability that \nthe random walker is still at a vertex $x$ after time $t$ without having jumped is given by $e^{-{\\mathrm{Deg}}(x)t}$. \nSee Section~7 in \\cite{KL10} for a further discussion of the connection between the heat semigroup and Markov\nprocesses.\nTherefore, at vertices with a large vertex degree, the process accelerates and thus will more quickly explore\nneighboring vertices. Hence, for the process, neighbors of vertices of large vertex degree are close\nwhich is consistent with the values of $\\varrho_\\sigma$.\n\n\nWe note that given an intrinsic metric $\\varrho$, we can always obtain an intrinsic metric of small\njump size merely by cutting from above. That is, if $\\varrho$ is an intrinsic metric and $C>0$, then \n$$\\varrho_C(x,y) = \\min \\{ \\varrho(x,y), C\\}$$\nis also intrinsic with jump size at most $C$. On the other hand, having a uniform lower bound from below\non the distance between neighbors is equivalent to bounded geometry as we now show.\n\\begin{proposition}\\label{p:intrinsic_2}\nLet $G$ be a weighted graph. There exists an intrinsic metric $\\varrho$ such that $\\varrho(x,y) \\geq C>0$\nfor all $x \\sim y$ if and only if ${\\mathrm{Deg}}$ is a bounded function on $X$.\n\\end{proposition}\n\\begin{proof}\nIf $\\varrho$ is an intrinsic metric with $\\varrho(x,y) \\geq C>0$ for all $x\\sim y$, then\n$$C^2 {\\mathrm{Deg}}(x)m(x) \\leq \\sum_{y \\in X} b(x,y)\\varrho^2(x,y) \\leq m(x)$$\nso that ${\\mathrm{Deg}}$ is bounded. \nConversely, if ${\\mathrm{Deg}}$ is bounded, then by Proposition~\\ref{p:intrinsic},\nit follows that the combinatorial graph distance is equivalent to an intrinsic metric $\\varrho$. In particular, there\nexists a constant $C>0$ such that $C d(x,y) \\leq \\varrho(x,y)$ for an intrinsic metric $\\varrho$. As $d(x,y)=1$\nfor all $x \\sim y$, the conclusion follows.\n\\end{proof}\n\nThus, we obtain two conditions involving the existence of intrinsic metrics which imply stochastic completeness.\n\\begin{corollary}[Stochastic completeness and intrinsic metrics]\nLet $G$ be a connected weighed graph. If either the combinatorial graph distance is equivalent\nto an intrinsic metric or if there exists an intrinsic metric which is uniformly bounded below\non neighbors, then $G$ is stochastically complete.\n\\end{corollary}\n\\begin{proof}\nThis follows immediately by combining Propositions~\\ref{p:intrinsic}~and~\\ref{p:intrinsic_2}\nwith Theorem~\\ref{t:bounded_SC}. \n\\end{proof}\n\n\\subsection{A word about essential self-adjointness and metric completeness}\\label{ss:metric_esa}\nWe briefly mention here some additional facts about metrics, geometry and analysis.\nIn Riemannian geometry, there is the famous Hopf-Rinow theorem, which gives a connection between\nmetric completeness, geodesic completeness and compactness of balls defined with respect to the geodesic metric,\nsee \\cite{doCar92} for example.\nFor locally finite graphs, a counterpart is shown in \\cite{HKMW13}, \nsee also \\cite{KM19} for a recent extension to a more general class of graphs. More specifically,\nTheorem A.1 in \\cite{HKMW13} shows that\nfor locally finite graphs and path metrics, the notions of metric completeness, geodesic completeness\nin the sense that all infinite geodesics have infinite length, and finiteness of balls are equivalent.\nFurthermore, if an intrinsic path metric on a locally finite graph\nsatisfies any of these equivalent conditions, then the restriction of the formal\nLaplacian to the finitely supported functions is essentially self-adjoint, see Theorem~2 in \\cite{HKMW13}. \nThus, metric completeness with respect to an intrinsic metric\nimplies that there exists a unique Laplacian, at least for locally finite graphs. \nThis corresponds to results known for Riemannian manifolds, see \\cite{Che73, Stri83}.\nFor a more general and thorough discussion which includes this question for magnetic Schr{\\\"o}dinger operators on graphs\nsee \\cite{Schm20}.\n\nSubsequently, the assumption that there exists an intrinsic metric for which balls are finite has\noften been used as a substitute for geodesic completeness in the graph setting. In particular, we will\nsee this assumption appearing in our criteria for stochastic completeness in the following sections.\n\n\n\\section{Uniqueness class, stochastic completeness and volume growth}\\label{s:volume}\nIn this section, we will discuss the connections between uniqueness class results for the heat\nequation, stochastic completeness and volume growth. As we have seen, using the combinatorial\ngraph metric gives a very different volume growth borderline for stochastic completeness compared to the manifold\nsetting. We will see that the results in these two settings can ultimately be reconciled via the use of intrinsic metrics.\n\n\\subsection{Uniqueness class}\nWe start by recalling Grigor$'$yan's uniqueness\nclass result on Riemannian manifolds. \nSpecifically, if $u$ is a solution of the heat equation on a Riemannian manifold with zero\ninitial condition and if there\nexists a monotone increasing function $f$ on $(0,\\infty)$ such that $\\int^\\infty {r}\/{f(r)}dr=\\infty$\nwhich dominates the growth of $u$ in the sense that\nfor all $r$ large\n$$\\int_0^T \\int_{B_r} u^2(x,t)\\  d\\mu \\ dt \\leq e^{f(r)},$$\nthen $u=0$, see Theorem~9.2 in \\cite{Gri99} or Theorem~11.9 in \\cite{Gri09} for a proof. \nBy Theorem~\\ref{t:sc_characterizations} above\nthis immediately implies stochastic completeness under a suitable volume growth restriction.\nMore precisely, if $u$ is a bounded solution of the heat equation\nwith bound given by $C$, then letting $f(r)=\\log(C^2 T V(r))$ shows that $u$ must be zero\nwhenever\n$$\\int^\\infty \\frac{r}{\\log V(r)}dr=\\infty.$$\nThus, the only bounded solution of the heat equation with trivial initial condition is trivial.\n\nHowever, in \\cite{Hua11b, Hua12}, there is already a counterexample to an analogue of this \nuniqueness class result when using intrinsic metrics. Namely, for the graph with\n$X={\\mathbb Z}$, \n$$b(x,y)=\\begin{cases}\n1 & \\textup{ if } |x-y|=1 \\\\\n0 & \\textup{ otherwise}\n\\end{cases}$$\nand counting measure $m=1$, there exists an explicit function $u$ which is non-zero, satisfies the\nheat equation with initial condition $0$ as well as the estimate\n$$\\int_0^T \\sum_{x \\in B_r}u^2(x,t) dt \\leq  e^{f(r)}$$\nfor all large $r$ with $f(r)=\\log T + C r \\log r$ for some constant $C$. \nIn particular, it is clear that\n$$\\int^\\infty \\frac{r}{f(r)}dr=\\infty.$$\nThus, no analogue to Grigor$'$yan's uniqueness class result can hold for all graphs, even when using\nintrinsic metrics.\n\nWe note that for this graph ${\\mathrm{Deg}}(x)=2$ for all $x \\in X$\nand thus this graph is stochastically complete by Theorem~\\ref{t:bounded_SC}. Therefore, this non-zero solution\ncannot be bounded by Theorem~\\ref{t:sc_characterizations}. Furthermore, we note by Proposition~\\ref{p:intrinsic}\nthat the combinatorial graph metric is equivalent to an intrinsic metric in this case and that the volume\ngrowth with respect to this metric is only quadratic. Finally, the constant $C$ appearing in the definition of $f$ \nin the example above is crucial\nas for $C<1\/2$, the uniqueness class result holds for all graphs, see Theorem~0.8 in \\cite{Hua12}. \n\nThe difference between these results in the discrete and continuous settings \nwas ultimately resolved in \\cite{HKS} by introducing a class of graphs for which a uniqueness\nclass result analogous to Grigor$'$yan's does hold.\nThese are the globally local\ngraphs which we introduce next. The idea for these graphs in the context of\na uniqueness class result is that the growth of the solution of the heat equation is balanced\nby a decay in the jump size of an intrinsic metric as we go further out in the graph.\nWe note that in the counter example to the uniqueness class result above\nwe use the combinatorial graph distance whose jump size is always one and thus does not decay.\n\n\\begin{definition}[Globally local graphs]\nLet $G$ be a weighted graph with a pseudo metric $\\rho$. Let $B_r$ denote the ball of radius\n$r$ with respect to $\\rho$ and let $j_r$ denote the jump size of $\\rho$ outside of $B_r$, that is,\n$$j_r = \\sup \\{\\rho(x,y) \\ | \\ x \\sim y, \\ x,y \\not \\in B_r\\}.$$\n$G$ is said to be \\emph{globally local} in $\\rho$ with respect to a monotone increasing function $f:(0,\\infty) \\longrightarrow (0,\\infty)$\nif $G$ has finite jump size, i.e., $j_0<\\infty$ and if there exists a constant $A>1$ such that\n$$\\limsup_{r \\to \\infty} \\frac{j_r f(Ar)}{r} < \\infty.$$\n\\end{definition}\n\nThus, globally local graphs not only have finite jump size but provided that $f$ has a certain growth,\nthe jump size must decay outside of balls. In the borderline case for the uniqueness class result, \n$f$ is of the order $r \\log r$ so that\n$j_r$ must take care of the growth of $\\log r$. We note, in particular, that the example mentioned above,\nthat the combinatorial metric used there as an intrinsic metric will not be globally local\nwith respect to $f(r)=r \\log r$.\n\nIn any case, with this notion of globally local graphs, Theorem~1.3 in \\cite{HKS} presents the following result.\n\\begin{theorem}[Uniqueness class for globally local graphs]\\label{t:uniqueness_class}\nLet $G$ be a weighted graph. Let $\\varrho$ be an intrinsic metric with finite balls $B_r^\\varrho$ and assume\nthat $G$ is globally local in $\\varrho$ with respect to a monotone increasing function $f:(0,\\infty) \\longrightarrow (0,\\infty)$\nsuch that\n$$\\int^\\infty \\frac{r}{f(r)}dr=\\infty.$$\nIf $u:X \\times [0,T] \\longrightarrow {\\mathbb R}$ is a solution of the heat equation with initial condition $0$ and\n$$\\int_0^T \\sum_{x \\in B_r^\\varrho} u^2(x,t) m(x) dt \\leq e^{f(r)}$$\nfor all $r >0$, then $u=0$.\n\\end{theorem}\nThe proof of Theorem~\\ref{t:uniqueness_class} can be found in Section~2 of \\cite{HKS}. Though rather\nlong and technical, the main\nidea is to estimate the size of a solution of the heat equation over a small ball at some time via the size of the\nsolution over a larger ball at an earlier time. Then one iterates this estimate down to time zero to show\nthat the solution must be trivial. Along the way, the use of cut-off functions involving intrinsic metrics\nis crucial. This is, in part, because of the fact that in the discrete setting there is no good substitute\nfor the chain rule which is used throughout the proof of Grigor$'$yan's uniqueness class\nresult for manifolds.\n\n\\subsection{Stochastic completeness and volume growth in intrinsic metrics}\nWe want to use the uniqueness class result above to establish stochastic completeness under a volume\ngrowth restriction which is valid for all graphs which allow for an intrinsic metric with finite distance balls. \nHowever, we note that the uniqueness class result above involves the additional assumption of being\nglobally local. Thus, some additional considerations are required in order\nto reduce from general graphs to the class of globally local ones.\n\nAs a first step, it turns out that one can reduce to the case of an intrinsic metric with finite jump size \nvia the notion of truncating the edge weights which\nis already contained in \\cite{GHM12}, see also \\cite{MU11}. \nMore specifically, if $\\varrho$ is an intrinsic metric for $G=(X,b,m)$, we define\nnew edge weights on $X$ via \n$$b_s(x,y) = \\begin{cases}\nb(x,y) & \\textup{ if } \\varrho(x,y) \\leq s \\\\\n0 & \\textup{ otherwise.}\n\\end{cases}$$\nThat is, we remove edges for which the distance between the adjacent vertices is large.\nIt follows that $\\varrho$ is also intrinsic for $G_s=(X,b_s,m)$ and $\\varrho$ now has finite jump size \nof at most $s$ on $G_s$.\nFurthermore, Lemma~3.4 in \\cite{HKS} gives that if $G_s$ is stochastically complete, then $G$ is stochastically\ncomplete. We note that $G_s$ is not necessarily connected even if we start with a connected graph; \nhowever, the equivalent notions of stochastic \ncompleteness presented in Theorem~\\ref{t:sc_characterizations} do not require connectedness of the graph.\nAs an alternate viewpoint, one may apply them on connected components of the graph. \nIn particular, Lemma~3.4 in \\cite{HKS}\nuses the weak Omori-Yau characterization of stochastic completeness, that is, condition (iv) in \nTheorem~\\ref{t:sc_characterizations} above to establish the result. \nSee also Theorem~2.2 in \\cite{GHM12} for a more general\nstatement involving Dirichlet forms associated to general jump processes. \n\nThus, without loss of generality, we may assume that the intrinsic metric has finite jump size. \nThe assumption of finite jump size along with finiteness of balls is easily seen to imply\nlocal finiteness of the graph, see, for example Lemma~3.5 in \\cite{Kel15}. Thus, we have reduced\nto the case of locally finite graphs with finite jump size and finite distance balls.\n\nFinally to reduce to the case of globally local graphs, the authors of \\cite{HKS} use the notion of refinements\nfor locally finite graphs found in \\cite{HS14}. The idea is to insert additional vertices within edges and extend the definitions\nof the edge weights, vertex measure and the intrinsic metric in such a way that both the finiteness of balls\nis preserved and that the measure of balls is only rescaled by a constant. Furthermore, as the inserted vertices\nare now closer together with respect to the new intrinsic metric, it follows by Lemma~3.3 in \\cite{HKS} \nthat this can be done in such a way that\nthe refined graph is globally local with respect to an arbitrarily chosen function. Finally, Theorem~1.5 in \\cite{HKS}\nshows that stochastic completeness is preserved during the process of refining the graph via the use\nof the weak Omori-Yau maximum principle.\n\nPutting everything together, we get the following analogue to Grigor$'$yan's volume growth result which\ncan be found as Theorem~1.1 in \\cite{HKS}.\n\\begin{theorem}[Volume growth and stochastic completeness]\\label{t:sc_volume1}\nLet $G$ be a weighted graph with an intrinsic metric $\\varrho$ with finite distance balls $B_r^\\varrho$.\nLet $V_\\varrho(r) = m(B_r^\\varrho)$ and let $\\log^{\\#}(x) = \\max\\{\\log(x), 1\\}$. If\n$$\\int^\\infty \\frac{r}{\\log^{\\#} V_\\varrho(r)}dr =\\infty,$$\nthen $G$ is stochastically complete.\n\\end{theorem}\n\\begin{proof}[Sketch of proof]\nFrom the discussion above, we can reduce to the case of finite jump size and finite balls and, thus, \nto locally finite graphs. In this case, the technique of refinements allows us to reduce to the case of graphs which are\nglobally local with respect to $f(r)=\\log^{\\#}V_\\varrho(r)$. Thus, given a bounded solution of the heat\nequation $u$ with initial condition 0 and bound $C$, we obtain\n$$\\int_0^T \\sum_{x \\in B_r^\\varrho}u^2(x,t) m(x) dt \\leq C^2T e^{f(r)} = e^{f(r)+C_1}$$\nfor some constant $C_1$.\nTherefore, by Theorem~\\ref{t:uniqueness_class}, we obtain that $u=0$ and by Theorem~\\ref{t:sc_characterizations}~(ii$'$)\nwe get that $G$ is stochastically complete.\n\\end{proof}\n\n\\begin{remark}\n\\begin{enumerate}\n\\item We note that the use of $\\log^{\\#}$ instead of just $\\log$ is to deal with the case when the measure\n$m$ is small. In particular, this covers the case when the entire vertex set has finite measure, that\nis, $m(X)<\\infty$.\nIn this case, stochastic completeness is actually equivalent to two other properties, namely, \nto recurrence and to form uniqueness, see Theorem~16\nin \\cite{Schm17} or Theorem~7.1 in \\cite{GHKLW15} for further details.  Thus, we see that in the case of finite measure,\nthe existence of an intrinsic metric which gives finite balls implies all three of these properties. A partial\nconverse to this result was recently proven in \\cite{Puch}. More specifically, if a graph is recurrent, then \nthere exists a finite measure and an intrinsic metric which has finite distance balls.\nFor a precise statement, see Theorem~11.6.15 in \\cite{Schm20}.\n\n\\item In the case of locally finite graphs with counting measure and an intrinsic metric with finite jump size\nand finite balls, \nTheorem~\\ref{t:sc_volume1} was first\nshown as Theorem~1.2 in \\cite{Fol14}. However, not only does the formulation have additional assumptions\non the graph,\nbut the proof is quite different from Grigor$'$yan's original proof on manifolds. Namely, the approach in \\cite{Fol14} is to synchronize the random\nwalk on the discrete graph with a random walk on a metric graph. Metric graphs are graphs where edges are \nintervals of real numbers. In particular, the energy form on a metric graph is a strongly local Dirichlet form. \nThus, the extension of Grigor$'$yan's result to strongly local Dirichlet forms shown in \\cite{Stu94} implies\nstochastic completeness given that the volume growth of the discrete graph is comparable to the volume\ngrowth of the metric graph. We note that the proof in \\cite{Stu94} also does not invoke the heat equation.\n\nAn analytic proof of the result in \\cite{Fol14} using the Omori-Yau maximum principle and \nwhich allows for an arbitrary vertex measure but still uses metric graphs and assumes local finiteness\nand finite jump size can be found in \\cite{Hua14b}. Thus,\nwhile an analogue of Grigor$'$yan's result was known for some classes of graphs since \\cite{Fol14}, the paper\n\\cite{HKS} contains the first proof which does not assume finite jump size nor local finiteness and does not\ninvoke metric graphs.\n\\end{enumerate}\n\\end{remark}\n\n\\subsection{Stochastic completeness and volume growth in the combinatorial graph metric} \nWe now briefly discuss how in the case of standard edge weights and counting measure the paper \\cite{GHM12}\nalready contains the optimal growth rate result for stochastic completeness when using the combinatorial\ngraph distance. More specifically, the authors of \\cite{GHM12} first use the method of \\cite{Dav85} from the\nstrongly local setting to establish that the volume growth condition\n$$\\liminf_{r \\to \\infty} \\frac{\\log V_{\\varrho_1}(r)}{r \\log r}<\\frac{1}{2}$$\nimplies stochastic completeness of general jump processes, see also \\cite{MUW12}\nwhere it is shown that the $1\/2$ on the right hand side can be replaced by $\\infty$. \nHere, the volume growth is defined with\nrespect to what the authors of \\cite{GHM12} call an \\emph{adapted} metric. \nThe idea for an adapted metric in \\cite{GHM12} is that the intrinsic condition\nneeds to be only satisfied for pairs of vertices that are close with respect to $\\varrho$, that is, \none truncates the metric before imposing the intrinsic condition.\nMore specifically, letting $\\varrho$ be a pseudo metric and $\\varrho_1 =\\min \\{ \\varrho, 1\\}$, then\n$\\varrho_1$ must satisfy\n$$\\sum_{y \\in X}b(x,y)\\varrho_1^2(x,y) \\leq m(x)$$\nfor all $x \\in X$ in order for $\\varrho$ to be called adapted.\n\nClearly any intrinsic metric is adapted. \nOn the other hand, we can also modify Example~\\ref{ex:intrinsic}~(2) in a natural way\nto get an adapted metric. \nMore specifically, by defining the length of an edge now by \n$$\\sigma_1(x,y) = \\left( {\\mathrm{Deg}}(x) \\vee {\\mathrm{Deg}}(y) \\right)^{-1\/2} \\wedge 1$$ \nwhere $a \\vee b = \\max \\{a, b \\}$ and $a \\wedge b = \\min \\{a,b\\}$ \nthen we can extend to paths to get an adapted metric $\\varrho_{\\sigma_1}$. Then, for locally finite graphs,\nCorollary~4.3 in \\cite{GHM12} gives that if $\\inf_{x \\in X} m(x)>0$ and\n$$V_{\\varrho_{\\sigma_1}}(r) \\leq e^{Cr}$$ \nfor $C>0$ and all large $r$, then $G$ is stochastically complete. \nWe note that the additional assumption that $\\inf_{x \\in X} m(x)>0$\nis used along with the volume growth assumption to show that balls defined with respect to $\\varrho_{\\sigma_1}$ are finite.\nThus, these assumptions are used as a replacement for the finiteness of balls assumption found \nin Theorem~\\ref{t:sc_volume1} above.\n\nAlthough clearly not optimal when compared with Theorem~\\ref{t:sc_volume1},\nit turns out that this volume growth result already gives an optimal volume growth condition in the case of standard edge\nweights, counting measure and combinatorial growth distance. More specifically, letting $B_r$ denote the ball\nof radius $r$ defined with respect to the combinatorial graph metric $d$ and $V(r)=m(B_r)$ which \nis the number of vertices in the ball in this case, \nTheorem~1.4 in \\cite{GHM12} gives the following result.\n\n\\begin{theorem}\\label{t:sc_volume_combinatorial}\nLet $G$ be a graph with standard edge weights and counting measure.\nIf \n$$V(r) \\leq C r^3$$\nfor some constant $C>0$ and all large $r$, then $G$ is stochastically complete.\n\\end{theorem}\n\\begin{proof}[Idea of proof]\nIt can be shown that under the assumption $V(r) \\leq C r^3$, there\nare sufficiently many vertices with small degree so that a ball defined with respect $\\varrho_{\\sigma_1}$ is contained\nin a ball of larger radius with respect to the combinatorial graph distance $d$. \nIn particular, the volume growth restriction on $V(r)$ can be used to get a volume growth restriction\non $V_{\\varrho_{\\sigma_1}}(r)$ and then stochastic completeness follows from the volume growth\ncriterion for $V_{\\varrho_{\\sigma_1}}(r)$ mentioned above.\nFor more details, see Section~4\nin \\cite{GHM12}.\n\\end{proof}\n\nWe note that the characterization of stochastic completeness of anti-trees provided in Corollary~\\ref{c:antitree_sc} above\ngives the sharpness of Theorem~\\ref{t:sc_volume_combinatorial} as for an anti-tree with sphere growth \n$a_r$ of the order $r^{1+\\epsilon}$ for any $\\epsilon>0$, we have that $V(r)$ grows like $r^{3+\\epsilon}$ and that the anti-tree is stochastically incomplete. Furthermore, in this case the weighted degree of vertices grows like\n$r^{2+\\epsilon}$ so that balls defined with respect to $\\varrho_{\\sigma_1}$ are not finite and thus the graph is not geodesically\ncomplete. Finally, we note that\nTheorem~\\ref{t:sc_volume_combinatorial} remains valid in the more general setting where the edge weights\nand vertex measure satisfy\n$$b(x,y) \\leq C m(x)m(y)$$\nfor all $x, y \\in X$ and some $C>0$, see Remark~4.4 in \\cite{GHM12}. This type of assumption is sometimes\ncalled an \\emph{ellipticity condition} on graphs, for more details and some applications of this condition\nin the context of curvature see \\cite{KM}.\n\n\\section{Stochastic completeness and curvature}\\label{s:curvature}\nIn recent years there has been a surge of interest in various notions of curvature in the discrete setting.\nWe do not even attempt to give a comprehensive overview of definitions nor of results. We rather confine\nourselves to two of the most prominent definitions of curvature and discuss results which relate curvature and stochastic \ncompleteness. The two notions of curvature that we will discuss are that of Bakry--{\\`E}mery \nwhich arises from the $\\Gamma$-calculus as outlined in \\cite{BE85}\nand Ollivier Ricci which arises from optimal transportation theory and is defined for general Markov\nchains in \\cite{Oll09}. \nWe will briefly introduce each\nand present the relevant results for stochastic completeness.\n\nWe note that, unlike in the manifold case where the Bishop--Gromov inequality relates lower bounds on Ricci curvature\nto volume growth, there is no analogous connection between lower curvature bounds and volume growth in\nfull generality in the discrete setting thus far. \nTherefore, we are not able to relate the \nvolume growth criteria for stochastic completeness presented in the previous section to the curvature\nconditions given in this section.\nHowever, for Bakry--{\\'E}mery curvature, see Theorem~1.8 in \\cite{HM} for some recent progress in connecting \ncurvature and volume growth for a specific class of graphs and Theorem~4.1 in \\cite{M} for\na connection between lower curvature bounds and the volume doubling property for finite\ngraphs. Furthermore, \\cite{AS} establishes comparisons between averaged inner and outer\ndegrees and volume growth and also gives an example of two graphs \nwith equal Olliver Ricci curvatures but different volume growths. \n\n\\subsection{Bakry--{\\'E}mery curvature and stochastic completeness}\nWe start with Bakry--{\\'E}mery curvature. This notion has origins in work on hypercontractive\nsemigroups found in \\cite{BE85}. For early manifestations in the graph setting, see \\cite{LY10,Schm96}. \nWe caution the reader at the outset that in the curvature on graphs community, one usually takes the Laplacian\nwith the opposite sign of ours.\n\nWe first introduce the $\\Gamma$-calculus. In order to take care of convergence of sums, we now assume\nthat all graphs are locally finite. In this case, we note that the domain of the formal Laplacian is the set of all functions\non $X$, that is, $\\mathcal{F} = C(X)$. For $f, g \\in C(X)$ and $x \\in X$, we let\n$$\\Gamma(f,g)(x) = -\\frac{1}{2} \\left(\\mathcal{L}(fg)-f \\mathcal{L} g - g \\mathcal{L} f\\right)(x).$$\nWe will follow convention and write $\\Gamma(f)$ for $\\Gamma(f,f)$. \nBy a direct calculation we then obtain\n$$\\Gamma(f)(x)=\\frac{1}{2m(x)}\\sum_{y \\in X}b(x,y)(f(x)-f(y))^2.$$\nIn particular, if $\\Gamma(f)=0$, then $f$ is constant on any connected component\nof the graph. In some sense, $\\Gamma$ can be thought of as an analogue to the norm squared of the gradient\nfrom the continuous setting.\n\nWe then define\n$$\\Gamma_2(f)= -\\frac{1}{2}\\mathcal{L} \\Gamma(f) + \\Gamma(f, \\mathcal{L} f).$$\nWith these notations $G$ is said to satisfy\n$CD(K,\\infty)$ at $x \\in X$ for $K \\in {\\mathbb R}$ if\n$$\\Gamma_2(f)(x) \\geq K \\Gamma(f)(x)$$\nfor all $f \\in C(X)$. The\n$CD(K,\\infty)$ condition on $G$ is then just the fact\nthat $G$ satisfies the conditions at all $x \\in X$.\n\n\\begin{definition}\nA locally finite weighted graph $G$ is said to satisfy\n$CD(K,\\infty)$ for $K \\in {\\mathbb R}$ if\n$$\\Gamma_2(f) \\geq K \\Gamma(f)$$\nfor all $f \\in C(X)$.\n\\end{definition}\n\nThe idea of the definition is to mimic the inequality obtained via Bochner's formula and a lower\nRicci curvature bound in the Riemannian manifold setting. The number $K$ is then thought\nto be a lower curvature bound on the graph.\n\nWe now work towards giving criteria for stochastic completeness involving Bakry--{\\'E}mery curvature\nconditions. We start with the main result found as Theorem~1.2 in \\cite{HL17} which gives stochastic completeness\nunder the condition $CD(K,\\infty)$, finiteness of distance balls and a uniform lower bound on the measure. \nWe recall that we denote the heat semigroup by \n$$P_t=e^{-tL}$$\nfor $t \\geq 0$. This semigroup is originally defined on $\\ell^2(X,m)$ via the spectral theorem and can then be extended to\nall $\\ell^p(X,m)$ spaces for $p \\in [1,\\infty]$ via monotone limits. \nThe heat semigroup is also strongly continuous, that is,\n$P_t f \\to f$ as $t \\to 0^+$ for all $f \\in \\ell^\\infty(X)$.\nFurthermore, $P_t$ is Markov, specifically,\nfor $0 \\leq f \\leq 1$, we have $0 \\leq P_t f \\leq 1$. Stochastic completeness is then equivalent to the \nfact that $P_t 1 =1$ for all $t \\geq 0$. \n\n\\begin{theorem}[Stochastic completeness and Bakry--{\\'E}mery curvature]\\label{t:sc_BE}\nLet $G$ be a locally finite graph connected with $\\inf_x m(x) >0$ \nand an intrinsic metric $\\varrho$ with finite distance balls.\nIf $G$ satisfies $CD(K,\\infty)$, then $G$ is stochastically complete.\n\\end{theorem}\n\\begin{proof}[Idea of proof]\nThe bulk of the work is in showing that $CD(K,\\infty)$ is actually equivalent to the \nfollowing gradient estimate on the heat semigroup\n$$ \\Gamma (P_t \\varphi) \\leq e^{-2Kt}P_t(\\Gamma(\\varphi))$$\nfor all $\\varphi \\in C_c(X)$ and $t \\geq 0$, see Theorem~4.1 in \\cite{HL17}. \nFurthermore, the finiteness of balls\nwith respect to an intrinsic metric allows for the construction of a sequence $\\varphi_n \\in C_c(X)$ such that\n$0 \\leq \\varphi_n \\leq 1$, $\\varphi_n(x) \\to 1$ as $n \\to \\infty$ for every $x \\in X$ and such that\n$$\\Gamma(\\varphi_n) \\leq \\frac{1}{n}$$\nfor all $n \\in {\\mathbb N}$.\nIn particular, one lets\n$$\\varphi_n(x) = \\left(\\frac{2n-\\varrho(x, x_0)}{n}\\right) \\vee 0 \\wedge 1$$\nwhere $x_0 \\in X$ is an arbitrary vertex, $a\\vee b = \\max \\{a,b\\}$ and $a\\wedge b = \\min\\{a, b\\}$.\nIt is then clear from the definition that $0 \\leq \\varphi_n \\leq 1$, $\\varphi_n =1$ on $B_n$, $\\varphi_n$ is supported\non $B_{2n}$, and thus $\\varphi_n \\in C_c(X)$ by the assumption of finite distance balls,\nand using the fact that $\\varrho$ is intrinsic, a direct calculation gives $\\Gamma(\\varphi_n) \\leq \\frac{1}{n}$.\n\nAs the semigroup is Markov on $C_c(X)$, we then obtain \n$$P_t (\\Gamma(\\varphi_n)) \\leq \\frac{1}{n}.$$\nGiven these ingredients, the proof is now straightforward as\n$P_t \\varphi_n(x) \\to P_t 1(x)$ for all $x \\in X$ by the monotone convergence theorem and thus\n\\begin{align*}\n\\Gamma(P_t 1)(x) &= \\lim_{n \\to \\infty} \\Gamma(P_t \\varphi_n)(x) \\leq \\liminf_{n \\to \\infty} e^{-2Kt}P_t(\\Gamma(\\varphi_n))(x) \\\\\n&\\leq \\liminf_{n \\to \\infty} e^{-2Kt} \\frac{1}{n} = 0\n\\end{align*}\nfor all $x \\in X$. \nThus, $P_t 1$ is constant for every $t \\geq 0$. \nFrom the heat equation we obtain\n$$\\partial_t P_t 1 = - \\mathcal{L} P_t 1 =0$$\nfor all $t >0$.\nAs $P_0 1 =1$ and the semigroup is strongly continuous\nwe obtain that $P_t 1 = 1$ for all $t \\geq 0$.\n\\end{proof}\n\n\\begin{remark}\nWe note that Theorem~\\ref{t:sc_BE} includes not only the assumption of finiteness of balls but also\nthe additional assumption that $\\inf_x m(x)>0$. Both of\nthese assumptions appear in the context of essential self-adjointness.\nIn particular, $\\inf_x m(x)>0$ implies both that $\\mathcal{L}$ maps $C_c(X)$ \ninto $\\ell^2(X,m)$ and that the restriction of $\\mathcal{L}$ to $C_c(X)$ is\nessentially self-adjoint, see Theorem~6 in \\cite{KL12}. In the context of the proof of \nTheorem~\\ref{t:sc_BE} in \\cite{HL17}, this assumption is used to establish the convergence of sums in the \nproof of the estimate $ \\Gamma (P_t \\varphi) \\leq e^{-2Kt}P_t(\\Gamma(\\varphi))$. However, it is not\nclear whether this assumption is really necessary for the ultimate result of stochastic completeness.\n\nOn the other hand, the result in \\cite{HL17} actually assumes a seemingly weaker assumption than finiteness\nof balls found in the formulation of Theorem~\\ref{t:sc_BE} above. Namely, Theorem~1.2 in \\cite{HL17} only\nassumes the existence of a sequence of finitely supported functions $(\\varphi_n)$ \nsuch that $0 \\leq \\varphi_n \\leq 1,\n\\varphi_n(x) \\to 1$ and $\\Gamma(\\varphi_n) \\leq 1\/n$ for all $n \\in {\\mathbb N}$. \nThis is sometimes referred to as a \\emph{completeness} assumption\non the graph as, in the manifold setting, the existence of such a sequence \nis known to be equivalent to geodesic completeness, see \\cite{BGL14,\nStri83}. It is clear in the proof above that the existence of an intrinsic metric with finite distance balls\nimplies the existence of such a sequence. On the other hand,\nMarcel Schmidt recently communicated to us that the converse is also true, that is, that\nthe existence of an intrinsic metric with finite distance balls is actually equivalent to completeness as defined above.\nFor a proof of this fact, see Appendix~A in the updated version of \\cite{LSW}. Furthermore, for graphs satisfying\nthe ellipticity condition $b(x,y) \\leq C m(x)m(y)$ for all $x, y \\in X$ and which are\nstochastically complete and satisfy an additional condition called the Feller property, see \\cite{PS12, Woj17} \nfor more details, $CD(0,\\infty)$ implies that the graph is complete, see Theorem~6.1 in \\cite{KM}.\n\\end{remark}\n\nTheorem~\\ref{t:sc_BE} gives stochastic completeness in the case of a uniform lower Bakry--{\\'E}mery curvature \nbound in the spirit of \\cite{Yau78}.\nHowever, looking at the optimal curvature results from the Riemannian setting mentioned in the introduction,\nwe would expect to allow for some decay of Ricci curvature as in \\cite{Var83, Hsu89}. One improvement of\nTheorem~\\ref{t:sc_BE} in this direction is contained for a special class of graphs in \\cite{HM}. More specifically,\nif $G$ is a graph with $X = {\\mathbb N}_0$, $x \\sim y$ if and only if $|x-y|=1$, $m$ is the counting measure\nand $\\varrho$ is an intrinsic metric on $G$, then letting\n$$\\kappa(x) = \\sup \\{K \\in {\\mathbb R} \\ | \\ G \\textup{ satisfies } CD(K,\\infty) \\textup{ at } x \\}$$\nif $\\kappa(x)$ decays like $-\\varrho^2(0,x)$, then $G$ is stochastically complete,\nsee Theorem~1.6 in \\cite{HM} for a more precise statement and proof.\n\n\n\n\n\\subsection{Ollivier Ricci curvature and stochastic completeness}\nWe now discuss a second commonly appearing manifestation of curvature in the discrete setting. \nThis formulation comes from optimal transport theory, see \\cite{Vill03} for a general background, and was defined for\nMarkov chains in \\cite{Oll07, Oll09}. For graphs, the basic idea is to transport a mass, which is given\nby the transition probability of a simple random walker starting at a vertex, to that of a mass at another vertex \nwith minimal effort. This definition\nwas then modified to give an infinitesimal version for the case of bounded degree in \\cite{LLY11} and extended\nto graphs with general measure and edge weights in \\cite{MW19}. \nWe note that, in contrast to the Bakry--{\\'E}mery formulation which\nis defined at a vertex, this curvature is defined for pairs of vertices.\n\nWe start with some basic definitions. As in the previous subsection, we assume that all graphs are locally finite.\nWe will further assume that all graphs are connected.\nFor a vertex $x \\in X$ and $\\epsilon>0$ small, one defines a transition probability distribution $\\mu_x^\\epsilon$ on $X$ via\n$$\\mu_x^\\epsilon(y)= \n\\begin{cases}\n1 - \\epsilon {\\mathrm{Deg}}(x) & \\textup{ if } y =x \\\\\n\\epsilon b(x,y)\/m(x)& \\textup{ otherwise}\n\\end{cases}\n$$\nwhere ${\\mathrm{Deg}}(x)$ is the weighted degree of $x$.\nThis is a probability distribution provided that $\\epsilon \\leq 1\/{\\mathrm{Deg}}(x)$. We note that a connection to the\nLaplacian is given by\n$$\\mu_x^\\epsilon(y)= 1_y(x) -\\epsilon L 1_y(x)$$\nas follows by a direct calculation.\n\nNow, for two vertices $x_1, x_2 \\in X$, we define the Wasserstein distance between $\\mu_{x_1}^\\epsilon$\nand $\\mu_{x_2}^\\epsilon$ via\n$$W(\\mu_{x_1}^\\epsilon, \\mu_{x_2}^\\epsilon) = \\inf_{\\pi} \\sum_{x,y \\in X}\\pi(x,y) d(x,y)$$\nwhere the infimum is taken over all $\\pi:X \\times X \\longrightarrow [0,1]$ with\n$\\sum_{y \\in X}\\pi(x,y) = \\mu_{x_1}^\\epsilon(x)$ and $\\sum_{x \\in X}\\pi(x,y) = \\mu_{x_2}^\\epsilon(y)$\nand $d$ is the combinatorial graph distance. The idea behind this is that $\\pi$ transports the mass\ndistribution from $\\mu_{x_1}^\\epsilon$ to $\\mu_{x_2}^\\epsilon$ and thus $W(\\mu_{x_1}^\\epsilon, \\mu_{x_2}^\\epsilon)$\nminimizes the effort required to carry out this transport.\n\nWe let $Lip(1)$ denote the set of functions with Lipshitz constant 1 with respect to the combinatorial\ngraph distance, that is,\n$$Lip(1) = \\{ f \\in C(X) \\ | \\ |f(x)-f(y)| \\leq d(x,y) \\textup{ for all } x, y \\in X \\}$$\nand let $\\ell^\\infty(X)$ denote the set of bounded functions.\nBy Kantorovich duality, see Theorem~1.14 in \\cite{Vill03}, we have\n$$W(\\mu_{x_1}^\\epsilon, \\mu_{x_2}^\\epsilon) = \n\\sup_{f \\in Lip(1) \\cap \\ell^\\infty(X)} \\sum_{x \\in X}f(x)(\\mu_{x_1}^\\epsilon(x)-\\mu_{x_2}^\\epsilon(x)).$$\nFinally, following \\cite{Oll09, LLY11}, in \\cite{MW19} we define the Ollivier Ricci curvature\nbetween two vertices as follows.\n\n\\begin{definition}[Ollivier Ricci curvature]\nFor vertices $x, y \\in X$ with $x \\neq y$, we let \n$$\\kappa^\\epsilon(x,y)=1- \\frac{W(\\mu_x^\\epsilon, \\mu_y^\\epsilon)}{d(x,y)}$$\nand define the \\emph{Ollivier Ricci curvature} as\n$$\\kappa(x,y)= \\lim_{\\epsilon \\to 0^+} \\frac{\\kappa^\\epsilon(x,y)}{\\epsilon}.$$\n\\end{definition}\n\n\\begin{remark}\nWe note that \\cite{Oll09} often considers the case when $m(x)=\\sum_{y \\in X}b(x,y)$\nand $\\epsilon=1$. In this case, ${\\mathrm{Deg}}(x)=1$ so that\n$$\\mu_x^1(y) = \\frac{b(x,y)}{\\sum_{z \\in X} b(x,z)}$$\nis just the one step transition probability at $x$ of the simple random walk on $G$.\nMore generally, when $\\epsilon<1$, we get that\n$$\\mu_x^\\epsilon(y) = \\begin{cases}\n1 - \\epsilon & \\textup{ if } y =x \\\\\n {\\epsilon b(x,y)}\/{\\sum_{z \\in X} b(x,z)} & \\textup{ otherwise}.\n\\end{cases}$$\nIn this case, the simple random walk is given a positive probability to remain at the vertex $x$. As such,\nthe constant $1- \\epsilon$ is sometimes referred to as the idleness parameter in this setting,\nsee \\cite{BCLMP18}.\n\nThe idea of letting $\\epsilon \\to 0^+$ in the case when $m(x) = \\sum_{y \\in X}b(x,y)$ then\nappears in \\cite{LLY11}. In particular, the concavity of the function $\\kappa^\\epsilon$ is used\nto establish the existence of the limit. However, as noted previously, the assumption\n$m(x) = \\sum_{y \\in X}b(x,y)$ gives that ${\\mathrm{Deg}}(x)=1$ and thus all such graphs are stochastically\ncomplete by Theorem~\\ref{t:bounded_SC}. \nThe contribution of \\cite{MW19} is to allow for this definition in the case of\npossibly unbounded vertex degree which makes the question of stochastic completeness\ninteresting. The existence of the limit as $\\epsilon \\to 0^+$ follows analogously to the argument\nin \\cite{LLY11}, see also \\cite{BCLMP18}.\n\\end{remark}\n\nWe note that the Ollivier Ricci curvature as defined above can be explicitly calculated in many cases.\nFor example, if the vertices $x\\sim y$ are not contained in any 3-, 4-, or 5-cycles, then\n$$\\kappa(x,y) = 2b(x,y)\\left( \\frac{1}{m(x)}+\\frac{1}{m(y)}\\right) - {\\mathrm{Deg}}(x) - {\\mathrm{Deg}}(y)$$\nsee Example~2.3 in \\cite{MW19}. As a concrete illustration, in the case of standard edge weights\nand counting measure, the curvature of an edge in a $k$-regular tree is \n$\\kappa(x,y)=4-2k$. This confirms the notion that regular trees of degree greater than 2 are the analogues\nof hyperbolic space as they have constant negative curvature.\n\nAs a second example, let $X={\\mathbb N}_0$ and let $x\\sim y$ if and only if $|x-y|=1$.\nSuch graphs will be referred to as \\emph{birth-death chains}.\nWe note that they automatically fall into the framework of weakly spherically symmetric graphs as \ndiscussed in Section~\\ref{s:symmetry}.\nLet $j, k \\in {\\mathbb N}_0$ with $k > j$. The Ollivier Ricci curvature on a birth-death chain can be calculated directly as\n$$\\kappa(j, k) = \\frac{1}{k-j} \\left( \\frac{b(j, j+1)-b(j,j-1)}{m(j)} - \\frac{b(k,k+1)-b(k,k-1)}{m(k)}\\right)$$\nsee Theorem~2.10 in \\cite{MW19}.  We note that for $j \\sim k$, this reduces to the formula above.\n\nBoth of the examples above are easily derived from the following\nformula which allows us to compute the curvature in terms of the Laplacian. \nTo state it, we let\n$$\\nabla_{xy}f= \\frac{f(x)-f(y)}{d(x,y)}$$\nfor $x \\neq y$.\nFor a proof of the following formula see Theorem~2.1 in \\cite{MW19}.\n\\begin{theorem}[Ollivier Ricci curvature and Laplacian]\\label{t:curvature_Lap}\nLet $G$ be a locally finite connected weighted graph. For vertices $x \\neq y$, we have\n$$\\kappa(x,y) =\\inf_{\\substack{f \\in Lip(1) \\\\ \\nabla_{xy}f=1}} \\nabla_{xy}\\mathcal{L} f\n =\\inf_{\\substack{f \\in Lip(1) \\cap C_c(X) \\\\ \\nabla_{xy}f=1}} \\nabla_{xy}\\mathcal{L} f.$$\n\\end{theorem}\n\nWe now show how Theorem~\\ref{t:curvature_Lap} allows us to prove a Laplacian\ncomparison result. In this result, we compare the Laplacian applied to a distance function\non a general graph to the Laplacian applied to a distance function on a birth-death chain.\nWe first define the notion of sphere curvature which will be involved. We recall that for a vertex $x_0 \\in X$, $S_r$ denotes\nthe sphere of radius $r$ around $x_0$ with respect to the combinatorial graph distance.\nWe then let\n$$\\kappa(r) = \\min_{y \\in S_r} \\max_{\\substack{x \\in S_{r-1} \\\\ x \\sim y}}\\kappa(x,y)$$\nfor $r\\in {\\mathbb N}$ with $\\kappa(0)=0$ and call $\\kappa$ the \\emph{sphere curvature}.\n\n\\begin{theorem}[Laplacian comparison]\\label{t:Lap_comp}\nLet $G$ be a locally finite connected weighted graph. If $x_0 \\in X$, $\\rho(x)=d(x,x_0)$ and $\\kappa$ denotes\nthe sphere curvature, then\n$$\\mathcal{L} \\rho(x) \\geq \\sum_{j=1}^{\\rho(x)} \\kappa(j) - {\\mathrm{Deg}}(x_0)$$\nfor all $x \\in X$.\n\\end{theorem}\n\\begin{proof}\nWe note that in general\n$$\\mathcal{L} \\rho(x) = {\\mathrm{Deg}}_-(x)-{\\mathrm{Deg}}_+(x)$$\nwhere ${\\mathrm{Deg}}_\\pm(x)$ are the outer and inner degrees as defined in Section~\\ref{s:symmetry}. \nIn particular, $\\mathcal{L} \\rho(x_0)=-{\\mathrm{Deg}}(x_0)$ so taking the\nsum to be zero in this case gives the statement for $x=x_0$.\n\nThe proof is now by induction on $r=\\rho(x)$ for $x \\in S_r$. Assume that the statement is true for $r-1$. \nLet $y \\in S_r$ and let $x\\in S_{r-1}$\nbe such that $\\kappa(x,y) \\geq \\kappa(x,z)$ for all $z \\sim x$ with $z \\in S_r$. We note that\n$\\kappa(r) \\leq \\kappa(x,y)$, $\\nabla_{yx}\\rho=1$ and $\\rho \\in Lip(1)$ \nso that Theorem~\\ref{t:curvature_Lap} gives \n\\begin{align*}\n\\kappa(r) \\leq \\kappa(x,y) &\\leq \\nabla_{yx}\\mathcal{L} \\rho =\\mathcal{L} \\rho(y) - \\mathcal{L} \\rho(x).\n\\end{align*}\nTherefore, by the inductive hypothesis,\n$$\\sum_{j=1}^{r} \\kappa(j) - {\\mathrm{Deg}}(x_0) = \\sum_{j=1}^{r-1} \\kappa(j) - {\\mathrm{Deg}}(x_0) + \\kappa(r) \\leq \\mathcal{L} \\rho(x) + \\kappa(r) \\leq \\mathcal{L} \\rho(y)$$\nwhich completes the proof.\n\\end{proof}\n\nWe note that the Laplacian comparison result is sharp on birth-death chains as\ncan be seen by a direct calculation. That is, for birth-death chains, we get\n$$\\mathcal{L} \\rho(x) = \\sum_{j=1}^{\\rho(x)} \\kappa(j) - {\\mathrm{Deg}}(x_0).$$\nThus, Theorem~\\ref{t:Lap_comp} compares the Laplacian of a distance function\non a general graph to that of a birth-death chain.\n\nWe now put the various pieces together to obtain a stochastic completeness result for Ollivier Ricci\ncurvature. This can be found as Theorem~4.11 in \\cite{MW19}.\n\\begin{theorem}\\label{t:SC_Oll_curv}\nLet $G$ be a locally finite connected weighted graph. If\n$$\\kappa(r) \\geq -C \\log r$$\nfor some constant $C>0$ and all large $r$, then $G$ is stochastically complete.\n\\end{theorem}\n\\begin{proof}\nLet $\\rho(x)=d(x,x_0)$. It follows from Theorem~\\ref{t:Lap_comp} that\n$$\\mathcal{L} \\rho(x) \\geq \\sum_{j=1}^{\\rho(x)} \\kappa(j) - {\\mathrm{Deg}}(x_0).$$\nNow, from the assumption that $\\kappa(r) \\geq -C \\log r$ we may choose an\nincreasing continuously differentiable function $f:[0,\\infty) \\longrightarrow (0,\\infty)$ such that\n$$\\mathcal{L} \\rho + f(\\rho) \\geq 0$$\nand \n$$\\int^\\infty \\frac{1}{f(r)}dr=\\infty.$$\nAs $\\rho(x_n) \\to \\infty$ along any sequence of vertices $(x_n)$ with\n${\\mathrm{Deg}}(x_n) \\to \\infty$, $G$ is stochastically complete by Theorem~\\ref{t:Khas}.\n\\end{proof}\n\nIt turns out that Theorem~\\ref{t:SC_Oll_curv} is sharp in the sense that for any $\\epsilon>0$,\nthere exists a stochastically incomplete graph with $\\kappa(r) \\geq -(\\log r)^{1+\\epsilon}$. This can already\nbe seen from the case of birth-death chains for which stochastic completeness is equivalent to\n$$\\sum_{r=0}^\\infty \\frac{r+1}{b(r,r+1)}=\\infty$$\nby Theorem~\\ref{t:sc_symmetry} above. For further details, see Theorem~4.11 in \\cite{MW19}.\n\nWe note that the optimal curvature criterion for stochastic completeness in terms of curvature in the \nmanifold setting gives the borderline for stochastic completeness around the curvature decay of order\n$-r^2$, see \\cite{Var83, Hsu89}.\nOne might be tempted to try and reconcile the difference between the manifold and graph setting\nby using intrinsic metrics as was successfully carried out in the case of volume growth, however, for Ollivier Ricci\ncurvature this approach turns out to not work, see Example~4.13 in \\cite{MW19}. \n\nWe note that\nthis is not the only difference between the continuous and discrete settings when it comes to curvature.\nAs another example,\nthere exist infinite graphs which have uniformly positive Ollivier Ricci curvature, see Example~4.18 in \\cite{MW19}.\nThis is known to be impossible in the manifold setting by the Bonnet--Myers theorem. Here,\nanti-trees also prove to be a source of counterexamples as they provide examples of infinite \ngraphs satisfying and $CD(K,\\infty)$ for $K>0$ and well as having uniformly positive Ollivier Ricci curvature, \nsee \\cite{CLMP20} in \\cite{KLW} for the Bakry--{\\'E}mery and Ollivier\nRicci curvature of anti-trees. However, as soon as one either imposes upper bounds on the vertex degree or\nlower bounds the measure, then a graph with uniformly positive lower curvature bounds \nmust be finite, see \\cite{LMP18} for the case of Bakry--{\\'E}mery\nand \\cite{MW19} for the case of Ollivier Ricci.\n\n\n\n\\subsection*{Acknowledgements} I would like to thank J{\\'o}zef Dodziuk\nfor suggesting such a fruitful area of study and for sustaining me\nover the years.  I am also happy to acknowledge the inspiration and support offered by\nIsaac Chavel and Leon Karp.\nFurthermore, I would like to thank Alexander Grigor$'$yan for encouragement and support,\nearly in my career up until the present moment. And also many thanks to my coauthors who contributed \nto this story and whom I also consider to be good friends. In alphabetical order:\nSebastian Haeseler, Bobo Hua, Xueping Huang, Matthias Keller, Daniel Lenz, Jun Masamune, \nFlorentin M{\\\"u}nch and Marcel Schmidt. \nFinally, I would like to thank Isaac Pesenson for the invitation to contribute\nthis article.\n\n\n\n\n\n\n\n\n\n\n\n\\begin{bibdiv}\n\\begin{biblist}\n\n\\bib{AS}{article}{\n   author={Adriani, Andrea},\n   author={Setti, Alberto G.},\n   title={Curvatures and volume of graphs},\n   eprint={arXiv:2009.12814 [math.DG]}\n}  \n\n\n\\bib{Az74}{article}{\n   author={Azencott, Robert},\n   title={Behavior of diffusion semi-groups at infinity},\n   journal={Bull. Soc. Math. 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Fac. Sci. Toulouse Math. (6)},\n   volume={24},\n   date={2015},\n   number={3},\n   pages={563--624},\n   issn={0240-2963},\n   review={\\MR{3403733}},\n   doi={10.5802\/afst.1456},\n}\n\n\\bib{BCLMP18}{article}{\n   author={Bourne, D. P.},\n   author={Cushing, D.},\n   author={Liu, S.},\n   author={M\\\"{u}nch, F.},\n   author={Peyerimhoff, N.},\n   title={Ollivier-Ricci idleness functions of graphs},\n   journal={SIAM J. Discrete Math.},\n   volume={32},\n   date={2018},\n   number={2},\n   pages={1408--1424},\n   issn={0895-4801},\n   review={\\MR{3815539}},\n   doi={10.1137\/17M1134469},\n}\n\t\n\n\\bib{BBI01}{book}{\n   author={Burago, Dmitri},\n   author={Burago, Yuri},\n   author={Ivanov, Sergei},\n   title={A course in metric geometry},\n   series={Graduate Studies in Mathematics},\n   volume={33},\n   publisher={American Mathematical Society, Providence, RI},\n   date={2001},\n   pages={xiv+415},\n   isbn={0-8218-2129-6},\n   review={\\MR{1835418}},\n   doi={10.1090\/gsm\/033},\n}\n\n\\bib{Che73}{article}{\n   author={Chernoff, Paul R.},\n   title={Essential self-adjointness of powers of generators of hyperbolic\n   equations},\n   journal={J. 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B.},\n   title={Heat kernels and spectral theory},\n   series={Cambridge Tracts in Mathematics},\n   volume={92},\n   publisher={Cambridge University Press},\n   place={Cambridge},\n   date={1990},\n   pages={x+197},\n   isbn={0-521-40997-7},\n   review={\\MR{1103113 (92a:35035)}},\n}\n\t\n\n\\bib{Dav92}{article}{\n   author={Davies, E. B.},\n   title={Heat kernel bounds, conservation of probability and the Feller\n   property},\n   note={Festschrift on the occasion of the 70th birthday of Shmuel Agmon},\n   journal={J. Anal. Math.},\n   volume={58},\n   date={1992},\n   pages={99--119},\n   issn={0021-7670},\n   review={\\MR{1226938}},\n   doi={10.1007\/BF02790359},\n}\n\n\\bib{doCar92}{book}{\n   author={do Carmo, Manfredo Perdig\\~{a}o},\n   title={Riemannian geometry},\n   series={Mathematics: Theory \\& Applications},\n   note={Translated from the second Portuguese edition by Francis Flaherty},\n   publisher={Birkh\\\"{a}user Boston, Inc., Boston, MA},\n   date={1992},\n   pages={xiv+300},\n   isbn={0-8176-3490-8},\n   review={\\MR{1138207}},\n   doi={10.1007\/978-1-4757-2201-7},\n}\n\n\n\\bib{Dod83}{article}{\n   author={Dodziuk, J{\\'o}zef},\n   title={Maximum principle for parabolic inequalities and the heat flow on\n   open manifolds},\n   journal={Indiana Univ. Math. 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Soc.},\n      place={Providence, RI},\n   },\n   date={1988},\n   pages={25--40},\n   review={\\MR{954626 (89h:58220)}},\n   doi={10.1090\/conm\/073\/954626},\n}\n\n\n\\bib{DM06}{article}{\n   author={Dodziuk, J{\\'o}zef},\n   author={Mathai, Varghese},\n   title={Kato's inequality and asymptotic spectral properties for discrete\n   magnetic Laplacians},\n   conference={\n      title={The ubiquitous heat kernel},\n   },\n   book={\n      series={Contemp. Math.},\n      volume={398},\n      publisher={Amer. Math. Soc.},\n      place={Providence, RI},\n   },\n   date={2006},\n   pages={69--81},\n   review={\\MR{2218014 (2007c:81054)}},\n}\n\n\\bib{Fel54}{article}{\n   author={Feller, William},\n   title={Diffusion processes in one dimension},\n   journal={Trans. Amer. Math. Soc.},\n   volume={77},\n   date={1954},\n   pages={1--31},\n   issn={0002-9947},\n   review={\\MR{63607}},\n   doi={10.2307\/1990677},\n}\n\n\\bib{Fel57}{article}{\n   author={Feller, William},\n   title={On boundaries and lateral conditions for the Kolmogorov\n   differential equations},\n   journal={Ann. of Math. (2)},\n   volume={65},\n   date={1957},\n   pages={527--570},\n   issn={0003-486X},\n   review={\\MR{90928}},\n   doi={10.2307\/1970064},\n}\n\n\\bib{Fol11}{article}{\n   author={Folz, Matthew},\n   title={Gaussian upper bounds for heat kernels of continuous time simple\n   random walks},\n   journal={Electron. J. Probab.},\n   volume={16},\n   date={2011},\n   pages={no. 62, 1693--1722},\n   issn={1083-6489},\n   review={\\MR{2835251}},\n   doi={10.1214\/EJP.v16-926},\n}\n\n\n\n\\bib{Fol14}{article}{\n   author={Folz, Matthew},\n   title={Volume growth and stochastic completeness of graphs},\n   journal={Trans. Amer. Math. Soc.},\n   volume={366},\n   date={2014},\n   number={4},\n   pages={2089--2119},\n   issn={0002-9947},\n   review={\\MR{3152724}},\n   doi={10.1090\/S0002-9947-2013-05930-2},\n}\n\n\\bib{FLW14}{article}{\n   author={Frank, Rupert L.},\n   author={Lenz, Daniel},\n   author={Wingert, Daniel},\n   title={Intrinsic metrics for non-local symmetric Dirichlet forms and\n   applications to spectral theory},\n   journal={J. Funct. Anal.},\n   volume={266},\n   date={2014},\n   number={8},\n   pages={4765--4808},\n   issn={0022-1236},\n   review={\\MR{3177322}},\n   doi={10.1016\/j.jfa.2014.02.008},\n}\n\n\\bib{FOT94}{book}{\n   author={Fukushima, Masatoshi},\n   author={{\\=O}shima, Y{\\=o}ichi},\n   author={Takeda, Masayoshi},\n   title={Dirichlet forms and symmetric Markov processes},\n   series={de Gruyter Studies in Mathematics},\n   volume={19},\n   publisher={Walter de Gruyter \\& Co.},\n   place={Berlin},\n   date={1994},\n   pages={x+392},\n   isbn={3-11-011626-X},\n   review={\\MR{1303354 (96f:60126)}},\n}\n\n\n\n\\bib{Gaf59}{article}{\n   author={Gaffney, Matthew P.},\n   title={The conservation property of the heat equation on Riemannian\n   manifolds},\n   journal={Comm. Pure Appl. Math.},\n   volume={12},\n   date={1959},\n   pages={1--11},\n   issn={0010-3640},\n   review={\\MR{102097}},\n   doi={10.1002\/cpa.3160120102},\n}\n\n\\bib{GHKLW15}{article}{\n   author={Georgakopoulos, Agelos},\n   author={Haeseler, Sebastian},\n   author={Keller, Matthias},\n   author={Lenz, Daniel},\n   author={Wojciechowski, Rados\\l aw K.},\n   title={Graphs of finite measure},\n   language={English, with English and French summaries},\n   journal={J. Math. Pures Appl. (9)},\n   volume={103},\n   date={2015},\n   number={5},\n   pages={1093--1131},\n   issn={0021-7824},\n   review={\\MR{3333051}},\n   doi={10.1016\/j.matpur.2014.10.006},\n}\n\n\\bib{Gol14}{article}{\n   author={Gol{\\'e}nia, Sylvain},\n   title={Hardy inequality and asymptotic eigenvalue distribution for\n   discrete Laplacians},\n   journal={J. Funct. 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(N.S.)},\n   volume={36},\n   date={1999},\n   number={2},\n   pages={135--249},\n   issn={0273-0979},\n   review={\\MR{1659871 (99k:58195)}},\n   doi={10.1090\/S0273-0979-99-00776-4},\n}\n\n\\bib{Gri09}{book}{\n   author={Grigor'yan, Alexander},\n   title={Heat kernel and analysis on manifolds},\n   series={AMS\/IP Studies in Advanced Mathematics},\n   volume={47},\n   publisher={American Mathematical Society, Providence, RI; International\n   Press, Boston, MA},\n   date={2009},\n   pages={xviii+482},\n   isbn={978-0-8218-4935-4},\n   review={\\MR{2569498 (2011e:58041)}},\n}\n\n\\bib{GHM12}{article}{\n   author={Grigor'yan, Alexander},\n   author={Huang, Xueping},\n   author={Masamune, Jun},\n   title={On stochastic completeness of jump processes},\n   journal={Math. Z.},\n   volume={271},\n   date={2012},\n   number={3-4},\n   pages={1211--1239},\n   issn={0025-5874},\n   review={\\MR{2945605}},\n   doi={10.1007\/s00209-011-0911-x},\n}\n\n\\bib{HK11}{article}{\n   author={Haeseler, Sebastian},\n   author={Keller, Matthias},\n   title={Generalized solutions and spectrum for Dirichlet forms on graphs},\n   conference={\n      title={Random walks, boundaries and spectra},\n   },\n   book={\n      series={Progr. Probab.},\n      volume={64},\n      publisher={Birkh\\\"{a}user\/Springer Basel AG, Basel},\n   },\n   date={2011},\n   pages={181--199},\n   review={\\MR{3051699}},\n}\n\n\\bib{HKLW12}{article}{\n   author={Haeseler, Sebastian},\n   author={Keller, Matthias},\n   author={Lenz, Daniel},\n   author={Wojciechowski, Rados{\\l}aw},\n   title={Laplacians on infinite graphs: Dirichlet and Neumann boundary\n   conditions},\n   journal={J. Spectr. Theory},\n   volume={2},\n   date={2012},\n   number={4},\n   pages={397--432},\n   issn={1664-039X},\n   review={\\MR{2947294}},\n}\n\n\\bib{Has60}{article}{\n   author={Has\\cprime minski\\u{\\i}, R. Z.},\n   title={Ergodic properties of recurrent diffusion processes and\n   stabilization of the solution of the Cauchy problem for parabolic\n   equations},\n   language={Russian, with English summary},\n   journal={Teor. Verojatnost. i Primenen.},\n   volume={5},\n   date={1960},\n   pages={196--214},\n   issn={0040-361x},\n   review={\\MR{0133871}},\n}\n\n\\bib{Hsu89}{article}{\n   author={Hsu, Pei},\n   title={Heat semigroup on a complete Riemannian manifold},\n   journal={Ann. Probab.},\n   volume={17},\n   date={1989},\n   number={3},\n   pages={1248--1254},\n   issn={0091-1798},\n   review={\\MR{1009455 (90j:58158)}},\n}\n\n\\bib{HH}{article}{\n   author={Hua, Bobo},\n   author={Huang, Xueping},\n   title={A survey on unbounded Laplacians and intrinsic metrics on graphs},\n}  \n\n\\bib{HL17}{article}{\n   author={Hua, Bobo},\n   author={Lin, Yong},\n   title={Stochastic completeness for graphs with curvature dimension\n   conditions},\n   journal={Adv. Math.},\n   volume={306},\n   date={2017},\n   pages={279--302},\n   issn={0001-8708},\n   review={\\MR{3581303}},\n   doi={10.1016\/j.aim.2016.10.022},\n}\n\n\\bib{HM}{article}{\n   author={Hua, Bobo},\n   author={M{\\\"u}nch, Florentin},\n   title={Ricci curvature on birth-death processes},\n   eprint={arXiv:1712.01494 [math.DG]}\n}  \n\n\n\n\\bib{Hua11a}{article}{\n   author={Huang, Xueping},\n   title={Stochastic incompleteness for graphs and weak Omori-Yau maximum\n   principle},\n   journal={J. Math. 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Probab.},\n   volume={27},\n   date={2014},\n   number={2},\n   pages={634--682},\n   issn={0894-9840},\n   review={\\MR{3195830}},\n   doi={10.1007\/s10959-012-0456-x},\n}\n\n\\bib{Hua14b}{article}{\n   author={Huang, Xueping},\n   title={A note on the volume growth criterion for stochastic completeness\n   of weighted graphs},\n   journal={Potential Anal.},\n   volume={40},\n   date={2014},\n   number={2},\n   pages={117--142},\n   issn={0926-2601},\n   review={\\MR{3152158}},\n   doi={10.1007\/s11118-013-9342-0},\n}\n\n\n\\bib{HKMW13}{article}{\n   author={Huang, Xueping},\n   author={Keller, Matthias},\n   author={Masamune, Jun},\n   author={Wojciechowski, Rados{\\l}aw K.},\n   title={A note on self-adjoint extensions of the Laplacian on weighted\n   graphs},\n   journal={J. Funct. Anal.},\n   volume={265},\n   date={2013},\n   number={8},\n   pages={1556--1578},\n   issn={0022-1236},\n   review={\\MR{3079229}},\n   doi={10.1016\/j.jfa.2013.06.004},\n}\n\n\\bib{HKS}{article}{\n   author={Huang, Xueping},\n   author={Keller, Matthias},\n   author={Schmidt, Marcel},\n   title={On the uniqueness class, stochastic completeness and volume growth for graphs},\n   journal={Trans. Amer. Math. Soc.},\n   date={to appear},\n   eprint={arXiv:1812.05386 [math.MG]}\n\n \n  \n}  \n\n\\bib{HS14}{article}{\n   author={Huang, Xueping},\n   author={Shiozawa, Yuichi},\n   title={Upper escape rate of Markov chains on weighted graphs},\n   journal={Stochastic Process. Appl.},\n   volume={124},\n   date={2014},\n   number={1},\n   pages={317--347},\n   issn={0304-4149},\n   review={\\MR{3131296}},\n   doi={10.1016\/j.spa.2013.08.004},\n}\n\n\\bib{Ichi82}{article}{\n   author={Ichihara, Kanji},\n   title={Curvature, geodesics and the Brownian motion on a Riemannian\n   manifold. II. 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Phenom.},\n   volume={5},\n   date={2010},\n   number={4},\n   pages={198--224},\n   issn={0973-5348},\n   review={\\MR{2662456}},\n   doi={10.1051\/mmnp\/20105409},\n}\n\n\\bib{KL12}{article}{\n   author={Keller, Matthias},\n   author={Lenz, Daniel},\n   title={Dirichlet forms and stochastic completeness of graphs and\n   subgraphs},\n   journal={J. Reine Angew. Math.},\n   volume={666},\n   date={2012},\n   pages={189--223},\n   issn={0075-4102},\n   review={\\MR{2920886}},\n   doi={10.1515\/CRELLE.2011.122},\n}\n\n\\bib{KLW13}{article}{\n   author={Keller, Matthias},\n   author={Lenz, Daniel},\n   author={Wojciechowski, Rados{\\l}aw K.},\n   title={Volume growth, spectrum and stochastic completeness of infinite\n   graphs},\n   journal={Math. 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Z.},\n   title={Ergodic properties of recurrent diffusion processes and\n   stabilization of the solution of the Cauchy problem for parabolic\n   equations},\n   language={Russian, with English summary},\n   journal={Teor. Verojatnost. i Primenen.},\n   volume={5},\n   date={1960},\n   pages={196--214},\n   issn={0040-361x},\n   review={\\MR{0133871 (24 \\#A3695)}},\n}\n\n\\bib{LSW}{article}{\n   author={Lenz, Daniel},\n   author={Schmidt, Marcel},\n   author={Wirth, Melchior},\n   title={Uniqueness of form extensions and domination of semigroups},\n   eprint={arXiv:1608.06798 [math.FA]}\n}  \n\n\\bib{LLY11}{article}{\n   author={Lin, Yong},\n   author={Lu, Linyuan},\n   author={Yau, Shing-Tung},\n   title={Ricci curvature of graphs},\n   journal={Tohoku Math. J. 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+{"text":"\\section{Introduction}\n\n\nThe Pipe nebula\\ is an ideal test site to study early cloud formation. From this massive, nearby, elongated cloud (\\tenpow{4}~$M_{\\odot}$; 145~pc away; 18~pc$\\times$3~pc in size) arises $^{12}$CO~1--0 emission that shows two filamentary structures: one elongated along the N--S direction at a $v_{\\rm LSR}$ range of 6--7~km~s$^{-1}$; and the other one along the E--W direction at 2--4~km~s$^{-1}$\\ \\citep{onishi99}.\nThe magnetic field is strikingly uniform along the N--S direction with optical polarization degrees up to an outstanding 15\\% value  (\\citealp{alves08}, hereafter AFG08; \\citealp{alves14}). Overall, the cloud seems to be magnetically dominated and turbulence seems to be sub-Alfv\\'enic \\citep{franco10}. Besides the simple structure, the cloud is at a very early stage of evolution with little to none star-formation feedback. Only 14 young stars have been detected, mostly in the B59 region at the west end, delivering a very low $\\sim$0.06\\%\\ star formation efficiency \\citep{forbrich09}. In addition, the cloud harbors more than 150 starless cores with very early chemistry that seem to be gravitationally unbound and pressure confined \\citep{lada08,rathborne08,roman09,roman10,frau10,frau12}. \n\n\n\n\n\n\n\n\n\n\n\n   \\begin{figure*}[t]\n   \\centering\n   \\includegraphics[width=\\textwidth,angle=0]{veloPol3.pdf}\n   \\caption{Velocity and polarization distribution of the Pipe nebula.\n   {\\it Color code}: blue and red for field groups NS and S, respectively.\n      {\\bf Central maps}: \n        {\\it Grayscale maps}: top and bottom panels show the regions with $^{13}$CO\\ $v_{\\rm m1}$\\ in the $>$4~km~s$^{-1}$\\ and $<$4~km~s$^{-1}$, respectively \\citep[fitted to the map by][]{onishi99}. The N--S filament is evident in $^{12}$CO emission \\citep{onishi99}.\n       {\\it Contours}: visual extinction map \\citep{lombardi06}. Levels are 5.5 and 20~mag.\n   {\\it Segments}: average polarization segment on each observed field (AFG08; \\citealp{franco10}). Length is proportional to $\\langle$P$_{\\rm deg}\\rangle$. \n {\\it Circles}: positions of the embedded dense cores with extinction peaks $>10$~mag \\citep{roman09,roman10}. \n   {\\bf Left-hand side panels}: data averaged over the fields observed by AFG08. {\\it Top panel}: $\\langle\\Delta$P$_{\\rm ang}\\rangle$\\ versus $\\langle$P$_{\\rm deg}\\rangle$\\ as in AFG08. {\\it Central panels}: $\\langle\\Delta$P$_{\\rm ang}\\rangle$\\ and $\\langle$P$_{\\rm deg}\\rangle$\\ versus $\\langle v_{\\rm m1}\\rangle$, where m1 stands for the first-order moment of the $^{13}$CO\\ emission.\n   {\\it Bottom panel}: average linewidth of the $^{13}$CO\\ velocity components as a function of $v_{i}$ for the different cloud regions. $v_{i}$ is sampled in 0.25~km~s$^{-1}$--wide bins. Error bars indicate 1-$\\sigma$ dispersion.\n   {\\bf Right-hand side panels}: data for every pixel separated by groups. {\\it Top panel}: P$_{\\rm ang}$\\ versus P$_{\\rm deg}$. {\\it Bottom panels}: P$_{\\rm ang}$\\ and P$_{\\rm deg}$\\ versus $v_{\\rm m1}$. \n   }\n   \\label{fig-vel}\n    \\end{figure*}\n\n\n\n\n\n\n\n\n\\section{Datasets and data process\\label{sec-data}}\n\nThe three datasets used in this work are published in different articles. We limit ourselves to provide the basic information and we refer the reader to the original publications.\n\nThe dust extinction maps were derived by \\citet{roman09,roman10} through the NICER infrared color excess technique \\citep{lombardi01}. The angular resolution is 20\\arcsec\\ and the typical rms is $<$1~mag. \n\nThe $^{13}$CO~1--0 map was presented by \\citet{onishi99} and has spectral and angular resolution of $\\sim$0.1~km~s$^{-1}$\\ and 4\\arcmin, respectively. The $^{13}$CO\\ emission is enclosed in the 5~mag contour of the $A_{v}$\\ map (Fig.~\\ref{fig-vel}). Hence, $^{13}$CO\\ is a good tracer of the gas kinematics (Section~~\\ref{ssec-resolution}).\n\nThe optical linear polarization data were published by AFG08 and \\citet{franco10}. They observed 46 fields of 12\\arcmin$\\times$12\\arcmin\\ in the $R$-band, covering most of the Pipe nebula. We used the $\\sim$6,600 stars detected with $P\/\\sigma P\\geq10$.\n\nOur data processing followed two steps. First, the $^{13}$CO\\ datacube was converted to spectra on a pixel-by-pixel basis. A semi-automatic algorithm detected the number of peaks in each spectrum and carried out a Gaussian fit to derive the peak velocity, $v_{i}$, and the linewidth, FWHM$_{i}$, of the different components. For each pixel, we also derived the first-order moment of the $^{13}$CO\\ emission ($v_{\\rm m1}$). Second, we paired the position of each $R$-band star to its pixel in the $A_{v}$\\ and $^{13}$CO\\ map. Upon positional match, we composed a set of properties for each pixel: $A_{v}$, $v_{\\rm m1}$, $v_{i}$, FWHM$_{i}$, and polarization angle (P$_{\\rm ang}$) and degree (P$_{\\rm deg}$).  The effects of the different angular resolutions are discussed in Section~\\ref{ssec-resolution}.\n\n\n\n\n\n   \\begin{figure}\n   \\centering\n   \\includegraphics[width=\\columnwidth,angle=0]{stemOscillationsPaper.pdf}\n   \\vskip .1cm\n   \\includegraphics[width=.95\\columnwidth,angle=0]{pipe2.png}\n \\caption{\n {\\bf Top panels}: projections of the position-position-velocity cube in two perpendicular position-velocity planes. The orientation of the projection planes is indicated as blue lines in the $A_{v}$\\ map subpanels. {\\it Bottom panel}: velocity as a function of position. The projection plane is at an angle of 124$^\\circ$\\ (shock direction: projection on the plane of the sky of the velocity difference of the filaments). {\\it Top panels}: $v_{i}$ and FWHM$_{i}$ as a function of position. The projection plane is at an angle of 34$^\\circ$\\ (perpendicular to the shock direction). {\\it Circles}: dense cores as in Fig.~\\ref{fig-vel}.\n{\\bf Cartoon}: proposed scenario. \n The antenna represents the observer. Blue and red cylinders represent the E--W and N--S filaments, respectively. The LOS velocities of the filaments are shown as arrows and the stem and bowl regions are labeled. The translucent plane shows the shock direction on the plane of the sky extended along the line-of-sight direction (blue line on bottom panel above). The velocities of the filaments along the shock are shown as arrows. Green arrows represent the tentative direction of the magnetic field (Section~\\ref{ssec-mag-field}). }\n   \\label{fig-ppv}\n    \\end{figure}\n\n\n\n\n\n\n\n\\section{Results and analysis\\label{sec-results}}\n\n\n\\subsection{Dual large-scale velocity structure}\n\nFigure~\\ref{fig-vel} outlines the main velocity structures of the Pipe nebula\\ derived from the $^{13}$CO\\ spectra. The cloud can be separated in two different filaments with narrow velocity ranges: one oriented E--W at $<$4~km~s$^{-1}$\\ and another one oriented N--S at $>$4~km~s$^{-1}$\\ \\citep[this filament is better seen in the $^{12}$CO\\ map: see][]{onishi99}. The spectra from the E--W filament typically have one velocity component, two in some cases. Instead, the spectra from the N--S filament show a richer velocity structure with up to four components. Both structures overlap along the line-of-sight (LOS) at the bowl (Fig.~\\ref{fig-vel}). The velocities at the overlapping region cover the entire range between those of the two filaments. This transition is illustrated in Fig.~\\ref{fig-ppv} through two orthogonal projections of the position-position-velocity cube \\citep[see][]{hacar13} that show distinct patterns (Section~\\ref{ssec-collision}).\n\nWe define two regions: the non-shocked gas --group NS-- that roughly includes B59 and the stem ($\\langle$P$_{\\rm deg}\\rangle$$\\la$5\\%) and the shocked gas --group S-- that roughly coincides with the bowl ($\\langle$P$_{\\rm deg}\\rangle$$\\ga$5\\%). These groups appear color-coded on Fig.~\\ref{fig-vel} and also have distinct properties in polarimetry, density, and turbulence (Section~\\ref{ssec-resPol}). \n\n\n\n\\subsection{Bimodal trends in velocity, polarization, and linewidth\\label{ssec-resPol}}\n\nThe top left panel on Fig.~\\ref{fig-vel} replicates Fig.~2 by \\citet{alves08}. They report a systematic anti-correlation between $\\langle\\Delta$P$_{\\rm ang}\\rangle$\\ and $\\langle$P$_{\\rm deg}\\rangle$. The color-coded NS and S groups appear clearly differentiated: S fields have high $\\langle$P$_{\\rm deg}\\rangle$\\ and low $\\langle\\Delta$P$_{\\rm ang}\\rangle$, oppositely to NS fields. This differentiation goes beyond field averages and holds at a single pixel level (top right panels). \n\nThe central left panels on Fig.~\\ref{fig-vel} show the distinct distribution of $\\langle\\Delta$P$_{\\rm ang}\\rangle$\\ and $\\langle$P$_{\\rm deg}\\rangle$\\ as a function of $\\langle v_{\\rm m1}\\rangle$. The bottom right panels of Fig.~\\ref{fig-vel} show similar plots but for all the pixels. These panels show that the clear polarization differences between group~S and group~NS are mostly independent of the velocity of the gas, thus suggesting different scenarios for the two regions. Group~S gas at 4--5~km~s$^{-1}$\\ shows the largest polarization degree.\n\nThe bottom left panel of Fig.~\\ref{fig-vel} and top panel of Fig.~\\ref{fig-ppv} show that the gas in the bowl has a larger linewidth than in B59 and the stem. Strikingly, the star-forming B59 complex has a smaller linewidth than group~S. This suggests that the Bowl gas is more disturbed than that of B59.\n\n\n\n\n\n\\section{Discussion\\label{sec-disc}}\n\n\n\\subsection{Data suitability and spatial resolution\\label{ssec-resolution}}\n\nThe polarization data trace material with $A_{v}$  $\\la$15~mag that includes 99.1\\% of the $A_{v}$\\ maps pixels, and \\citet{franco10} show that P$_{\\rm deg}$\\ and $A_{v}$\\ grow roughly proportionally. Hence, optical data are good tracers of the magnetic field of the cloud. The NICER $A_{v}$\\ maps are very accurate: a comparison to 1.2~mm emission maps of dense cores shows a perfect within-the-errors agreement for up to 45~mag \\citep{frauthesis}. The spatial resolution of the $A_{v}$\\ map is $\\sim$0.015~pc. This is significantly smaller than the typical 0.1~pc diameter of dense cores, and thus, the map resolution is more than enough for the large scales. $^{13}$CO\\ is a good tracer of the dense gas (Fig.~\\ref{fig-vel}). The $^{13}$CO\\ pixel size is $\\sim$0.17~pc, comparable to a dense core. With a mean $^{13}$CO\\ linewidth of 0.6$\\pm$0.3~km~s$^{-1}$, the pixel crossing time is $\\sim$0.28~Myr. The dense core formation timescale is 0.5--1~Myr \\citep{bergin07}, and thus, it seems unlikely that any velocity substructure would form. Therefore, the $^{13}$CO\\ map should suffice to reflect all significant variations in the gas velocity.\n\n\n\n\\subsection{Interaction of gas filaments and core formation\\label{ssec-collision}}\n\nThe two filaments overlap at the bowl and the gas velocities cover the entire range between the original velocities (Fig.~\\ref{fig-ppv}). This fact points to an interaction, possibly a collision, as reported toward denser and more massive IRDCs \\citep{duartecabral11,henshaw13,nakamura14}. The sound speed at 12~K is $c_{s}$=0.27~km~s$^{-1}$\\ and the pre-shock Alfv\\'en speed\\footnote{Assuming a magnetic field on the plane of the sky with $B_{\\rm stem}$=30~$\\mu$G, and a density of \\scitenpow{3}{3}~cm$^{-3}$\\  (AFG08).} is $v_{\\rm A}$=0.78~km~s$^{-1}$. The magnetosonic speed $v_{m} = \\sqrt{c_{s}^{2}+v_{A}^{2}}$ is 0.83~km~s$^{-1}$, smaller than the velocity separation of the filaments. Therefore, an MHD wave is slower than the velocity difference between the filaments and a shock would indeed form. Hence, in the Pipe nebula\\ it is possible to compare the evolution from non-shocked (stem) to shocked (bowl) gas.\n\nThe two orthogonal projections of the position-position-velocity cube show that LOS velocities are anisotropic (Fig.~\\ref{fig-ppv}). On the one hand, the bottom panel shows highly ordered velocities likely to be the velocity gradient generated along the direction of the collision. On the other hand, the middle panel shows a large velocity dispersion likely to be the projection across the shock. This identified direction for the plausible shock is the projection on the plane of the sky of the velocity difference between the filaments. In addition, for the collision to happen, the N--S filament must be closer to us because it is going away faster than the E--W filament. The total shock velocity would be a combination of two components: (a) the line-of-sight velocity and (b) the plane-of-the-sky velocity whose direction is deduced from the projections of the position-position-velocity cube. Figure~\\ref{fig-ppv} shows a cartoon of the proposed scenario. \n\nFigure~\\ref{fig-vel} shows that linewidths increase {\\it only} at the bowl {\\it and} for the velocities between the two filaments (roughly 4 to 6~km~s$^{-1}$). This points to a more turbulent region where the interaction takes place. The interaction would settle the gas at the observed intermediate velocities and result in sub- and trans-Alfv\\'enic gas motions \\citep{franco10}. The accumulation of material would cause the increase in density and would lead to the formation of the bowl. In fact, the average column density of 6.0$\\pm$1.8~mag at the bowl is twice as high as the average 3.0$\\pm$1.3~mag at the stem. This increase reflects an increase in volume density due to the plausible filament collision.\n\nThe bowl is the region with the largest column density and the most fertile in number of dense cores (Fig.~\\ref{fig-vel}). B59, the other region rich in dense cores, is also proposed to be formed through an external shock that triggered star formation \\citep{peretto12}. Cores have velocities in good agreement with gas velocities. At the bowl, cores cover the entire velocity range between the filaments, follow the velocity gradient along the shock, and tend to cluster at the larger velocities (bottom panel in Fig.~\\ref{fig-ppv}). This combined structure of cores and gas is the ``molecular ring'' reported by \\citet{muench07}. There are other clusters of cores, one in B59 and two in the stem (middle panel in Fig.~\\ref{fig-ppv}). The gas surrounding these clusters shows similar properties: a larger dispersion of velocities, and increases in linewidth, $^{13}$CO\\ brightness, and column density. These properties resemble those of the shocked bowl but with smaller values and at a smaller spatial scale. In general, it seems that in the slowed, shocked gas the higher density and sub-Alfv\\'enic motions enhance the creation of cores.\n\n\n\n\\subsection{Magnetic field compression and polarization increase\\label{ssec-mag-field}}\n\nThe high P$_{\\rm deg}$\\ values --close to the theoretical maximum \\citep{whittet03}-- suggest that the magnetic field direction is close to the plane of the sky AFG08. The higher P$_{\\rm deg}$\\ values are detected towards the bowl, where the two filaments interact. The shock can have reoriented the magnetic field placing it closer to the plane-of-the-sky (Fig.~\\ref{fig-ppv}). This scenario could explain the variation of P$_{\\rm deg}$. Moreover, the peak of P$_{\\rm deg}$\\ is at a velocity of $\\sim$4.5~km~s$^{-1}$, the intermediate velocity between the filaments where the shock would have the strongest effects.\n\nThe material traced by the optical polarization is dominated by magnetic field over gravity and turbulence \\citep{franco10}. This low-density gas is in flux-freezing conditions, this is, matter and magnetic field are tied. As expected in this regime, the large-scale gas compression that causes the column density to double also doubles the magnetic field strength: from 30~$\\mu$G at the stem to 65~$\\mu$G at the bowl (AFG08). This implies that the magnetic energy density is four times higher in the bowl than in the stem ($E_{\\rm mag}^{\\rm stem}$$\\simeq$\\scitenpow{3.6}{-11}~erg~cm$^{-3}$\\ and $E_{\\rm mag}^{\\rm bowl}$$\\simeq$\\scitenpow{1.7}{-10}~erg~cm$^{-3}$). Similarly, the larger line widths in the bowl than in the stem (0.6$\\pm$0.3 and 0.48$\\pm$0.19~km~s$^{-1}$, respectively) give a turbulent kinetic energy density about 40\\% higher in the bowl ($E_{\\rm kin}^{\\rm bowl}$$\\simeq$\\scitenpow{2.3}{-11}~erg~cm$^{-3}$). This generates an increase of the surface pressure on dense cores, reported to be $\\sim$60\\% higher in the bowl with respect to the stem \\citep{lada08}. The energy increase can be explained by the plausible shock scenario: assuming an average volume density of \\scitenpow{3}{3}~cm$^{-3}$\\ (AFG08) and that the shock velocity is the velocity difference between the two filaments, 2.5~km~s$^{-1}$, then the shock energy density is $E_{\\rm shock}$$\\simeq$\\scitenpow{3.7}{-10}~erg~cm$^{-3}$, twice the energy increase from the stem to the bowl. Consequently, the kinetic energy of the collision could increase the measured turbulence levels, and then drag and compress the magnetic field, transferring and storing some of the kinetic energy into magnetic field energy. \n\nThe Alfv\\'enic Mach number $M_{A}=v_{\\rm shock}\/v_{A}$ of the plausible shock is $\\sim$3.0. Simulations of low $M_{A}$ (highly magnetized) collisions predict the formation of shocks \\citep{lesaffre13}. Indeed, models with low Alfv\\'enic Mach numbers ($M_{A}$$\\sim$4--5) drive more energy into compressing the magnetic field \\citep{pon12}. In particular, for clouds of similar temperature and density to the Pipe nebula\\ colliding at 2--3~km~s$^{-1}$, the density and magnetic field strength increase by a factor similar to $M_{\\rm A}$. Moreover, 20--50\\% of the shock energy is stored in the magnetic field. These predictions match our results at the Pipe nebula, also if taking the factor of 2 error in the measured quantities into account. This provides a plausible theoretical framework that reproduces our observations. The majority of the remaining energy is expected to be radiated away through mid-J CO lines that can be used to further test this scenario.\n\n\n\n\\section{Conclusions\\label{sec-concl}}\n\nWe combined visual extinction, optical polarimetry, and $^{13}$CO\\ data of the Pipe nebula. Here, we summarize the conclusions.\n\n\\begin{enumerate}\n\n\\item The velocity patterns of the gas suggest that the N--S and E--W filaments are colliding along the NW-SE direction at the bowl, where turbulence increases.\n\n\\item There is a good correlation between polarization properties and velocity of the gas, suggesting that magnetization increases at the interaction site. The column density and magnetic field strength double with respect to the non-shocked gas, pointing to a highly magnetized, continuous collision under flux-freezing.\n\n\\item Cores in the bowl tend to cluster in space and tend to follow the $^{13}$CO\\ velocity gradient. Dense cores tend to form in regions similar to the bowl where the cloud gas has large velocity dispersion, linewidths, column density, and $^{13}$CO\\ emission. It seems that in the slowed, shocked gas the higher density and sub-Alfv\\'enic motions enhance the creation of cores.\n\n\\end{enumerate}\n\n\n\n\n\n\n\\begin{acknowledgements}\n\nWe thank Daniele Galli and Andy Pon for useful comments and suggestions that are greatly appreciated. P.F. is supported by the CONSOLIDER project CSD2009-00038, Spain. P.F. and J.M.G. are supported by the MINECO project AYA2011-30228-C03-02, Spain. G.A.P.F. is supported by the CNPq agency, Brazil. CRZ is supported by program CONACYT CB-2010 152160, M\\'exico.\n\n\\end{acknowledgements}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nParticles that carry a fraction of the electron charge, $Q=\\epsilon e$, also called millicharged particles (mCPs) are an elegant extension to the standard model (SM). The discovery of their simplest iteration, without the presence of a dark photon, would be a violation of the charge quantization hypothesis. Alternatively, mCPs could also arise as charges quantized under a dark force which kinematically mixes with the SM photon \\cite{Izaguirre:2015eya}. \n\nMillicharged particles have attracted significant interest and there have been many attempts to detect them.\nCollider and beam-dump experiments have been performed to search for mCPs and null results have lead to strong limits on their parameter space in the MeV-TeV mass range. These include LEP~\\cite{akers1995search}, the SLAC millicharge experiment~\\cite{Prinz:1998ua}, neutrino experiments~\\cite{Magill:2018tbb}, the Argoneut experiment~\\cite{Acciarri:2019jly}, MilliQan pathfinder experiment at the LHC~\\cite{Ball:2020dnx} and the BEBC beam dump experiment~\\cite{Marocco:2020dqu}. A plethora of experiments have been proposed to improve these tests and search for still smaller charges~\\cite{Ball:2016zrp,Berlin:2018bsc,Kelly:2018brz,Harnik:2019zee}. At lower masses, stringent limits arise from absence of anomalous emission in stellar environments~\\cite{Davidson:2000hf,Chang:2018rso} at masses below an MeV. The prospect of mCPs making up some or all of the dark matter (DM) has also received a lot of interest over the years. There are robust predictions for their relic density from the early universe~\\cite{Dvorkin:2019zdi,Creque-Sarbinowski:2019mcm}, as well as numerous ways to detect them~\\cite{Knapen:2017ekk,Blanco:2019lrf,Essig:2019kfe,Berlin:2019uco,Kurinsky:2019pgb,Barak:2020fql,Griffin:2020lgd} depending on their mass $m_Q$ and charge. Subcomponent millicharge DM (mCDM) has also been invoked to explain several recent experimental anomalies~\\cite{Barkana:2018qrx,Munoz:2018pzp,Liu:2019knx,Kurinsky:2020dpb,Bloch:2020uzh,Harnik:2020ugb}. \n\nSince mCPs interact with the SM particles through a massless mediator, their transfer cross-section can be large at small velocities. As a result, mCDM with charge large enough to scatter in the atmosphere and Earth overburden of direct detection experiments, rapidly loses its virial kinetic energy and thermalizes with the environment. When it eventually reaches a direct detection experiment, it does not possess enough energy to deposit in the detector and can not be observed at direct detection experiments~\\cite{Emken:2019tni}. However, these mCPs which are now cooled to the ambient temperature, get trapped due to Earth's gravity and build up for the duration of the Earth's existence. This can lead to mCP densities on Earth up to fourteen orders of magnitude larger than that of the virial population in the galaxy \\cite{Pospelov:2020ktu}. This slow, albeit dense population requires novel detection strategies, some of which include mCP particle-antiparticle annihilation in a large-volume detector~\\cite{Pospelov:2020ktu} and accelerating mCPs present in electrostatic accelerator tubes to higher energies, sufficient for subsequent direct detection~\\cite{Pospelov:2020ktu}. \nIf the charge is large enough, negatively charged mCPs can bind with large positively charged SM nuclei, thereby creating fractional charge for a macroscopic material. Searches for such fractional charges bound to a sample material include Millikan-like oil-drop experiments~\\cite{kim2007search}, as well as more recent levitation experiments with microspheres~\\cite{moore2014search,afek2020limits}. However, these limits hinge critically on the assumption that the negatively charged mCPs bind to matter, thereby restricting their validity to only large charges and masses. \n\nIn this work, we point out an alternate search strategy; the remarkably stable trapped ions developed for metrology and quantum information science are the ideal targets to detect an ambient thermalized mCP population. There is a long history of using trapped charged SM  particles for particle physics applications. Trapped ions have been used to measure the anomalous magnetic moment of the electron~\\cite{hanneke2011cavity}, the electron's electric dipole moment~\\cite{cairncross2017precision}, proton and antiproton magnetic moments \\cite{schneider2017double,smorra2017parts}, as well as time variation in fundamental constants which can be induced by ultralight bosonic dark matter~\\cite{Roberts_2020}. Ion traps are also the preeminent candidate for qubits to realize quantum computing \\cite{haffner2008quantum}. \n\nFor all of the applications listed above and especially for magnetic moment measurements and quantum computing, it is important to keep the ion trapped for sufficiently long duration without heating from the surroundings. In the last couple of decades, there has been remarkable progress in reduction of the measured heating rate of trapped ions \\cite{hite2012100}. Further progress is expected and is an active area of research in order to achieve scalability of multi-ion systems and increased sampling rate in precision measurements. We briefly explain why the above properties make ion traps the ideal candidate for detecting mCPs next. \n\n\n\n\\subsection{Summary of Findings}\n\\label{sec:exec}\n\nWe propose the use of ion traps as detectors for millicharged particle (mCP) dark matter.  While an individual ion constitutes a much smaller target mass than any other dark matter direct detection experiment, a trapped ion has significant advantages for detection of mCPs including isolation from the environment, a lower energy threshold for detection, and a larger scattering cross section with mCPs.  These advantages outweigh the small target mass, allowing ion traps to reach many orders of magnitude past other detection methods for mCPs. \n\nSignificant effort has been put into isolating trapped ions from their environments.\nThus trapped ions are now sensitive to small energy depositions down to $\\sim$ neV.  \nThe dense, thermal gas mCPs, if they exist on Earth, would permeate the detector and can scatter off the ion.  \nDepending on the nature of the experiment, it may be possible to measure individual scattering events or just an overall heating rate of the ion.\nThe mCPs are thermalized with the walls of the detector which is held at a temperature much higher than the temperature of the trapped ion. Thus, the higher-energy mCPs can transfer kinetic energy which can be detected either as a single jump of the trapped ion, or an accumulation of several scatters resulting in heating of the ion.  We will discuss both types of signals.\nThe high degree of isolation of the ion achievable in these traps makes them sensitive to scattering rates for mCDM over a wide range of parameter space.\n\nIon traps also make excellent mCDM detectors because of their low energy thresholds that are set by the energy-level spacings in the trap that can be as low as $\\sim \\text{neV}$.  This allows detection of single scattering events with this energy and also allows such low-energy scatters to contribute to the heating rate.\n\nFurthermore, because they are charged, ions make excellent targets for mCP scattering.  mCPs scatter with ions via Rutherford scattering which is greatly enhanced at small relative velocities and momentum transfers.  Since the mCPs are thermalized with the walls, they generally have much lower velocities than virialized dark matter. The corresponding boost to the scattering cross section combined with the large number density of the thermalized mCPs makes direct detection with single ions viable. \n\n\n\nThe rest of this paper is organized as follows.\nIn Sec.\\,\\ref{sec:traps} we provide a description of ion traps. In Sec.\\,\\ref{sec:mcpdyn}, we provide an overview of the mCP density on Earth, as well as their dynamics, including propagation through the trap. The detection signals are explained in Sec.\\,\\ref{sec:signals} and results and projections are presented in Sec.\\,\\ref{sec:results}. We conclude with a discussion in Sec.\\,\\ref{sec:discussion}. \n\n\\section{Ion traps}\n\\label{sec:traps}\nTraps for (single) charged particles belong to the basic tool-set of atomic, molecular, and optical physics \\cite{safronova2018search}. They have wide applications in determinations of atomic masses and fundamental constants,  in precision measurements to test fundamental symmetries, and in quantum information technology. \nTypical ingredients common to all trap experiments are shown in Fig$.\\,$\\ref{fig:Trap}. The core of the experiments are usually sets of electrodes supplied by AC and DC voltages, optionally placed in strong magnetic fields.  \n \\begin{figure}[h!]\n\\centering\n\n    \\centering\n    \\includegraphics[width=\\linewidth]{figs\/fig1.pdf} \n\n\n    \\caption{Elements common to typical trap experiments. A set of electrodes supplied by AD and DC voltages is mounted in a vacuum chamber, optionally in a strong magnetic field. Cryogenic traps have in addition thermal shielding and cryogenic vacuum chambers at low temperatures in their surroundings. The particles are manipulated with laser beams, microwave and radio-frequency generators. Signals are read-out by detecting image currents or fluorescent light. }\n    \\label{fig:Trap}\n\\end{figure}\nThe electrodes are mounted in vacuum chambers, in some cryogenic trap experiments pressures on the level of 10$^{-18}\\,$mbar are achieved,  which provides ultra-long particle storage times \\cite{sellner2017improved} and enables non-destructive long-term studies at low background. Thermal shielding and an additional insulation vacuum chamber usually surround the inner vacuum chamber. The trapped particles are manipulated, cooled and excited via laser, microwave and radio-frequency-drives. The experimental signals for ultra-sensitive precision studies, frequency measurements, and monitoring of quantum information-processing protocols are either image-currents picked up by sensitive detection circuits, or fluorescence signals. The signals are acquired and processed with spectrum analyzers and charge coupled device (CCD) cameras, respectively.\n\n\\begin{table*}\n\\begin{center}\n\\begin{tabular}{ ||c|c|c|c|c|c|c|c| }\n\\hline\n Experiment & Type & Ion & $V_z$  &  $T_{\\rm wall}$ & $\\omega_p$ [neV] & $T_{\\rm ion}$[neV] & Heating Rate (neV\/s) \\\\ \n \\hline \n Hite et al, 2012 \\cite{hite2012100}& Paul &$^9\\text{Be}^+$ & 0.1 V&300 K&$\\omega_z=14.8$ &$14.8$&640 \\\\\n Goodwin et al, 2016\\,\\cite{goodwin2016resolved} &Penning&$^{40}\\text{Ca}^+$ & 175\\,V  &300\\,K& $\\omega_z=1.24$ & $1.24$ & 0.37 \\\\\n Borchert et al, 2019\\,\\cite{borchert2019measurement}&Penning&$\\bar{p}$& 0.633\\,V& 5.6\\,K & $\\omega_+=77.4$&7240 & 0.13 \\\\\n &&& & & $\\omega_-=0.050$& &  \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\caption{List of ion traps and the relevant experimental parameters used for setting limits in this paper. The ion used, $V_z$,  potential barrier in the axial direction and $T_{\\rm wall}$, the temperature of the walls of the trap and $\\omega_p$, the fundamental frequency of the trap in the relevant direction are listed. Also listed are $T_{\\rm ion}$, the temperature of the ion in the trap and the measured heating rate. }\n\\label{table:data}\n\\end{table*}\n\n\n\\subsection{Penning Traps}\nIn a Penning trap, there is a strong magnetic field $B_0$ superimposed with an electrostatic quadrupolar potential $\\Phi(z,\\rho)$, which is attractive along the magnetic field axis. The motion of a charged particle in such crossed static fields is composed of harmonic oscillator modes of three independent types. The modified cyclotron and the magnetron modes correspond to oscillations perpendicular to the axis, while the axial mode corresponds to oscillations along the magnetic field lines. Associated with the mode oscillations are the three trap eigenfrequencies $\\nu_+$, $\\nu_-$, and $\\nu_z$, respectively. Room-temperature traps such as that in Ref.\\,\\cite{goodwin2016resolved}, cool and trap ions such as $^{40}\\textrm{Ca}^+$ or $^{9}\\textrm{Be}^+$ to the axial ground state using optical sideband cooling~\\cite{mavadia2014optical}. A delay period after cooling is followed by subsequent spectroscopy which determines the final state of the ion, thus measuring the heating rate. The lowest heating rate achieved thus far at room temperature is reported in Ref.\\,\\cite{goodwin2016resolved} where the increase in the number of phonons of the axial mode was reported to be $\\dot{n}=0.3\/\\textrm{s}$ with the axial frequency $\\nu_z=0.3\\,\\textrm{MHz}$, see  Tab.\\,~\\ref{table:data}. \\\\\nParticularly interesting for the detection of mode energy changes are experiments dedicated to direct measurements of nuclear magnetic moments such as those of the proton \\cite{schneider2017300}, the antiproton \\cite{smorra2017parts}, or $^3$He$^{2+}$ \\cite{mooser2018new}. These experiments operate advanced cryogenic Penning-trap systems consisting of multi-trap assemblies, common to all of them is a so-called analysis trap with a strong superimposed magnetic inhomogeneity $B(z)=B_0+B_2 z^2$, where $B_2$ characterizes its strength \\cite{ulmer2011observation}. \nThe interaction of the magnetic bottle with the particle's magnetic moment $\\mu_z=\\mu_++\\mu_-+\\mu_s$ results in a magnetostatic axial energy $E_{B,z}=-\\mu_z B_z$, where $\\mu_+$ and $\\mu_-$ are the orbital angular magnetic moments associated with the modified cyclotron and the magnetron mode, while $\\mu_s$ is the spin magnetic moment. \nAs a result, the particle's axial frequency $\\nu_z=\\nu_{z,0}+\\Delta\\nu_z (n_+,n_-,m_s)$ becomes a function of the radial trap eigenstates $n_+$ and $n_-$, as well as the spin eigenstate $m_s$ with\n\\begin{align}\n\\Delta\\nu_z=\\frac{h\\nu_+}{4\\pi^2 m \\nu_z}\\frac{B_2}{B_0} \\left(\\left(n_++\\frac{1}{2}\\right)+\\frac{\\nu_-}{\\nu_+}  \\left(n_-+\\frac{1}{2}\\right)+\\frac{g}{2} m_s\\right).\n\\end{align}\nMeasurements of single-particle magnetic moments in Penning traps rely on the detection of axial frequency shifts $\\Delta\\nu_{z,\\text{SF}}$  induced by driven spin quantum transitions $\\Delta m_s=1$. Since nuclear magnetic moments are about three orders of magnitude smaller than the Bohr magneton, these experiments require highest sensitivity with respect to magnetic moments, which is usually achieved by the utilization of magnetic bottle strengths of the order $B_2\\approx100\\,$kT\/m$^2$ to $B_2\\approx400\\,$kT\/m$^2$. Combined with continuous measurements of the axial frequency, such strong magnetic bottles provide excellent resolution of the radial mode energies with \n\\begin{eqnarray}\n\\frac{(\\Delta\\nu_z)}{\\Delta E_\\rho}=\\frac{1}{4\\pi^2m \\nu_z}\\frac{B_2}{B_0}\\approx 1 \\frac{\\text{Hz}}{\\mu \\textrm{eV}}, \n\\end{eqnarray}\nwhile in axial frequency measurements resolutions of order  200$\\,$mHz$\\cdot\\sqrt{\\text{s}}\/\\sqrt{t_\\text{avg}}$ are achieved, $t_\\text{avg}$ being the averaging time which is typically on the order of several tens of seconds. Transition rates in the radial modes\n$(dn_{+,-})\/dt\\propto(n_{+,-}\/\\omega_{+,-} ) S_E (\\omega_{+,-})$ lead to random walks in radial energy space and to axial frequency diffusion. Here  $S_E (\\omega_{+,-})$ is the power spectral density of a noisy background drive, and $n_{+,-}$ is the principal quantum number of the modified cyclotron ($n_+$) \/ magnetron ($n_-$) oscillator. The scaling of the heating rate with $n_{+,-}$ is related to eigenstate-overlap of harmonic oscillator states \\cite{ulmer2011observation}. \\\\ \nBy analyzing time sequences of axial frequency measurements $\\nu_z (t)$, the average radial quantum transition rates are obtained. With a highly optimized trap setup with which the antiproton magnetic moment was measured with 1.5 parts per billion precision, the BASE experiment at CERN reports on the observation of absolute cyclotron transition rates of 6(1) quanta per hour~\\cite{borchert2019measurement}. Together with the determination of the $n_+$ state during the recorded measurement, this result is  consistent with a projected ground state heating rate of 0.1 cyclotron quantum transitions per hour, setting an upper limit which is by a factor of 1800 lower than the best reported Paul-trap heating rates, and by a factor of 230 lower than the best room-temperature Penning trap. These numbers are summarized in Tab.\\,\\ref{table:data}. Note that the antiproton experiments are conducted in a background vacuum of $\\approx 10^{-18}\\,$mbar \\cite{sellner2017improved}, constraining parasitic heating induced by collisions with  background-gas to a level of $4\\times10^{-9}$\/s. \n\n\n\\subsection{Paul Traps}\nPaul traps or radiofrequency traps utilize an oscillating voltage to confine in the perpendicular direction instead of the magnetic field used for the same purpose in the Penning trap. Paul traps have a rich history of being used as mass spectrometers and more recently in building quantum computers~\\cite{brownnutt2015ion}. \n\nThe effective potential in the presence of both DC and AC potentials can be written as\n\\begin{equation}\n    \\psi(x,y,z,t)=\\left(U_{\\rm DC}+V_{\\rm AC} \\cos \\Omega t\\right)\\frac{r^2+2 (z_0^2-z^2)}{r_0^2+2z_0^2} .\n\\end{equation}\nHere $z$ is the axial direction, $r$ the radial direction with the distance to the electrodes given by $r_0$ and $z_0$. The rapidly oscillating potential creates a pseudopotential for charges of both signs and this leads to approximately simple harmonic motion very close to the trap center.  After laser cooling to the ground state, the total heating rate can be measured via Raman sideband technique~\\cite{turchette2000heating}. There has been extensive study of the heating rate in Paul traps and its dependence on distance to electrodes, wall temperature, trap temperature~\\cite{brownnutt2015ion} as well as ion beam treatment of electrodes~\\cite{hite2012100}. Electric field noise from the electrodes has been identified as the dominant heating source, with the dependence on distance scaling as $d^{-2}$~\\cite{brownnutt2015ion} to $d^{-4}$~\\cite{daniilidis2011fabrication}. Although heating rates are lower for bigger traps~\\cite{poulsen2012efficient}, smaller sized traps employ shallower potential wells $\\approx0.1~\\textrm{V}$. As we shall see, this allows mCPs with large charge to reach the trap and hence provide complementary reach at large charge parameter space. Hence we reinterpret limits only from a microtrap~\\cite{hite2012100}.  The heating rates reported in ~\\cite{hite2012100} are $\\dot{n}=43\/\\textrm{s}$ for the axial frequency $\\nu_z=3.6~\\textrm{MHz}$. These numbers are tabulated in Tab.\\,~\\ref{table:data}.\n\n\n\n\\section{Millicharged Particle Dynamics}\n\\label{sec:mcpdyn}\n\\subsection{Terrestrial Accumulation} \nIf mCDM exists and is virialized in the galaxy, there is a non-zero flux of mCPs flowing through the Earth at all times.  This mCDM stops in the atmosphere or rock overburden for large enough charge, and can accumulate on Earth. This process was treated in detail in~\\cite{Emken:2019tni} and the subsequent accumulation in~\\cite{Pospelov:2020ktu}. We provide here a summary of the relevant results of these papers that are used in Section.\\,\\ref{sec:results} and refer the reader to ~\\cite{Pospelov:2020ktu} for details.\n\nFollowing~\\cite{Pospelov:2020ktu} we consider effectively millicharge particles mediated by a dark photon which kinematically mixes with the SM photon. The dark photon mass is taken to be large enough ($m_{A'} \\gtrsim 10^{-12}~\\textrm{eV}$) such that the effect of large-scale electric and magnetic fields can be ignored while considering mCP propagation\\footnote{We leave to upcoming work the calculation of the accumulation of millicharged dark matter in cases where the electric and magnetic fields are relevant \\cite{blprfuture}. Once such calculations are complete, our given limits on the number density in the lab can be translated to limits on dark matter fraction in theses cases.}.  In this limit, the mCPs with mass $m_Q\\ge 1$\\,GeV are stuck on the Earth after thermalization.\\footnote{Millicharged particles with masses below 1\\,GeV can accumulate on Earth temporarily before evaporating. We leave limits on these masses for future work.} The volume averaged DM number density on Earth, $\\langle n_Q \\rangle$ can be several orders of magnitude larger than the virial DM density. It is given by, \n\\begin{align}\n\\label{start}\n\\langle n_Q \\rangle&= \\frac{\\pi R_{\\oplus}^2 v_{\\rm vir}  t_{\\oplus}}{4\/3 \\pi R_{\\oplus}^3}n_{\\rm vir} \\nonumber \\\\\n&\\approx \\frac{3\\times 10^{15} }{\\textrm{cm}^3} \\frac{t_{\\oplus}}{10^{10}\\,\\textrm{y}} f_Q \\frac{\\rm GeV}{m_Q},\n\\end{align}\nwhere, $R_{\\oplus}$ and $t_{\\oplus}$ are the radius and age of the Earth and $n_{\\rm vir}$ and $v_{\\rm vir}$ are the galactic virial number density and velocity. However, the equilibrium density profile is peaked at the Earth core. This density $n_{\\rm static}$ was calculated by taking into account the Earth's temperature and density variations in \\cite{Neufeld:2018slx}. However, the sinking to the Earth's core is not immediate, and there is a dynamic population, the so-called ``traffic-jam\" density, $n_{\\rm tj}$, given by,\n\\begin{equation}\n    n_{\\rm tj}=n_{\\rm vir}\\frac{v_{\\rm vir}}{v_{\\rm term}} .\n\\end{equation}\nHere, $v_{\\rm term}$ is the terminal velocity in rock, given in \\cite{Pospelov:2020ktu}. Finally, the density in the laboratory, $n_{\\rm lab}$ is given by, \n\\begin{equation}\nn_{\\rm lab} = \\textrm{Max}\\left( n_{\\rm static}, \\textrm{Min}\\left( n_{\\rm tj} , \\langle n_Q \\rangle \\right)\\right)\n\\label{eq:regimes}\n\\end{equation}\n\nThese results are applicable only for $\\epsilon$ large enough, such that it stops in the corresponding overburden. According to \\cite{Pospelov:2020ktu} this is valid only for\n\\begin{align}\n\\epsilon&\\gtrsim 2\\times10^{-4} \\sqrt{\\frac{m_Q}{\\rm GeV}} \\quad &&\\textrm{surface,} \\nonumber \\\\\n&\\gtrsim 3\\times 10^{-6}\\sqrt{\\frac{m_Q}{\\rm GeV}} \\quad &&\\textrm{1 km mine.} \n\\label{eq:applicable}\n\\end{align}\n\nIt is also important to comment on the asymmetry of the mCP population. If the mCDM is a symmetric population with equal number of particles and antiparticles, accumulation on Earth can result in Sommerfeld-enhanced annihilations which prevent build up. We consider here the asymmetric case such that opposite charges are carried by different species just like the the SM proton and electron, such that annihilations are absent. In this scenario, for large enough $\\epsilon$, the negatively charged mCPs can form deep bound states with the large positively nuclei, whereas the positive mCPs can only bind with the less massive electrons which also have smaller charge compared to the heavy SM nuclei. In ref.~\\cite{Pospelov:2020ktu} it was pointed out that positive mCPs with $\\epsilon \\ge 0.042$  bind to electrons at 300 K (room temperature). We find that such bound states are temporary with electrons rapidly preferring to bind with ions which possess larger charge than the positive mCPs. Thus the terrestrial population of positive mCPs remains free of binding for $\\epsilon\\lesssim 1$ and is present as a locally thermalized population that is diffusing everywhere. Thus, the results we derive apply to all positive charges with $\\epsilon \\lesssim 1$ and negative charges which do not bind with nuclei, i.e \\cite{Pospelov:2020ktu} $\\epsilon < \\frac{m_e}{\\mu_{Q,N}}$ where $\\mu_{Q,N}$ is the reduced mass of the mCDM-nuclear system.  \n\n\nIt is important to emphasize that the limits put on the ambient number density are applicable to mCPs charged directly under the SM photon as well as to ones mediated by a dark photon as long as the dark photon mass is below the relevant momentum transfer for scattering with ions which is around 1 eV.   \n\n\\subsection{Passage Through Apparatus}\n\n\\label{trappassage}\nWe next turn to the trajectory of mCPs through the trap peripherals in order to reach the trapped ion.  We need to know the density and temperature of the mCPs reaching the ion.  In this subsection we explain the main factors entering the calculation, but as the calculation itself is somewhat involved we leave the details to Appendix \\ref{Appendix:passage}.  Our main result is in Eq.\\,\\eqref{eqn:qion} relating the number density of mCPs at the position of the ion, $n^Q_{\\rm ion}$, to the ambient number density on the Earth $n_{\\rm lab}$.  \nThis is the equation we use to set our limits on mCPs.\n\nAt equilibrium, mCPs are expected to have roughly uniform density near the Earth's surface, including permeating all materials.  However the conditions of the experiment can affect this naive expectation for several reasons.\n\nFirst, some of the ion traps we consider are cryogenic.  Over essentially all of our parameter space the mCPs have a short interaction length in material and so will rapidly thermalize to the cryostat temperature as they enter the experiment.  By itself this would lead to an increase in mCP density by a factor linear in mCP velocity ($\\propto \\sqrt{T}$) because the fluxes entering and leaving the cryostat must be equal in equilibrium.\n\nSecond, the ion traps are surrounded by metal which has a work function that can affect the passage of mCPs.  This is only relevant for mCPs of relatively large charge ($\\epsilon \\gtrsim 10^{-2}$) but for those it can be a significant effect.  The work function for mCPs, which we will call $\\epsilon \\phi$, is not simply $\\epsilon$ times the work function for electrons.  For electrons the work function arises from several contributions of varying signs including e.g.~the binding to the lattice of nuclei, the Fermi sea of other electrons, and surface effects such as the ``double layer\" or the image charge potential.  Several of these do not apply or are negligible for mCPs.  Recall we are only considering positively charged mCPs since the negative ones may be stuck deeply bound to some nucleus somewhere on the Earth.  Thus the main effects are repulsion by the double layer and possibly also repulsion from the nuclei.   We consider the work function for mCPs in more detail in  Appendix \\ref{Appendix:work function}.  Our conclusion is that the work function is repulsive for positively charged mCPs and thus every metal sheet provides a barrier for mCPs to cross. \nThe size of the potential barrier that has to be crossed is $\\phi \\sim \\text{few} \\, \\text{eV}$.\nNote that for an experiment at room temperature ($T \\sim 0.03 \\, \\text{eV}$) the metal barriers are then irrelevant for charges $\\epsilon \\lesssim 10^{-1}$ because the Boltzmann tail easily pushes a fast enough rate of mCPs over the barrier.  For a cryogenic experiment at $T \\sim 6 \\, \\text{K} \\sim 5 \\times 10^{-4} \\, \\text{eV}$, the metal barrier will be relevant for charges $\\epsilon \\gtrsim 10^{-3}$ and essentially insurmountable for charges $\\epsilon \\gtrsim 10^{-2}$.  As we show in Appendix \\ref{Appendix:passage}, the most important effect comes from an experiment encased in two different metals where the work function for mCPs rises from the outer metal to the inner metal.  We will take this difference to be $\\Delta \\phi = 3 \\, \\text{eV}$ for all experiments we consider since this will be a conservative estimate as we show in Appendix \\ref{Appendix:work function}.\n\n\nThird, the ions are always in a region of ultrahigh vacuum.  This means that pumps were used to remove the Standard Model (SM) particles.  Given a short interaction length of the mCP in materials, these pumps could remove the mCPs from the ion chamber as well.  In one of the experiments we consider (Goodwin et al.~\\cite{goodwin2016resolved}) this effect is not relevant because the trap is at room temperature and the region of sensitivity is at low enough $\\epsilon$ that the millicharges pass easily through the walls\\footnote{The sensitivity is limited to be below $10^{-3}$ charge because of the large trap potential as will be discussed in Section \\ref{sec:results}.}.  In the other two experiments we consider (Hite et al.~\\cite{hite2012100} and Borchert et al.~\\cite{borchert2019measurement}), the vacuum pumps are turned off well before the actual data taking is begun.  In the case of Borchert this is at least a year, while for Hite we conservatively assume it is only a day\\footnote{The final answer is only logarithmically sensitive to this timescale anyway.}.  And of course mCPs are always continually flowing in from the walls of the vacuum chamber.  This would rapidly refill the trap region and this effect would not be relevant, except for the largest charges where the refill can be slow because of the work function of the surrounding metal.  This does mean that, depending on the parameters of the mCP and of the ion trap, the number density in the trap may be either in the equilibrium regime or in the filling regime.  This is why Eq.\\,\\eqref{eqn:final trap density} has two different regimes.  Eq.\\,\\eqref{eqn:final trap density} relates the number density of mCPs inside the trap $n_{\\rm trap}$ to the ambient number density on the Earth $n_{\\rm lab}$.  There is then one remaining step to find the number density of mCPs at the position of the ion.\n\nFourth, the ion trap itself has applied electromagnetic fields to trap the ion.  These can affect the passage of the mCPs, though again this is only relevant for larger charges $\\epsilon \\gtrsim 10^{-3}$.\nTab.\\,\\ref{table:data} lists parameters of the various experiments we consider.  All the experiments use an electric DC potential to confine the ion in the axial direction.  The height of the potential barrier along the axial direction is listed as $V_z$.  The mCPs in the trap are thermalized to the temperature of the walls of the trap $T_{\\rm wall}$.  Starting from the number density of mCPs calculated inside the trap $n_{\\rm trap}$ in Eq.\\,\\eqref{eqn:final trap density}, we take a Boltzmann suppression on the number density which can make it up the axial barrier height.  Thus we take the final number density at the position of the ion to be given by Eq.\\,\\eqref{eqn:qion}.  For the experiments we consider this Boltzmann suppression is relevant for setting the ceiling (largest $\\epsilon$ values) of the Goodwin regions in Figure \\ref{fig:evsm}, but is irrelevant for the other regions. \nThe Penning traps use a magnetic field to confine the ions in the radial direction.  This B-field will cause mCPs with a large enough charge to circle around the axial B-field lines but they are still free to move along the axis.  The magnetic field does not change the phase-space density of the mCPs and so it does not significantly affect our signal.  The Paul traps use an RF potential to confine the ion in the radial direction.  This potential around the minimum is locally attractive for the positive mCP independent of the sign of the charge of the trapped ion.  An mCP coming from far outside the RF fields in a radial direction could give a barrier in principle.  However for mCPs approaching along the axial direction, the RF pseudopotential in the axial direction is very weak and negligible.  And then  such mCPs will actually be concentrated in the radial direction towards the ion at the center of the trap since the potential is locally attractive.  We conservatively ignore this possible (Sommerfeld-like) enhancement though it could be large. For large enough mCP masses, $m_Q \\gtrsim 10^{10} \\textrm{GeV}$, the free-fall under gravity \ncan generate velocities much larger than the thermal velocities assumed. This can increase the heating rate further. We conservatively ignore this effect and leave its consideration for future work.\n\n\\section{Observables for Millicharged Particles in Ion Traps}\n\\label{sec:signals}\nIn this section we consider the interaction of ambient mCPs with ion traps to identify observables for detection. As seen in Section.~\\ref{trappassage}, the mCPs enter the ion trap with effective temperature $T_Q \\ge T_{\\rm wall}$, where $T_{\\rm wall}$ is the wall temperature. $T_{\\rm wall}= 5.6$ K for the cryogenic trap we consider~\\cite{borchert2019measurement} and $T_{\\rm wall}=T_{\\rm room}\\approx 300$ K for room temperature traps. The ions in the trap are at a much colder temperature $T_{\\rm ion} \\ll T_{\\rm wall}$. With this hierarchy of temperatures, the mCPs can cause two types of signal.\n\nThe first signal involves individual scattering events that impart energy $E_{\\rm ion}$ to the ion thus leading to a change in its harmonic oscillator quantum number. This signal is very similar to dark matter scattering in a conventional dark matter detector. The rate of events has to be slower than the rate at which the ions are interrogated. $E_{\\rm ion}$ also needs to exceed the energy resolution $E_{\\rm res}$ for detection. \n\nThe second type of signal is the heating of the trapped ion due to collisions with multiple mCPs. In this case, the individual hits $E_{\\rm ion}$ can be smaller than $E_{\\rm res}$ and only the sum needs to exceed this resolution. \n\nFor both of these signal types, we only consider individual energy transfers $E_{\\rm ion}$ much larger than the typical energy spacing of the trap $\\omega$, for concreteness $E_{\\rm ion} \\ge 10\\omega$. \\footnote{In principle, energy transfers $E_{\\rm ion} =n\\times \\omega$ where all integers $n\\ge 1$ are allowed. We restrict $E_{\\rm ion} \\ge 10 \\times \\omega$, a conservative choice that helps avoid form-factor calculations.} In this limit, the trapped ion can be approximated as a free particle with initial energy $T_{\\rm ion}$ and final energy $T_{\\rm ion}+E_{\\rm ion}$. Equivalently, for energy transfers much larger than the spacing, the form-factor that incorporates wave-function overlap can be approximated to unity.  \nWe start quantifying both of these signals by the angular differential cross-section given by the Rutherford formula,\n\\begin{equation}\n    \\frac{d\\sigma}{d\\Omega}=\\frac{2\\pi\\alpha^2 \\epsilon^2}{\\mu^2 v_{\\rm rel}^4\\left(1-\\cos\\theta\\right)^2} ,\n\\end{equation}\nwhere $\\theta$ is the scattering angle. We next introduce kinematic variables that  simplify the computation of the scattering rate. The results are presented here, with the details of the derivation presented in Appendix.~\\ref{app1}. The incoming mCP and trapped ion velocities are assumed to be $\\mathbf{v_Q}$ and $\\mathbf{v_{\\rm ion}}$ respectively. \nThe center of mass (CM) velocity $\\mathbf{v_{CM}}$ is given by,\n\\begin{equation}\n    \\mathbf{v_{CM}}=\\frac{\\left(m_{\\rm ion} \\mathbf{v_{\\rm ion}}+m_Q \\mathbf{v_Q}\\right)}{m_{\\rm ion}+m_Q} .\n\\end{equation}\n\nThe change in velocity of the ion, $\\mathbf{\\Delta v_{\\rm ion}}$ is given by,\n\\begin{align}\n    \\mathbf{\\Delta v_{\\rm ion}}=\\frac{m_Q}{m_{\\rm ion}+m_Q}\n    \\left[\\left(\\cos \\theta-1 \\right)\n    \\left(\\mathbf{v_{\\rm ion}}-\\mathbf{v_Q}\\right)\n    \\right. \n    \\nonumber \\\\ \n    \\left.\n    +\\sin \\theta |\\mathbf{v_{\\rm ion}}-\\mathbf{v_Q}|\n    \\mathbf{n_{\\perp}}\n    \\right] .\n\\end{align}\nThe transferred energy $E_{\\rm ion}$ is given by,\n\\begin{equation}\n    E_{\\rm ion} = m_{\\rm ion} \\mathbf{v_{CM}}.\\mathbf{\\Delta v_{\\rm ion}} .\n\\end{equation}\n\nGiven a threshold $E_{\\rm thr}$, the single event rate $R_{\\rm single}$ with energy transfer $E_{\\rm ion}$ above this threshold is,\n\\begin{align}\n    R_{\\rm single}\\left(E_{\\rm ion}\\ge E_{\\rm thr}\\right)=n^Q_{\\rm ion} \\int d^3 \\mathbf{v_Q} g_Q  \\int d^3 \\mathbf{v_{\\rm ion}} g_{\\rm ion}  \\int d \\Omega  \\nonumber \\\\ |v_Q-v_{\\rm ion}| \\frac{d\\sigma} {d \\Omega} \\Theta \\left(|E_{\\rm ion}|-E_{\\rm thr}\\right) . \n\\end{align}\nHere $n^Q_{\\rm ion} $ is the number density of mCPs at the ion position and is given by Eq.\\,\\eqref{eqn:qion}, $g_{\\textrm{ion}(Q)}$ is the Maxwell--Boltzmann distribution for the ion\/mCP with temperatures $T_{\\rm ion}$ and $T_{\\rm wall}$, respectively. \n\n\nThe heating rate per ion, $\\dot{H}$ can be computed through,\n\\begin{align}\n    \\dot{H}&=n^Q_{\\rm ion}  \\int d^3 \\mathbf{v_Q} g_Q  \\int d^3 \\mathbf{v_{\\rm ion}} g_{\\rm ion}  \\int d \\Omega |v_Q-v_{\\rm ion}|  \\nonumber \\\\ &\\frac{d\\sigma} {d \\Omega}E_{\\rm ion} \\Theta \\left(|E_{\\rm ion}|-E_{\\rm thr}\\right) \\Theta\\left(E_{\\rm samp}-|E_{\\rm ion}|\\right) . \n\\end{align}\nThe Heaviside theta function ensures the inequality $E_{\\rm thr}\\le E_{\\rm ion} \\le E_{\\rm samp}$. Here $E_{\\rm samp}$ is defined so as to prevent the average heating rate from including contribution from extremely rare events. It is defined through,\n\\begin{equation}\nR_{\\rm single}\\left(E_{\\rm ion}\\ge E_{\\rm samp}\\right) t_{\\rm obs}=1 ,\n\\end{equation}\nwhere $t_{\\rm obs}$ is the total observation time. \n\n\n \\section{Results}\n \\label{sec:results}\n \n\n\\begin{figure*}\n\\centering\n\\begin{subfigure}{0.48\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figs\/comparetraps1e3.pdf} \n\n\\end{subfigure}\n\\begin{subfigure}{0.48\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figs\/evsmmainall.pdf} \n \n\\end{subfigure}\n    \\caption{Compilation of new limits using existing heating measurements from various traps in Table \\ref{table:data}: a room temperature Paul trap, Hite et al.~\\cite{hite2012100}, a room temperature Penning trap, Goodwin et al.~\\cite{goodwin2016resolved} and a cryogenic penning trap, Borchert et al. \\cite{borchert2019measurement}. {\\bf (left)} Comparison between traps for an ambient density $n_{\\rm lab}=10^3 \\textrm{cm}^{-3}$; {\\bf (right)} Combined limits from the three traps for different  $n_{\\rm lab}$} \n    \\label{fig:evsm}\n\\end{figure*}\n In this section, we set constraints on mCPs and make projections for the future by using the expressions for the signal rates derived in Sec.\\,\\ref{sec:signals}. \n \n \\subsection{Limits from existing measurements}\n To obtain existing limits, we use the data presented in Tab.\\,\\ref{table:data}. All of the trap parameters for the $^9\\text{Be}^+$  trap are taken from \\cite{hite2012100} while the trap depth is conservatively taken to be $V_z=0.1$~V, an order of magnitude larger than the typical potential depths in microtraps\\footnote{see for e.g.~\\cite{warring2013techniques} from the same group where the potential depth is report to be $V_z$=5 mV}. While the rest of the parameters for the $^{40}\\textrm{Ca}^+$ experiment are provided in ~\\cite{goodwin2016resolved}, we obtained the potential depth $V_z=175$~V from the authors. Finally, for ref.\\,\\cite{borchert2019measurement}, we use the parameters $T_{\\rm wall}=5.6$~K and $V_z$=0.6~V. While the analysis in ~\\cite{borchert2019measurement} dealt with measuring the cyclotron mode ($\\omega_+)$, the observable heating rate is equivalently a limit on the magnetron mode ($\\omega_-$) also. While individual jumps in $\\omega_-$ are unobservable with existing precision, the frequency shift due to the heating rate accumulates. Since the Rutherford cross-section increases for smaller energy transfers, we use $\\omega_-$ to convert existing heating limits into limits on mCPs. \n\n We start by plotting existing limits in Fig.\\ref{fig:evsm} in the millicharge $\\epsilon$ vs mass $m_Q$ parameter space for contours of constant ambient density $n_{\\rm lab}$. As mentioned earlier, for mCDM, the lab density is expected to be orders of magnitude larger than the virial density, i.e. $n_{\\rm lab} \\gg n_{\\rm vir}$. In the left panel, limits arising from the antiproton trap \\cite{borchert2019measurement} in blue, $^{40}\\textrm{Ca}^+$~\\cite{goodwin2016resolved} in red and $^{9}\\textrm{Be}$ in green \\cite{hite2012100} for an ambient density of $n_{\\rm lab}=10^3\/\\textrm{cm}^3$. The dominant limits arise from the antiproton trap owing to its superior heating rate as seen in Tab.\\,\\ref{table:data}. However, since it is a cryogenic trap, the limits disappear at $\\epsilon \\approx 10^{-2}$ owing to the suppression arising from mCPs finding it increasingly difficult to penetrate metals due to their work function. As explained in Section.~\\ref{sec:mcpdyn} and Appendix.~\\ref{Appendix:passage}, this factor roughly scales as $e^{-\\frac{\\epsilon \\Delta \\phi}{T_{\\rm wall}}}$ where $\\Delta \\phi$ is the difference in work functions between two adjacent metals. Hence this suppression is ameliorated for room temperature traps like the $^9\\textrm{Be}^+$ trap \\cite{hite2012100} in green which extends to $\\epsilon\\approx 0.1$. While the $^{40}\\textrm{Ca}^+$ trap in red is also at room temperature, the trap is at a potential of 175~V and thus mCP charges above $\\epsilon \\approx 10^{-3}$ do not reach the ion. However, there is reach to higher mCP masses for this trap as a result of the ion being more massive. The right panel of Fig.~\\ref{fig:evsm}\n corresponds to combined limits from these three experiments for different $n_{\\rm lab}$.  \n \n \\begin{figure}\n\\centering\n\n    \\includegraphics[width=\\linewidth]{figs\/compwithexisting.pdf} \n\n    \\caption\n    Comparison of limits derived in this work from ~\\cite{hite2012100,goodwin2016resolved,borchert2019measurement} with existing limits from Oil Drop~\\cite{kim2007search}, levitation experiments~\\cite{moore2014search,afek2020limits} and LEP~\\cite{Marocco:2020dqu} for $m_Q=10$ GeV.}\n    \\label{fig:compexist}\n\\end{figure}\n \n \n \n As is clear from both figures, existing data for anomalous heating in traps sets exquisite bounds on mCPs thermalized locally. Bounds are applicable to orders of magnitude in the mCP mass $m_Q$ as well as many orders of magnitude in charge. Number densities as small as $n_{\\rm lab}=1\\, \\textrm{cm}^{-3}$ are ruled out around the $\\epsilon\\approx 10^{-3}$ and $m_Q\\approx 10\\, \\textrm{GeV}$ parameter point. \n \nIn order to compare these limits on the ambient mCP population to ones that already exist in literature, we fix the mCP mass $m_Q=10$\\,GeV and show limits in the $n_{\\rm lab}$ vs $\\epsilon $ plane in Fig.~\\ref{fig:compexist}. The same color coding as Fig.~\\ref{fig:evsm} is followed. In gray we show limits from LEP~\\cite{Marocco:2020dqu}, as well as limits on mCPs bound in matter arising from Oil drop experiments~\\cite{kim2007search}, and levitation experiments~\\cite{moore2014search,afek2020limits}. As noted earlier, the limits on mCPs bound in matter are applicable only to negative mCPs with large enough charge such that binding with SM nuclei is possible. Furthermore, if mCP-SM bound states exist, there is no guarantee for these bound states to be evenly distributed all over the Earth. However, for the positive mCPs none of these caveats apply and they thermalize and distribute themselves over the entire Earth volume. Regardless, as seen in Fig.~\\ref{fig:compexist} the limits obtained from ion traps are orders of magnitude stronger than the levitation experiments. For $\\epsilon\\approx 3\\times 10^{-3}$, lab densities as small as $n_{\\rm lab} \\gtrsim 1 \\textrm{cm}^{-3}$ are ruled out by the measured heating rate at the antiproton experiment \\cite{borchert2019measurement}. \n \\begin{figure}\n\\centering\n\n    \\centering\n    \\includegraphics[width=\\linewidth]{figs\/dmplot.pdf} \n  \n\n    \\caption{Limits and projections on the virial DM fraction in mCPs as a function of $\\epsilon$ and $m_Q$. Since the current experiments were all conducted at the surface, the robust limits are only above the ``$>$1m Thermalization\" line where the colored contours are solid. For a hypothetical deep mine experiment, the dashed part of the contours are also accessible.}\n    \\label{fig:virial}\n\\end{figure}\n\nNext, in Fig.~\\ref{fig:virial} we convert limits on $n_{\\rm lab}$ into limits on the fraction of virial DM existing in mCPs, $f_Q= \\frac{\\rho_Q}{\\rho_{\\rm DM}}$. For this purpose we use $n_{\\rm lab}$ from Eq.\\,\\ref{eq:regimes}. Existing limits from colliders are shown in gray. The solid parts of the colored contours correspond to the region where the incoming virial DM gets thermalized within 1 meter and hence the robust current limits we put are restricted to this region i.e. above the top black line. The dashed lines show the reach for an identical heating rate experiment that is conducted in a deep mine at 1km depth. Virial DM fractions as small as $f_Q\\approx 10^{-12}$ are already ruled using existing heating data for DM masses in the 1-10 GeV mass range. For heavier masses, the terminal velocity is larger and hence the traffic jam densities are smaller. Nonetheless we set limits on DM fractions as small as $f_Q=10^{-3}$ for masses as large as $m_Q\\approx 10^5$~GeV.\n\nThe limits presented above were derived using data from existing experiments that measure the anomalous heating rate. We next make projections for the future to capture improvements in reducing the heating rate as well as to incorporate mCP detection specific modifications. \n\n\\subsection{Projections}\n\\begin{figure}\n\\centering\n    \\includegraphics[width=\\linewidth]{figs\/future1e3.pdf} \n    \n    \\caption{Event Rate limits and projections for $n_{\\rm lab}=10^3 \\textrm{cm}^{-3}$. Existing limits from heating of the $\\omega^-$ mode are shown (brown shaded) from Fig \\ref{fig:evsm}.  The projection for a search for single events in the BASE experiment \\cite{borchert2019measurement} with energy deposit above $10\\omega^+$ and rate 0.1 event\/hr are shown (dark blue). Here $\\omega_+=77.4$ neV and $\\omega_-= 0.050$ neV.  Next, projections are also shown assuming sensitivity to 1 Event\/year event rates. In pink, we show sensitivity from the existing setup of Ref.~\\cite{borchert2019measurement}. The light blue curve corresponds to swapping a single (anti)-proton with fully ionized Calcium or equivalently trapping 400 Calcium ions in a Coulomb crystal. We also show the reach for a futuristic experiment with energy thresholds of $E_{\\rm min}=10\\omega_{-}$ in green. Finally, reach from a hypothetical electron trap with trap frequency $\\omega=200$~neV and $E_{\\rm min}=2~\\mu$eV is shown in orange.\n    }\n    \\label{fig:futureproj}\n\\end{figure}\n\n\nLimits shown thus far arise from the cumulative heating rate measured. Another promising avenue, is the (non)-observation of individual event rates. In Fig.~\\ref{fig:futureproj}, we compare projections from a non-observation of single event with $E_{\\rm ion}\\ge 10\\omega_+$ with the same parameters as existing data in Ref.~\\cite{borchert2019measurement} in dark blue with the heating of the much smaller $\\omega_-$  in brown for $n_{\\rm lab}=10^3~\\textrm{cm}^{-3}$. \n\nWe find that this projection is near-identical to the heating limit in brown. Both the heating limits as well as the event rate sensitivity are expected to improve in the future. For e.g. heating rates are known to decrease with larger electrode distance or increasing the frequency of the trapped ion~\\cite{brownnutt2015ion}. Whereas the mCP search with the heating rate in the current BASE apparatus is already background limited, the event rate analysis is not. \n\nIt is unclear what the limiting background will be for events with $E_{\\min}=10\\omega$. The harmonic oscillator selection rules prevent excitation of $E_{\\rm ion}>\\omega$. Background gas particles at the existing pressure of $3\\times 10^{-18}$~mbar and $100\\textrm{\\AA}^2$ cross-section, correspond to 1 event every 5000 years. The limiting rate will perhaps be electric field noise, whose estimate is unknown. In Fig.~\\ref{fig:futureproj} we make projections for various experimental choices for an optimistic choice of $\\textrm{1 yr}^{-1}$ event rates for $n_{\\rm lab}=10^3 ~\\textrm{cm}^{-3}$. In pink, we show projections for a trapped proton keeping the existing energy threshold $E_{\\rm min}=10\\omega_+$ and other parameters in \\cite{borchert2019measurement}. \nNext, we explore the reach for highly charged ions as well as ions in a lattice (see for e.g. Ref.~\\cite{micke2020coherent} for recent heating limits from highly charged $^{40}\\textrm{Ar}^{13+}$ ions in a lattice). In light blue, we consider the same set up as \\cite{borchert2019measurement} but consider the reach for fully ionized Calcium, which enhances the Rutherford scattering cross-section due to large ionic charge. This limit is also  equivalent to 400 ions in a Coulomb crystal observed for 1 year.  In green, we consider the effect of a vast and futuristic improvement in sensitivity to the energy jump of a single event, $E_{\\rm min}=10\\omega_-$. Finally we consider a hypothetical electron trap, with trap frequency $\\omega=0.3$~GHz and $E_{\\rm min}$=3~GHz. Despite the large trap frequency required for electron trapping, it is competitive with ions in the small $m_Q$ regime. This is because the momentum transfer is much smaller for electron targets compared to ions for the same energy transfer. \n\n\n\\section{Discussion}\n\\label{sec:discussion}\nGeneric cosmologies should produce non-trivial abundances of well motivated stable particles that make up some or all of the observed DM density provided the reheat temperature is high enough. Such a cosmic density of millicharge particles, with large enough charge, can get stopped on Earth and accumulate through the planet's history, forming an overdense, locally thermalized population. In this work, we analyzed the utility of ion traps as detectors of such an mCP population. Ions trapped in harmonic potentials can detect energy deposits as small as neV with low intrinsic backgrounds. We have shown that the existing measurement of heating rates in Penning and Paul traps~\\cite{borchert2019measurement,goodwin2016resolved,hite2012100} sets strong limits on a wide range of mCP masses and charges as seen in Fig.\\,\\ref{fig:evsm}. These limits on the ambient thermalized population improve on existing limits from levitated spheres by several orders of magnitude with no assumptions about binding of mCPs in material. This can be seen in Fig.\\,\\ref{fig:compexist}.\n\nThese limits can in turn be interpreted as limits on the fraction of virial DM existing in mCPs, $f_Q= \\frac{\\rho_Q}{\\rho_{\\rm DM}}$. We find in Fig.\\,\\ref{fig:virial} that fractions as small as $f_Q\\gtrsim 10^{-12}$ are ruled out for masses around 1-100 GeV for millicharge around $10^{-3}$. Smaller charges can be probed with a similar setup conducted deep underground. \n\nTurning to future prospects, while modest improvements in the observed heating rates are expected, greater strides can be made with single event observation as seen in Fig.\\,\\ref{fig:futureproj}. With event\/year sensitivities, a single (anti)proton can improve upon the existing bounds from heating by one order of magnitude with energy thresholds of $\\approx 100$ neV. Reducing energy threshold can further increase the parameter space that can be probed.\n\nAnother viable direction is using a single fully ionized heavy ion ~\\cite{micke2020coherent} which increases the Rutherford cross-section with mCPs. Multiple ions that form a Coulomb crystal also result in increased sensitivity. While we made projections only for the single event rate for a Coulomb crystal, it is reasonable to expect better signal\/background discrimination by considering in detail the selection rules for collective excitations of the crystal. Heavier ions will however require improvements in energy resolution in order to be sensitive to the same quantum jump as an (anti)proton experiment. \n\nIon experiments have been shown to be transportable~\\cite{cao2016transportable,delehaye2018single,gellesch2020transportable} and can thus be conducted at different altitudes; deep underground in mines, at high altitudes or in space will have drastically different mCP induced heating rate due to the large densities at mines and virial densities in space. This can be used for sensitivity to smaller ambient mCP densities as well as a viable background discrimination tool.\n\n\n\nFinally, electrons can be stably trapped in deep potential wells. Due to its lower mass, an electron can extract $\\approx\\frac{m_p}{m_e}$ more energy than a proton for the same momentum transfer. For masses around 1\\,GeV and below, electron traps might be a promising alternative to probe low charges. It is important to emphasize the complementarity of different traps. Existing Traps of larger sizes have lower heating rates but feature deeper potentials. Hence, they are suited for probing small charges, whereas microtraps are ideal for larger charges. The choice of the trapped charged SM particle, a heavy ion, a proton or an electron can provide optimal sensitivity to different masses due to kinematic matching. For mCP detection purposes, a large trap with shallow trapping potentials will be optimal. \n\nIon trapping has myriad applications including the realization of qubits for a quantum computer. There are significant resources invested in this endeavor which should translate to longer stability, reduced heating rates, scaling to large numbers of ions, as well as the realization of long-term stability in electrons. It is exciting that these improvements translate directly into increased sensitivity of dark matter detection.  \n\\\\\n\\\\\n\\textbf{Note Added:} While this paper was in preparation, ideas related to detecting mCPs using ion traps appeared in Ref~\\cite{Carney:2021irt}. Event rate measurements were explored, which is a subset of the observables discussed in this work. While future projections were made in Ref.~\\cite{Carney:2021irt}, our work additionally provides existing limits from the heating rate, which is already measured in many traps. \nWhile we agree qualitatively with the broad conclusions of the Ref.~\\cite{Carney:2021irt} for the future, that ion traps are ideal mCP detectors, we disagree with the quantitative results.\nIn particular, for the future projections using their parameters, we find different answers for the reach.  The discrepancy may arise from the following differences.\nFirst, in Ref.~\\cite{Carney:2021irt} the scattering is treated  as a  Rutherford scatter on a free, stationary ion. Taking into account the initial energy of the ion makes a significant difference. Second, in Ref.~\\cite{Carney:2021irt}, the ambient number density is conservatively assumed to be the virial one, whereas, our work incorporates the orders of magnitude increase in local density. Third, we analyze the effect of work functions, trapping potentials and mean-free-paths which are very important at large charge. This  provides a maximum charge above which there is no reach. Fourth, we point out that experiments performed near the surface of the earth have access to only a constrained region in charge vs mass, which can be expanded if the experiment was to instead be performed in a deep mine. \nThe initial arxiv version of the publication \\cite{Carney:2021irt} takes into account only virialized velocity distributions, so it did not apply to the parameter space we consider where current ion traps have sensitivity.  The subsequent slowdown to room and cryogenic temperatures were incorporated in the published version which just appeared during the final stages of this work, though with the differences noted above.\n\n\n\\acknowledgments\nWe would like to thank Richard Thompson for useful discussions regarding the particulars of the experimental data in ~\\cite{goodwin2016resolved}.  The work of DB and FSK was supported by the Cluster of Excellence ``Precision Physics, Fundamental Interactions, and Structure of Matter'' (PRISMA+ EXC 2118\/1) funded by the German Research Foundation (DFG) within the German Excellence Strategy (Project ID 39083149), by the European Research Council (ERC) under the European Union Horizon 2020 research and innovation program (project Dark-OST, grant agreement No 695405), and by the DFG Reinhart Koselleck project. SU acknowleges support by RIKEN and the Max Planck, RIKEN, PTB Center for Time, Constants and Fundamental Symmetries. \nCS acknowledges support by the ERC (project STEP, grant agreement No 852818) and the Institute of Physics in Mainz. \nPWG and HR acknowledge support from the Simons Investigator Award 824870, DOE Grant DE-SC0012012, NSF Grant PHY-2014215, DOE HEP QuantISED award \\#100495, and the Gordon and Betty Moore Foundation Grant GBMF7946.  This work was also supported by the U.S.~Department of Energy, Office of Science, National Quantum Information (NQI) Science Research Centers through the Fermilab SQMS NQI Center.\n\n\n\\onecolumngrid\n\n\n\\section{Introduction}\nParticles that carry a fraction of the electron charge, $Q=\\epsilon e$, also called millicharged particles (mCPs) are an elegant extension to the standard model (SM). The discovery of their simplest iteration, without the presence of a dark photon, would be a violation of the charge quantization hypothesis. Alternatively, mCPs could also arise as charges quantized under a dark force which kinematically mixes with the SM photon \\cite{Izaguirre:2015eya}. \n\nMillicharged particles have attracted significant interest and there have been many attempts to detect them.\nCollider and beam-dump experiments have been performed to search for mCPs and null results have lead to strong limits on their parameter space in the MeV-TeV mass range. These include LEP~\\cite{akers1995search}, the SLAC millicharge experiment~\\cite{Prinz:1998ua}, neutrino experiments~\\cite{Magill:2018tbb}, the Argoneut experiment~\\cite{Acciarri:2019jly}, MilliQan pathfinder experiment at the LHC~\\cite{Ball:2020dnx} and the BEBC beam dump experiment~\\cite{Marocco:2020dqu}. A plethora of experiments have been proposed to improve these tests and search for still smaller charges~\\cite{Ball:2016zrp,Berlin:2018bsc,Kelly:2018brz,Harnik:2019zee}. At lower masses, stringent limits arise from absence of anomalous emission in stellar environments~\\cite{Davidson:2000hf,Chang:2018rso} at masses below an MeV. The prospect of mCPs making up some or all of the dark matter (DM) has also received a lot of interest over the years. There are robust predictions for their relic density from the early universe~\\cite{Dvorkin:2019zdi,Creque-Sarbinowski:2019mcm}, as well as numerous ways to detect them~\\cite{Knapen:2017ekk,Blanco:2019lrf,Essig:2019kfe,Berlin:2019uco,Kurinsky:2019pgb,Barak:2020fql,Griffin:2020lgd} depending on their mass $m_Q$ and charge. Subcomponent millicharge DM (mCDM) has also been invoked to explain several recent experimental anomalies~\\cite{Barkana:2018qrx,Munoz:2018pzp,Liu:2019knx,Kurinsky:2020dpb,Bloch:2020uzh,Harnik:2020ugb}. \n\nSince mCPs interact with the SM particles through a massless mediator, their transfer cross-section can be large at small velocities. As a result, mCDM with charge large enough to scatter in the atmosphere and Earth overburden of direct detection experiments, rapidly loses its virial kinetic energy and thermalizes with the environment. When it eventually reaches a direct detection experiment, it does not possess enough energy to deposit in the detector and can not be observed at direct detection experiments~\\cite{Emken:2019tni}. However, these mCPs which are now cooled to the ambient temperature, get trapped due to Earth's gravity and build up for the duration of the Earth's existence. This can lead to mCP densities on Earth up to fourteen orders of magnitude larger than that of the virial population in the galaxy \\cite{Pospelov:2020ktu}. This slow, albeit dense population requires novel detection strategies, some of which include mCP particle-antiparticle annihilation in a large-volume detector~\\cite{Pospelov:2020ktu} and accelerating mCPs present in electrostatic accelerator tubes to higher energies, sufficient for subsequent direct detection~\\cite{Pospelov:2020ktu}. \nIf the charge is large enough, negatively charged mCPs can bind with large positively charged SM nuclei, thereby creating fractional charge for a macroscopic material. Searches for such fractional charges bound to a sample material include Millikan-like oil-drop experiments~\\cite{kim2007search}, as well as more recent levitation experiments with microspheres~\\cite{moore2014search,afek2020limits}. However, these limits hinge critically on the assumption that the negatively charged mCPs bind to matter, thereby restricting their validity to only large charges and masses. \n\nIn this work, we point out an alternate search strategy; the remarkably stable trapped ions developed for metrology and quantum information science are the ideal targets to detect an ambient thermalized mCP population. There is a long history of using trapped charged SM  particles for particle physics applications. Trapped ions have been used to measure the anomalous magnetic moment of the electron~\\cite{hanneke2011cavity}, the electron's electric dipole moment~\\cite{cairncross2017precision}, proton and antiproton magnetic moments \\cite{schneider2017double,smorra2017parts}, as well as time variation in fundamental constants which can be induced by ultralight bosonic dark matter~\\cite{Roberts_2020}. Ion traps are also the preeminent candidate for qubits to realize quantum computing \\cite{haffner2008quantum}. \n\nFor all of the applications listed above and especially for magnetic moment measurements and quantum computing, it is important to keep the ion trapped for sufficiently long duration without heating from the surroundings. In the last couple of decades, there has been remarkable progress in reduction of the measured heating rate of trapped ions \\cite{hite2012100}. Further progress is expected and is an active area of research in order to achieve scalability of multi-ion systems and increased sampling rate in precision measurements. We briefly explain why the above properties make ion traps the ideal candidate for detecting mCPs next. \n\n\n\n\\subsection{Summary of Findings}\n\\label{sec:exec}\n\nWe propose the use of ion traps as detectors for millicharged particle (mCP) dark matter.  While an individual ion constitutes a much smaller target mass than any other dark matter direct detection experiment, a trapped ion has significant advantages for detection of mCPs including isolation from the environment, a lower energy threshold for detection, and a larger scattering cross section with mCPs.  These advantages outweigh the small target mass, allowing ion traps to reach many orders of magnitude past other detection methods for mCPs. \n\nSignificant effort has been put into isolating trapped ions from their environments.\nThus trapped ions are now sensitive to small energy depositions down to $\\sim$ neV.  \nThe dense, thermal gas mCPs, if they exist on Earth, would permeate the detector and can scatter off the ion.  \nDepending on the nature of the experiment, it may be possible to measure individual scattering events or just an overall heating rate of the ion.\nThe mCPs are thermalized with the walls of the detector which is held at a temperature much higher than the temperature of the trapped ion. Thus, the higher-energy mCPs can transfer kinetic energy which can be detected either as a single jump of the trapped ion, or an accumulation of several scatters resulting in heating of the ion.  We will discuss both types of signals.\nThe high degree of isolation of the ion achievable in these traps makes them sensitive to scattering rates for mCDM over a wide range of parameter space.\n\nIon traps also make excellent mCDM detectors because of their low energy thresholds that are set by the energy-level spacings in the trap that can be as low as $\\sim \\text{neV}$.  This allows detection of single scattering events with this energy and also allows such low-energy scatters to contribute to the heating rate.\n\nFurthermore, because they are charged, ions make excellent targets for mCP scattering.  mCPs scatter with ions via Rutherford scattering which is greatly enhanced at small relative velocities and momentum transfers.  Since the mCPs are thermalized with the walls, they generally have much lower velocities than virialized dark matter. The corresponding boost to the scattering cross section combined with the large number density of the thermalized mCPs makes direct detection with single ions viable. \n\n\n\nThe rest of this paper is organized as follows.\nIn Sec.\\,\\ref{sec:traps} we provide a description of ion traps. In Sec.\\,\\ref{sec:mcpdyn}, we provide an overview of the mCP density on Earth, as well as their dynamics, including propagation through the trap. The detection signals are explained in Sec.\\,\\ref{sec:signals} and results and projections are presented in Sec.\\,\\ref{sec:results}. We conclude with a discussion in Sec.\\,\\ref{sec:discussion}. \n\n\\section{Ion traps}\n\\label{sec:traps}\nTraps for (single) charged particles belong to the basic tool-set of atomic, molecular, and optical physics \\cite{safronova2018search}. They have wide applications in determinations of atomic masses and fundamental constants,  in precision measurements to test fundamental symmetries, and in quantum information technology. \nTypical ingredients common to all trap experiments are shown in Fig$.\\,$\\ref{fig:Trap}. The core of the experiments are usually sets of electrodes supplied by AC and DC voltages, optionally placed in strong magnetic fields.  \n \\begin{figure}[h!]\n\\centering\n\n    \\centering\n    \\includegraphics[width=\\linewidth]{figs\/fig1.pdf} \n\n\n    \\caption{Elements common to typical trap experiments. A set of electrodes supplied by AD and DC voltages is mounted in a vacuum chamber, optionally in a strong magnetic field. Cryogenic traps have in addition thermal shielding and cryogenic vacuum chambers at low temperatures in their surroundings. The particles are manipulated with laser beams, microwave and radio-frequency generators. Signals are read-out by detecting image currents or fluorescent light. }\n    \\label{fig:Trap}\n\\end{figure}\nThe electrodes are mounted in vacuum chambers, in some cryogenic trap experiments pressures on the level of 10$^{-18}\\,$mbar are achieved,  which provides ultra-long particle storage times \\cite{sellner2017improved} and enables non-destructive long-term studies at low background. Thermal shielding and an additional insulation vacuum chamber usually surround the inner vacuum chamber. The trapped particles are manipulated, cooled and excited via laser, microwave and radio-frequency-drives. The experimental signals for ultra-sensitive precision studies, frequency measurements, and monitoring of quantum information-processing protocols are either image-currents picked up by sensitive detection circuits, or fluorescence signals. The signals are acquired and processed with spectrum analyzers and charge coupled device (CCD) cameras, respectively.\n\n\\begin{table*}\n\\begin{center}\n\\begin{tabular}{ ||c|c|c|c|c|c|c|c| }\n\\hline\n Experiment & Type & Ion & $V_z$  &  $T_{\\rm wall}$ & $\\omega_p$ [neV] & $T_{\\rm ion}$[neV] & Heating Rate (neV\/s) \\\\ \n \\hline \n Hite et al, 2012 \\cite{hite2012100}& Paul &$^9\\text{Be}^+$ & 0.1 V&300 K&$\\omega_z=14.8$ &$14.8$&640 \\\\\n Goodwin et al, 2016\\,\\cite{goodwin2016resolved} &Penning&$^{40}\\text{Ca}^+$ & 175\\,V  &300\\,K& $\\omega_z=1.24$ & $1.24$ & 0.37 \\\\\n Borchert et al, 2019\\,\\cite{borchert2019measurement}&Penning&$\\bar{p}$& 0.633\\,V& 5.6\\,K & $\\omega_+=77.4$&7240 & 0.13 \\\\\n &&& & & $\\omega_-=0.050$& &  \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\caption{List of ion traps and the relevant experimental parameters used for setting limits in this paper. The ion used, $V_z$,  potential barrier in the axial direction and $T_{\\rm wall}$, the temperature of the walls of the trap and $\\omega_p$, the fundamental frequency of the trap in the relevant direction are listed. Also listed are $T_{\\rm ion}$, the temperature of the ion in the trap and the measured heating rate. }\n\\label{table:data}\n\\end{table*}\n\n\n\\subsection{Penning Traps}\nIn a Penning trap, there is a strong magnetic field $B_0$ superimposed with an electrostatic quadrupolar potential $\\Phi(z,\\rho)$, which is attractive along the magnetic field axis. The motion of a charged particle in such crossed static fields is composed of harmonic oscillator modes of three independent types. The modified cyclotron and the magnetron modes correspond to oscillations perpendicular to the axis, while the axial mode corresponds to oscillations along the magnetic field lines. Associated with the mode oscillations are the three trap eigenfrequencies $\\nu_+$, $\\nu_-$, and $\\nu_z$, respectively. Room-temperature traps such as that in Ref.\\,\\cite{goodwin2016resolved}, cool and trap ions such as $^{40}\\textrm{Ca}^+$ or $^{9}\\textrm{Be}^+$ to the axial ground state using optical sideband cooling~\\cite{mavadia2014optical}. A delay period after cooling is followed by subsequent spectroscopy which determines the final state of the ion, thus measuring the heating rate. The lowest heating rate achieved thus far at room temperature is reported in Ref.\\,\\cite{goodwin2016resolved} where the increase in the number of phonons of the axial mode was reported to be $\\dot{n}=0.3\/\\textrm{s}$ with the axial frequency $\\nu_z=0.3\\,\\textrm{MHz}$, see  Tab.\\,~\\ref{table:data}. \\\\\nParticularly interesting for the detection of mode energy changes are experiments dedicated to direct measurements of nuclear magnetic moments such as those of the proton \\cite{schneider2017300}, the antiproton \\cite{smorra2017parts}, or $^3$He$^{2+}$ \\cite{mooser2018new}. These experiments operate advanced cryogenic Penning-trap systems consisting of multi-trap assemblies, common to all of them is a so-called analysis trap with a strong superimposed magnetic inhomogeneity $B(z)=B_0+B_2 z^2$, where $B_2$ characterizes its strength \\cite{ulmer2011observation}. \nThe interaction of the magnetic bottle with the particle's magnetic moment $\\mu_z=\\mu_++\\mu_-+\\mu_s$ results in a magnetostatic axial energy $E_{B,z}=-\\mu_z B_z$, where $\\mu_+$ and $\\mu_-$ are the orbital angular magnetic moments associated with the modified cyclotron and the magnetron mode, while $\\mu_s$ is the spin magnetic moment. \nAs a result, the particle's axial frequency $\\nu_z=\\nu_{z,0}+\\Delta\\nu_z (n_+,n_-,m_s)$ becomes a function of the radial trap eigenstates $n_+$ and $n_-$, as well as the spin eigenstate $m_s$ with\n\\begin{align}\n\\Delta\\nu_z=\\frac{h\\nu_+}{4\\pi^2 m \\nu_z}\\frac{B_2}{B_0} \\left(\\left(n_++\\frac{1}{2}\\right)+\\frac{\\nu_-}{\\nu_+}  \\left(n_-+\\frac{1}{2}\\right)+\\frac{g}{2} m_s\\right).\n\\end{align}\nMeasurements of single-particle magnetic moments in Penning traps rely on the detection of axial frequency shifts $\\Delta\\nu_{z,\\text{SF}}$  induced by driven spin quantum transitions $\\Delta m_s=1$. Since nuclear magnetic moments are about three orders of magnitude smaller than the Bohr magneton, these experiments require highest sensitivity with respect to magnetic moments, which is usually achieved by the utilization of magnetic bottle strengths of the order $B_2\\approx100\\,$kT\/m$^2$ to $B_2\\approx400\\,$kT\/m$^2$. Combined with continuous measurements of the axial frequency, such strong magnetic bottles provide excellent resolution of the radial mode energies with \n\\begin{eqnarray}\n\\frac{(\\Delta\\nu_z)}{\\Delta E_\\rho}=\\frac{1}{4\\pi^2m \\nu_z}\\frac{B_2}{B_0}\\approx 1 \\frac{\\text{Hz}}{\\mu \\textrm{eV}}, \n\\end{eqnarray}\nwhile in axial frequency measurements resolutions of order  200$\\,$mHz$\\cdot\\sqrt{\\text{s}}\/\\sqrt{t_\\text{avg}}$ are achieved, $t_\\text{avg}$ being the averaging time which is typically on the order of several tens of seconds. Transition rates in the radial modes\n$(dn_{+,-})\/dt\\propto(n_{+,-}\/\\omega_{+,-} ) S_E (\\omega_{+,-})$ lead to random walks in radial energy space and to axial frequency diffusion. Here  $S_E (\\omega_{+,-})$ is the power spectral density of a noisy background drive, and $n_{+,-}$ is the principal quantum number of the modified cyclotron ($n_+$) \/ magnetron ($n_-$) oscillator. The scaling of the heating rate with $n_{+,-}$ is related to eigenstate-overlap of harmonic oscillator states \\cite{ulmer2011observation}. \\\\ \nBy analyzing time sequences of axial frequency measurements $\\nu_z (t)$, the average radial quantum transition rates are obtained. With a highly optimized trap setup with which the antiproton magnetic moment was measured with 1.5 parts per billion precision, the BASE experiment at CERN reports on the observation of absolute cyclotron transition rates of 6(1) quanta per hour~\\cite{borchert2019measurement}. Together with the determination of the $n_+$ state during the recorded measurement, this result is  consistent with a projected ground state heating rate of 0.1 cyclotron quantum transitions per hour, setting an upper limit which is by a factor of 1800 lower than the best reported Paul-trap heating rates, and by a factor of 230 lower than the best room-temperature Penning trap. These numbers are summarized in Tab.\\,\\ref{table:data}. Note that the antiproton experiments are conducted in a background vacuum of $\\approx 10^{-18}\\,$mbar \\cite{sellner2017improved}, constraining parasitic heating induced by collisions with  background-gas to a level of $4\\times10^{-9}$\/s. \n\n\n\\subsection{Paul Traps}\nPaul traps or radiofrequency traps utilize an oscillating voltage to confine in the perpendicular direction instead of the magnetic field used for the same purpose in the Penning trap. Paul traps have a rich history of being used as mass spectrometers and more recently in building quantum computers~\\cite{brownnutt2015ion}. \n\nThe effective potential in the presence of both DC and AC potentials can be written as\n\\begin{equation}\n    \\psi(x,y,z,t)=\\left(U_{\\rm DC}+V_{\\rm AC} \\cos \\Omega t\\right)\\frac{r^2+2 (z_0^2-z^2)}{r_0^2+2z_0^2} .\n\\end{equation}\nHere $z$ is the axial direction, $r$ the radial direction with the distance to the electrodes given by $r_0$ and $z_0$. The rapidly oscillating potential creates a pseudopotential for charges of both signs and this leads to approximately simple harmonic motion very close to the trap center.  After laser cooling to the ground state, the total heating rate can be measured via Raman sideband technique~\\cite{turchette2000heating}. There has been extensive study of the heating rate in Paul traps and its dependence on distance to electrodes, wall temperature, trap temperature~\\cite{brownnutt2015ion} as well as ion beam treatment of electrodes~\\cite{hite2012100}. Electric field noise from the electrodes has been identified as the dominant heating source, with the dependence on distance scaling as $d^{-2}$~\\cite{brownnutt2015ion} to $d^{-4}$~\\cite{daniilidis2011fabrication}. Although heating rates are lower for bigger traps~\\cite{poulsen2012efficient}, smaller sized traps employ shallower potential wells $\\approx0.1~\\textrm{V}$. As we shall see, this allows mCPs with large charge to reach the trap and hence provide complementary reach at large charge parameter space. Hence we reinterpret limits only from a microtrap~\\cite{hite2012100}.  The heating rates reported in ~\\cite{hite2012100} are $\\dot{n}=43\/\\textrm{s}$ for the axial frequency $\\nu_z=3.6~\\textrm{MHz}$. These numbers are tabulated in Tab.\\,~\\ref{table:data}.\n\n\n\n\\section{Millicharged Particle Dynamics}\n\\label{sec:mcpdyn}\n\\subsection{Terrestrial Accumulation} \nIf mCDM exists and is virialized in the galaxy, there is a non-zero flux of mCPs flowing through the Earth at all times.  This mCDM stops in the atmosphere or rock overburden for large enough charge, and can accumulate on Earth. This process was treated in detail in~\\cite{Emken:2019tni} and the subsequent accumulation in~\\cite{Pospelov:2020ktu}. We provide here a summary of the relevant results of these papers that are used in Section.\\,\\ref{sec:results} and refer the reader to ~\\cite{Pospelov:2020ktu} for details.\n\nFollowing~\\cite{Pospelov:2020ktu} we consider effectively millicharge particles mediated by a dark photon which kinematically mixes with the SM photon. The dark photon mass is taken to be large enough ($m_{A'} \\gtrsim 10^{-12}~\\textrm{eV}$) such that the effect of large-scale electric and magnetic fields can be ignored while considering mCP propagation\\footnote{We leave to upcoming work the calculation of the accumulation of millicharged dark matter in cases where the electric and magnetic fields are relevant \\cite{blprfuture}. Once such calculations are complete, our given limits on the number density in the lab can be translated to limits on dark matter fraction in theses cases.}.  In this limit, the mCPs with mass $m_Q\\ge 1$\\,GeV are stuck on the Earth after thermalization.\\footnote{Millicharged particles with masses below 1\\,GeV can accumulate on Earth temporarily before evaporating. We leave limits on these masses for future work.} The volume averaged DM number density on Earth, $\\langle n_Q \\rangle$ can be several orders of magnitude larger than the virial DM density. It is given by, \n\\begin{align}\n\\label{start}\n\\langle n_Q \\rangle&= \\frac{\\pi R_{\\oplus}^2 v_{\\rm vir}  t_{\\oplus}}{4\/3 \\pi R_{\\oplus}^3}n_{\\rm vir} \\nonumber \\\\\n&\\approx \\frac{3\\times 10^{15} }{\\textrm{cm}^3} \\frac{t_{\\oplus}}{10^{10}\\,\\textrm{y}} f_Q \\frac{\\rm GeV}{m_Q},\n\\end{align}\nwhere, $R_{\\oplus}$ and $t_{\\oplus}$ are the radius and age of the Earth and $n_{\\rm vir}$ and $v_{\\rm vir}$ are the galactic virial number density and velocity. However, the equilibrium density profile is peaked at the Earth core. This density $n_{\\rm static}$ was calculated by taking into account the Earth's temperature and density variations in \\cite{Neufeld:2018slx}. However, the sinking to the Earth's core is not immediate, and there is a dynamic population, the so-called ``traffic-jam\" density, $n_{\\rm tj}$, given by,\n\\begin{equation}\n    n_{\\rm tj}=n_{\\rm vir}\\frac{v_{\\rm vir}}{v_{\\rm term}} .\n\\end{equation}\nHere, $v_{\\rm term}$ is the terminal velocity in rock, given in \\cite{Pospelov:2020ktu}. Finally, the density in the laboratory, $n_{\\rm lab}$ is given by, \n\\begin{equation}\nn_{\\rm lab} = \\textrm{Max}\\left( n_{\\rm static}, \\textrm{Min}\\left( n_{\\rm tj} , \\langle n_Q \\rangle \\right)\\right)\n\\label{eq:regimes}\n\\end{equation}\n\nThese results are applicable only for $\\epsilon$ large enough, such that it stops in the corresponding overburden. According to \\cite{Pospelov:2020ktu} this is valid only for\n\\begin{align}\n\\epsilon&\\gtrsim 2\\times10^{-4} \\sqrt{\\frac{m_Q}{\\rm GeV}} \\quad &&\\textrm{surface,} \\nonumber \\\\\n&\\gtrsim 3\\times 10^{-6}\\sqrt{\\frac{m_Q}{\\rm GeV}} \\quad &&\\textrm{1 km mine.} \n\\label{eq:applicable}\n\\end{align}\n\nIt is also important to comment on the asymmetry of the mCP population. If the mCDM is a symmetric population with equal number of particles and antiparticles, accumulation on Earth can result in Sommerfeld-enhanced annihilations which prevent build up. We consider here the asymmetric case such that opposite charges are carried by different species just like the the SM proton and electron, such that annihilations are absent. In this scenario, for large enough $\\epsilon$, the negatively charged mCPs can form deep bound states with the large positively nuclei, whereas the positive mCPs can only bind with the less massive electrons which also have smaller charge compared to the heavy SM nuclei. In ref.~\\cite{Pospelov:2020ktu} it was pointed out that positive mCPs with $\\epsilon \\ge 0.042$  bind to electrons at 300 K (room temperature). We find that such bound states are temporary with electrons rapidly preferring to bind with ions which possess larger charge than the positive mCPs. Thus the terrestrial population of positive mCPs remains free of binding for $\\epsilon\\lesssim 1$ and is present as a locally thermalized population that is diffusing everywhere. Thus, the results we derive apply to all positive charges with $\\epsilon \\lesssim 1$ and negative charges which do not bind with nuclei, i.e \\cite{Pospelov:2020ktu} $\\epsilon < \\frac{m_e}{\\mu_{Q,N}}$ where $\\mu_{Q,N}$ is the reduced mass of the mCDM-nuclear system.  \n\n\nIt is important to emphasize that the limits put on the ambient number density are applicable to mCPs charged directly under the SM photon as well as to ones mediated by a dark photon as long as the dark photon mass is below the relevant momentum transfer for scattering with ions which is around 1 eV.   \n\n\\subsection{Passage Through Apparatus}\n\n\\label{trappassage}\nWe next turn to the trajectory of mCPs through the trap peripherals in order to reach the trapped ion.  We need to know the density and temperature of the mCPs reaching the ion.  In this subsection we explain the main factors entering the calculation, but as the calculation itself is somewhat involved we leave the details to Appendix \\ref{Appendix:passage}.  Our main result is in Eq.\\,\\eqref{eqn:qion} relating the number density of mCPs at the position of the ion, $n^Q_{\\rm ion}$, to the ambient number density on the Earth $n_{\\rm lab}$.  \nThis is the equation we use to set our limits on mCPs.\n\nAt equilibrium, mCPs are expected to have roughly uniform density near the Earth's surface, including permeating all materials.  However the conditions of the experiment can affect this naive expectation for several reasons.\n\nFirst, some of the ion traps we consider are cryogenic.  Over essentially all of our parameter space the mCPs have a short interaction length in material and so will rapidly thermalize to the cryostat temperature as they enter the experiment.  By itself this would lead to an increase in mCP density by a factor linear in mCP velocity ($\\propto \\sqrt{T}$) because the fluxes entering and leaving the cryostat must be equal in equilibrium.\n\nSecond, the ion traps are surrounded by metal which has a work function that can affect the passage of mCPs.  This is only relevant for mCPs of relatively large charge ($\\epsilon \\gtrsim 10^{-2}$) but for those it can be a significant effect.  The work function for mCPs, which we will call $\\epsilon \\phi$, is not simply $\\epsilon$ times the work function for electrons.  For electrons the work function arises from several contributions of varying signs including e.g.~the binding to the lattice of nuclei, the Fermi sea of other electrons, and surface effects such as the ``double layer\" or the image charge potential.  Several of these do not apply or are negligible for mCPs.  Recall we are only considering positively charged mCPs since the negative ones may be stuck deeply bound to some nucleus somewhere on the Earth.  Thus the main effects are repulsion by the double layer and possibly also repulsion from the nuclei.   We consider the work function for mCPs in more detail in  Appendix \\ref{Appendix:work function}.  Our conclusion is that the work function is repulsive for positively charged mCPs and thus every metal sheet provides a barrier for mCPs to cross. \nThe size of the potential barrier that has to be crossed is $\\phi \\sim \\text{few} \\, \\text{eV}$.\nNote that for an experiment at room temperature ($T \\sim 0.03 \\, \\text{eV}$) the metal barriers are then irrelevant for charges $\\epsilon \\lesssim 10^{-1}$ because the Boltzmann tail easily pushes a fast enough rate of mCPs over the barrier.  For a cryogenic experiment at $T \\sim 6 \\, \\text{K} \\sim 5 \\times 10^{-4} \\, \\text{eV}$, the metal barrier will be relevant for charges $\\epsilon \\gtrsim 10^{-3}$ and essentially insurmountable for charges $\\epsilon \\gtrsim 10^{-2}$.  As we show in Appendix \\ref{Appendix:passage}, the most important effect comes from an experiment encased in two different metals where the work function for mCPs rises from the outer metal to the inner metal.  We will take this difference to be $\\Delta \\phi = 3 \\, \\text{eV}$ for all experiments we consider since this will be a conservative estimate as we show in Appendix \\ref{Appendix:work function}.\n\n\nThird, the ions are always in a region of ultrahigh vacuum.  This means that pumps were used to remove the Standard Model (SM) particles.  Given a short interaction length of the mCP in materials, these pumps could remove the mCPs from the ion chamber as well.  In one of the experiments we consider (Goodwin et al.~\\cite{goodwin2016resolved}) this effect is not relevant because the trap is at room temperature and the region of sensitivity is at low enough $\\epsilon$ that the millicharges pass easily through the walls\\footnote{The sensitivity is limited to be below $10^{-3}$ charge because of the large trap potential as will be discussed in Section \\ref{sec:results}.}.  In the other two experiments we consider (Hite et al.~\\cite{hite2012100} and Borchert et al.~\\cite{borchert2019measurement}), the vacuum pumps are turned off well before the actual data taking is begun.  In the case of Borchert this is at least a year, while for Hite we conservatively assume it is only a day\\footnote{The final answer is only logarithmically sensitive to this timescale anyway.}.  And of course mCPs are always continually flowing in from the walls of the vacuum chamber.  This would rapidly refill the trap region and this effect would not be relevant, except for the largest charges where the refill can be slow because of the work function of the surrounding metal.  This does mean that, depending on the parameters of the mCP and of the ion trap, the number density in the trap may be either in the equilibrium regime or in the filling regime.  This is why Eq.\\,\\eqref{eqn:final trap density} has two different regimes.  Eq.\\,\\eqref{eqn:final trap density} relates the number density of mCPs inside the trap $n_{\\rm trap}$ to the ambient number density on the Earth $n_{\\rm lab}$.  There is then one remaining step to find the number density of mCPs at the position of the ion.\n\nFourth, the ion trap itself has applied electromagnetic fields to trap the ion.  These can affect the passage of the mCPs, though again this is only relevant for larger charges $\\epsilon \\gtrsim 10^{-3}$.\nTab.\\,\\ref{table:data} lists parameters of the various experiments we consider.  All the experiments use an electric DC potential to confine the ion in the axial direction.  The height of the potential barrier along the axial direction is listed as $V_z$.  The mCPs in the trap are thermalized to the temperature of the walls of the trap $T_{\\rm wall}$.  Starting from the number density of mCPs calculated inside the trap $n_{\\rm trap}$ in Eq.\\,\\eqref{eqn:final trap density}, we take a Boltzmann suppression on the number density which can make it up the axial barrier height.  Thus we take the final number density at the position of the ion to be given by Eq.\\,\\eqref{eqn:qion}.  For the experiments we consider this Boltzmann suppression is relevant for setting the ceiling (largest $\\epsilon$ values) of the Goodwin regions in Figure \\ref{fig:evsm}, but is irrelevant for the other regions. \nThe Penning traps use a magnetic field to confine the ions in the radial direction.  This B-field will cause mCPs with a large enough charge to circle around the axial B-field lines but they are still free to move along the axis.  The magnetic field does not change the phase-space density of the mCPs and so it does not significantly affect our signal.  The Paul traps use an RF potential to confine the ion in the radial direction.  This potential around the minimum is locally attractive for the positive mCP independent of the sign of the charge of the trapped ion.  An mCP coming from far outside the RF fields in a radial direction could give a barrier in principle.  However for mCPs approaching along the axial direction, the RF pseudopotential in the axial direction is very weak and negligible.  And then  such mCPs will actually be concentrated in the radial direction towards the ion at the center of the trap since the potential is locally attractive.  We conservatively ignore this possible (Sommerfeld-like) enhancement though it could be large. For large enough mCP masses, $m_Q \\gtrsim 10^{10} \\textrm{GeV}$, the free-fall under gravity \ncan generate velocities much larger than the thermal velocities assumed. This can increase the heating rate further. We conservatively ignore this effect and leave its consideration for future work.\n\n\\section{Observables for Millicharged Particles in Ion Traps}\n\\label{sec:signals}\nIn this section we consider the interaction of ambient mCPs with ion traps to identify observables for detection. As seen in Section.~\\ref{trappassage}, the mCPs enter the ion trap with effective temperature $T_Q \\ge T_{\\rm wall}$, where $T_{\\rm wall}$ is the wall temperature. $T_{\\rm wall}= 5.6$ K for the cryogenic trap we consider~\\cite{borchert2019measurement} and $T_{\\rm wall}=T_{\\rm room}\\approx 300$ K for room temperature traps. The ions in the trap are at a much colder temperature $T_{\\rm ion} \\ll T_{\\rm wall}$. With this hierarchy of temperatures, the mCPs can cause two types of signal.\n\nThe first signal involves individual scattering events that impart energy $E_{\\rm ion}$ to the ion thus leading to a change in its harmonic oscillator quantum number. This signal is very similar to dark matter scattering in a conventional dark matter detector. The rate of events has to be slower than the rate at which the ions are interrogated. $E_{\\rm ion}$ also needs to exceed the energy resolution $E_{\\rm res}$ for detection. \n\nThe second type of signal is the heating of the trapped ion due to collisions with multiple mCPs. In this case, the individual hits $E_{\\rm ion}$ can be smaller than $E_{\\rm res}$ and only the sum needs to exceed this resolution. \n\nFor both of these signal types, we only consider individual energy transfers $E_{\\rm ion}$ much larger than the typical energy spacing of the trap $\\omega$, for concreteness $E_{\\rm ion} \\ge 10\\omega$. \\footnote{In principle, energy transfers $E_{\\rm ion} =n\\times \\omega$ where all integers $n\\ge 1$ are allowed. We restrict $E_{\\rm ion} \\ge 10 \\times \\omega$, a conservative choice that helps avoid form-factor calculations.} In this limit, the trapped ion can be approximated as a free particle with initial energy $T_{\\rm ion}$ and final energy $T_{\\rm ion}+E_{\\rm ion}$. Equivalently, for energy transfers much larger than the spacing, the form-factor that incorporates wave-function overlap can be approximated to unity.  \nWe start quantifying both of these signals by the angular differential cross-section given by the Rutherford formula,\n\\begin{equation}\n    \\frac{d\\sigma}{d\\Omega}=\\frac{2\\pi\\alpha^2 \\epsilon^2}{\\mu^2 v_{\\rm rel}^4\\left(1-\\cos\\theta\\right)^2} ,\n\\end{equation}\nwhere $\\theta$ is the scattering angle. We next introduce kinematic variables that  simplify the computation of the scattering rate. The results are presented here, with the details of the derivation presented in Appendix.~\\ref{app1}. The incoming mCP and trapped ion velocities are assumed to be $\\mathbf{v_Q}$ and $\\mathbf{v_{\\rm ion}}$ respectively. \nThe center of mass (CM) velocity $\\mathbf{v_{CM}}$ is given by,\n\\begin{equation}\n    \\mathbf{v_{CM}}=\\frac{\\left(m_{\\rm ion} \\mathbf{v_{\\rm ion}}+m_Q \\mathbf{v_Q}\\right)}{m_{\\rm ion}+m_Q} .\n\\end{equation}\n\nThe change in velocity of the ion, $\\mathbf{\\Delta v_{\\rm ion}}$ is given by,\n\\begin{align}\n    \\mathbf{\\Delta v_{\\rm ion}}=\\frac{m_Q}{m_{\\rm ion}+m_Q}\n    \\left[\\left(\\cos \\theta-1 \\right)\n    \\left(\\mathbf{v_{\\rm ion}}-\\mathbf{v_Q}\\right)\n    \\right. \n    \\nonumber \\\\ \n    \\left.\n    +\\sin \\theta |\\mathbf{v_{\\rm ion}}-\\mathbf{v_Q}|\n    \\mathbf{n_{\\perp}}\n    \\right] .\n\\end{align}\nThe transferred energy $E_{\\rm ion}$ is given by,\n\\begin{equation}\n    E_{\\rm ion} = m_{\\rm ion} \\mathbf{v_{CM}}.\\mathbf{\\Delta v_{\\rm ion}} .\n\\end{equation}\n\nGiven a threshold $E_{\\rm thr}$, the single event rate $R_{\\rm single}$ with energy transfer $E_{\\rm ion}$ above this threshold is,\n\\begin{align}\n    R_{\\rm single}\\left(E_{\\rm ion}\\ge E_{\\rm thr}\\right)=n^Q_{\\rm ion} \\int d^3 \\mathbf{v_Q} g_Q  \\int d^3 \\mathbf{v_{\\rm ion}} g_{\\rm ion}  \\int d \\Omega  \\nonumber \\\\ |v_Q-v_{\\rm ion}| \\frac{d\\sigma} {d \\Omega} \\Theta \\left(|E_{\\rm ion}|-E_{\\rm thr}\\right) . \n\\end{align}\nHere $n^Q_{\\rm ion} $ is the number density of mCPs at the ion position and is given by Eq.\\,\\eqref{eqn:qion}, $g_{\\textrm{ion}(Q)}$ is the Maxwell--Boltzmann distribution for the ion\/mCP with temperatures $T_{\\rm ion}$ and $T_{\\rm wall}$, respectively. \n\n\nThe heating rate per ion, $\\dot{H}$ can be computed through,\n\\begin{align}\n    \\dot{H}&=n^Q_{\\rm ion}  \\int d^3 \\mathbf{v_Q} g_Q  \\int d^3 \\mathbf{v_{\\rm ion}} g_{\\rm ion}  \\int d \\Omega |v_Q-v_{\\rm ion}|  \\nonumber \\\\ &\\frac{d\\sigma} {d \\Omega}E_{\\rm ion} \\Theta \\left(|E_{\\rm ion}|-E_{\\rm thr}\\right) \\Theta\\left(E_{\\rm samp}-|E_{\\rm ion}|\\right) . \n\\end{align}\nThe Heaviside theta function ensures the inequality $E_{\\rm thr}\\le E_{\\rm ion} \\le E_{\\rm samp}$. Here $E_{\\rm samp}$ is defined so as to prevent the average heating rate from including contribution from extremely rare events. It is defined through,\n\\begin{equation}\nR_{\\rm single}\\left(E_{\\rm ion}\\ge E_{\\rm samp}\\right) t_{\\rm obs}=1 ,\n\\end{equation}\nwhere $t_{\\rm obs}$ is the total observation time. \n\n\n \\section{Results}\n \\label{sec:results}\n \n\n\\begin{figure*}\n\\centering\n\\begin{subfigure}{0.48\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figs\/comparetraps1e3.pdf} \n\n\\end{subfigure}\n\\begin{subfigure}{0.48\\textwidth}\n    \\centering\n    \\includegraphics[width=\\linewidth]{figs\/evsmmainall.pdf} \n \n\\end{subfigure}\n    \\caption{Compilation of new limits using existing heating measurements from various traps in Table \\ref{table:data}: a room temperature Paul trap, Hite et al.~\\cite{hite2012100}, a room temperature Penning trap, Goodwin et al.~\\cite{goodwin2016resolved} and a cryogenic penning trap, Borchert et al. \\cite{borchert2019measurement}. {\\bf (left)} Comparison between traps for an ambient density $n_{\\rm lab}=10^3 \\textrm{cm}^{-3}$; {\\bf (right)} Combined limits from the three traps for different  $n_{\\rm lab}$} \n    \\label{fig:evsm}\n\\end{figure*}\n In this section, we set constraints on mCPs and make projections for the future by using the expressions for the signal rates derived in Sec.\\,\\ref{sec:signals}. \n \n \\subsection{Limits from existing measurements}\n To obtain existing limits, we use the data presented in Tab.\\,\\ref{table:data}. All of the trap parameters for the $^9\\text{Be}^+$  trap are taken from \\cite{hite2012100} while the trap depth is conservatively taken to be $V_z=0.1$~V, an order of magnitude larger than the typical potential depths in microtraps\\footnote{see for e.g.~\\cite{warring2013techniques} from the same group where the potential depth is report to be $V_z$=5 mV}. While the rest of the parameters for the $^{40}\\textrm{Ca}^+$ experiment are provided in ~\\cite{goodwin2016resolved}, we obtained the potential depth $V_z=175$~V from the authors. Finally, for ref.\\,\\cite{borchert2019measurement}, we use the parameters $T_{\\rm wall}=5.6$~K and $V_z$=0.6~V. While the analysis in ~\\cite{borchert2019measurement} dealt with measuring the cyclotron mode ($\\omega_+)$, the observable heating rate is equivalently a limit on the magnetron mode ($\\omega_-$) also. While individual jumps in $\\omega_-$ are unobservable with existing precision, the frequency shift due to the heating rate accumulates. Since the Rutherford cross-section increases for smaller energy transfers, we use $\\omega_-$ to convert existing heating limits into limits on mCPs. \n\n We start by plotting existing limits in Fig.\\ref{fig:evsm} in the millicharge $\\epsilon$ vs mass $m_Q$ parameter space for contours of constant ambient density $n_{\\rm lab}$. As mentioned earlier, for mCDM, the lab density is expected to be orders of magnitude larger than the virial density, i.e. $n_{\\rm lab} \\gg n_{\\rm vir}$. In the left panel, limits arising from the antiproton trap \\cite{borchert2019measurement} in blue, $^{40}\\textrm{Ca}^+$~\\cite{goodwin2016resolved} in red and $^{9}\\textrm{Be}$ in green \\cite{hite2012100} for an ambient density of $n_{\\rm lab}=10^3\/\\textrm{cm}^3$. The dominant limits arise from the antiproton trap owing to its superior heating rate as seen in Tab.\\,\\ref{table:data}. However, since it is a cryogenic trap, the limits disappear at $\\epsilon \\approx 10^{-2}$ owing to the suppression arising from mCPs finding it increasingly difficult to penetrate metals due to their work function. As explained in Section.~\\ref{sec:mcpdyn} and Appendix.~\\ref{Appendix:passage}, this factor roughly scales as $e^{-\\frac{\\epsilon \\Delta \\phi}{T_{\\rm wall}}}$ where $\\Delta \\phi$ is the difference in work functions between two adjacent metals. Hence this suppression is ameliorated for room temperature traps like the $^9\\textrm{Be}^+$ trap \\cite{hite2012100} in green which extends to $\\epsilon\\approx 0.1$. While the $^{40}\\textrm{Ca}^+$ trap in red is also at room temperature, the trap is at a potential of 175~V and thus mCP charges above $\\epsilon \\approx 10^{-3}$ do not reach the ion. However, there is reach to higher mCP masses for this trap as a result of the ion being more massive. The right panel of Fig.~\\ref{fig:evsm}\n corresponds to combined limits from these three experiments for different $n_{\\rm lab}$.  \n \n \\begin{figure}\n\\centering\n\n    \\includegraphics[width=\\linewidth]{figs\/compwithexisting.pdf} \n\n    \\caption\n    Comparison of limits derived in this work from ~\\cite{hite2012100,goodwin2016resolved,borchert2019measurement} with existing limits from Oil Drop~\\cite{kim2007search}, levitation experiments~\\cite{moore2014search,afek2020limits} and LEP~\\cite{Marocco:2020dqu} for $m_Q=10$ GeV.}\n    \\label{fig:compexist}\n\\end{figure}\n \n \n \n As is clear from both figures, existing data for anomalous heating in traps sets exquisite bounds on mCPs thermalized locally. Bounds are applicable to orders of magnitude in the mCP mass $m_Q$ as well as many orders of magnitude in charge. Number densities as small as $n_{\\rm lab}=1\\, \\textrm{cm}^{-3}$ are ruled out around the $\\epsilon\\approx 10^{-3}$ and $m_Q\\approx 10\\, \\textrm{GeV}$ parameter point. \n \nIn order to compare these limits on the ambient mCP population to ones that already exist in literature, we fix the mCP mass $m_Q=10$\\,GeV and show limits in the $n_{\\rm lab}$ vs $\\epsilon $ plane in Fig.~\\ref{fig:compexist}. The same color coding as Fig.~\\ref{fig:evsm} is followed. In gray we show limits from LEP~\\cite{Marocco:2020dqu}, as well as limits on mCPs bound in matter arising from Oil drop experiments~\\cite{kim2007search}, and levitation experiments~\\cite{moore2014search,afek2020limits}. As noted earlier, the limits on mCPs bound in matter are applicable only to negative mCPs with large enough charge such that binding with SM nuclei is possible. Furthermore, if mCP-SM bound states exist, there is no guarantee for these bound states to be evenly distributed all over the Earth. However, for the positive mCPs none of these caveats apply and they thermalize and distribute themselves over the entire Earth volume. Regardless, as seen in Fig.~\\ref{fig:compexist} the limits obtained from ion traps are orders of magnitude stronger than the levitation experiments. For $\\epsilon\\approx 3\\times 10^{-3}$, lab densities as small as $n_{\\rm lab} \\gtrsim 1 \\textrm{cm}^{-3}$ are ruled out by the measured heating rate at the antiproton experiment \\cite{borchert2019measurement}. \n \\begin{figure}\n\\centering\n\n    \\centering\n    \\includegraphics[width=\\linewidth]{figs\/dmplot.pdf} \n  \n\n    \\caption{Limits and projections on the virial DM fraction in mCPs as a function of $\\epsilon$ and $m_Q$. Since the current experiments were all conducted at the surface, the robust limits are only above the ``$>$1m Thermalization\" line where the colored contours are solid. For a hypothetical deep mine experiment, the dashed part of the contours are also accessible.}\n    \\label{fig:virial}\n\\end{figure}\n\nNext, in Fig.~\\ref{fig:virial} we convert limits on $n_{\\rm lab}$ into limits on the fraction of virial DM existing in mCPs, $f_Q= \\frac{\\rho_Q}{\\rho_{\\rm DM}}$. For this purpose we use $n_{\\rm lab}$ from Eq.\\,\\ref{eq:regimes}. Existing limits from colliders are shown in gray. The solid parts of the colored contours correspond to the region where the incoming virial DM gets thermalized within 1 meter and hence the robust current limits we put are restricted to this region i.e. above the top black line. The dashed lines show the reach for an identical heating rate experiment that is conducted in a deep mine at 1km depth. Virial DM fractions as small as $f_Q\\approx 10^{-12}$ are already ruled using existing heating data for DM masses in the 1-10 GeV mass range. For heavier masses, the terminal velocity is larger and hence the traffic jam densities are smaller. Nonetheless we set limits on DM fractions as small as $f_Q=10^{-3}$ for masses as large as $m_Q\\approx 10^5$~GeV.\n\nThe limits presented above were derived using data from existing experiments that measure the anomalous heating rate. We next make projections for the future to capture improvements in reducing the heating rate as well as to incorporate mCP detection specific modifications. \n\n\\subsection{Projections}\n\\begin{figure}\n\\centering\n    \\includegraphics[width=\\linewidth]{figs\/future1e3.pdf} \n    \n    \\caption{Event Rate limits and projections for $n_{\\rm lab}=10^3 \\textrm{cm}^{-3}$. Existing limits from heating of the $\\omega^-$ mode are shown (brown shaded) from Fig \\ref{fig:evsm}.  The projection for a search for single events in the BASE experiment \\cite{borchert2019measurement} with energy deposit above $10\\omega^+$ and rate 0.1 event\/hr are shown (dark blue). Here $\\omega_+=77.4$ neV and $\\omega_-= 0.050$ neV.  Next, projections are also shown assuming sensitivity to 1 Event\/year event rates. In pink, we show sensitivity from the existing setup of Ref.~\\cite{borchert2019measurement}. The light blue curve corresponds to swapping a single (anti)-proton with fully ionized Calcium or equivalently trapping 400 Calcium ions in a Coulomb crystal. We also show the reach for a futuristic experiment with energy thresholds of $E_{\\rm min}=10\\omega_{-}$ in green. Finally, reach from a hypothetical electron trap with trap frequency $\\omega=200$~neV and $E_{\\rm min}=2~\\mu$eV is shown in orange.\n    }\n    \\label{fig:futureproj}\n\\end{figure}\n\n\nLimits shown thus far arise from the cumulative heating rate measured. Another promising avenue, is the (non)-observation of individual event rates. In Fig.~\\ref{fig:futureproj}, we compare projections from a non-observation of single event with $E_{\\rm ion}\\ge 10\\omega_+$ with the same parameters as existing data in Ref.~\\cite{borchert2019measurement} in dark blue with the heating of the much smaller $\\omega_-$  in brown for $n_{\\rm lab}=10^3~\\textrm{cm}^{-3}$. \n\nWe find that this projection is near-identical to the heating limit in brown. Both the heating limits as well as the event rate sensitivity are expected to improve in the future. For e.g. heating rates are known to decrease with larger electrode distance or increasing the frequency of the trapped ion~\\cite{brownnutt2015ion}. Whereas the mCP search with the heating rate in the current BASE apparatus is already background limited, the event rate analysis is not. \n\nIt is unclear what the limiting background will be for events with $E_{\\min}=10\\omega$. The harmonic oscillator selection rules prevent excitation of $E_{\\rm ion}>\\omega$. Background gas particles at the existing pressure of $3\\times 10^{-18}$~mbar and $100\\textrm{\\AA}^2$ cross-section, correspond to 1 event every 5000 years. The limiting rate will perhaps be electric field noise, whose estimate is unknown. In Fig.~\\ref{fig:futureproj} we make projections for various experimental choices for an optimistic choice of $\\textrm{1 yr}^{-1}$ event rates for $n_{\\rm lab}=10^3 ~\\textrm{cm}^{-3}$. In pink, we show projections for a trapped proton keeping the existing energy threshold $E_{\\rm min}=10\\omega_+$ and other parameters in \\cite{borchert2019measurement}. \nNext, we explore the reach for highly charged ions as well as ions in a lattice (see for e.g. Ref.~\\cite{micke2020coherent} for recent heating limits from highly charged $^{40}\\textrm{Ar}^{13+}$ ions in a lattice). In light blue, we consider the same set up as \\cite{borchert2019measurement} but consider the reach for fully ionized Calcium, which enhances the Rutherford scattering cross-section due to large ionic charge. This limit is also  equivalent to 400 ions in a Coulomb crystal observed for 1 year.  In green, we consider the effect of a vast and futuristic improvement in sensitivity to the energy jump of a single event, $E_{\\rm min}=10\\omega_-$. Finally we consider a hypothetical electron trap, with trap frequency $\\omega=0.3$~GHz and $E_{\\rm min}$=3~GHz. Despite the large trap frequency required for electron trapping, it is competitive with ions in the small $m_Q$ regime. This is because the momentum transfer is much smaller for electron targets compared to ions for the same energy transfer. \n\n\n\\section{Discussion}\n\\label{sec:discussion}\nGeneric cosmologies should produce non-trivial abundances of well motivated stable particles that make up some or all of the observed DM density provided the reheat temperature is high enough. Such a cosmic density of millicharge particles, with large enough charge, can get stopped on Earth and accumulate through the planet's history, forming an overdense, locally thermalized population. In this work, we analyzed the utility of ion traps as detectors of such an mCP population. Ions trapped in harmonic potentials can detect energy deposits as small as neV with low intrinsic backgrounds. We have shown that the existing measurement of heating rates in Penning and Paul traps~\\cite{borchert2019measurement,goodwin2016resolved,hite2012100} sets strong limits on a wide range of mCP masses and charges as seen in Fig.\\,\\ref{fig:evsm}. These limits on the ambient thermalized population improve on existing limits from levitated spheres by several orders of magnitude with no assumptions about binding of mCPs in material. This can be seen in Fig.\\,\\ref{fig:compexist}.\n\nThese limits can in turn be interpreted as limits on the fraction of virial DM existing in mCPs, $f_Q= \\frac{\\rho_Q}{\\rho_{\\rm DM}}$. We find in Fig.\\,\\ref{fig:virial} that fractions as small as $f_Q\\gtrsim 10^{-12}$ are ruled out for masses around 1-100 GeV for millicharge around $10^{-3}$. Smaller charges can be probed with a similar setup conducted deep underground. \n\nTurning to future prospects, while modest improvements in the observed heating rates are expected, greater strides can be made with single event observation as seen in Fig.\\,\\ref{fig:futureproj}. With event\/year sensitivities, a single (anti)proton can improve upon the existing bounds from heating by one order of magnitude with energy thresholds of $\\approx 100$ neV. Reducing energy threshold can further increase the parameter space that can be probed.\n\nAnother viable direction is using a single fully ionized heavy ion ~\\cite{micke2020coherent} which increases the Rutherford cross-section with mCPs. Multiple ions that form a Coulomb crystal also result in increased sensitivity. While we made projections only for the single event rate for a Coulomb crystal, it is reasonable to expect better signal\/background discrimination by considering in detail the selection rules for collective excitations of the crystal. Heavier ions will however require improvements in energy resolution in order to be sensitive to the same quantum jump as an (anti)proton experiment. \n\nIon experiments have been shown to be transportable~\\cite{cao2016transportable,delehaye2018single,gellesch2020transportable} and can thus be conducted at different altitudes; deep underground in mines, at high altitudes or in space will have drastically different mCP induced heating rate due to the large densities at mines and virial densities in space. This can be used for sensitivity to smaller ambient mCP densities as well as a viable background discrimination tool.\n\n\n\nFinally, electrons can be stably trapped in deep potential wells. Due to its lower mass, an electron can extract $\\approx\\frac{m_p}{m_e}$ more energy than a proton for the same momentum transfer. For masses around 1\\,GeV and below, electron traps might be a promising alternative to probe low charges. It is important to emphasize the complementarity of different traps. Existing Traps of larger sizes have lower heating rates but feature deeper potentials. Hence, they are suited for probing small charges, whereas microtraps are ideal for larger charges. The choice of the trapped charged SM particle, a heavy ion, a proton or an electron can provide optimal sensitivity to different masses due to kinematic matching. For mCP detection purposes, a large trap with shallow trapping potentials will be optimal. \n\nIon trapping has myriad applications including the realization of qubits for a quantum computer. There are significant resources invested in this endeavor which should translate to longer stability, reduced heating rates, scaling to large numbers of ions, as well as the realization of long-term stability in electrons. It is exciting that these improvements translate directly into increased sensitivity of dark matter detection.  \n\\\\\n\\\\\n\\textbf{Note Added:} While this paper was in preparation, ideas related to detecting mCPs using ion traps appeared in Ref~\\cite{Carney:2021irt}. Event rate measurements were explored, which is a subset of the observables discussed in this work. While future projections were made in Ref.~\\cite{Carney:2021irt}, our work additionally provides existing limits from the heating rate, which is already measured in many traps. \nWhile we agree qualitatively with the broad conclusions of the Ref.~\\cite{Carney:2021irt} for the future, that ion traps are ideal mCP detectors, we disagree with the quantitative results.\nIn particular, for the future projections using their parameters, we find different answers for the reach.  The discrepancy may arise from the following differences.\nFirst, in Ref.~\\cite{Carney:2021irt} the scattering is treated  as a  Rutherford scatter on a free, stationary ion. Taking into account the initial energy of the ion makes a significant difference. Second, in Ref.~\\cite{Carney:2021irt}, the ambient number density is conservatively assumed to be the virial one, whereas, our work incorporates the orders of magnitude increase in local density. Third, we analyze the effect of work functions, trapping potentials and mean-free-paths which are very important at large charge. This  provides a maximum charge above which there is no reach. Fourth, we point out that experiments performed near the surface of the earth have access to only a constrained region in charge vs mass, which can be expanded if the experiment was to instead be performed in a deep mine. \nThe initial arxiv version of the publication \\cite{Carney:2021irt} takes into account only virialized velocity distributions, so it did not apply to the parameter space we consider where current ion traps have sensitivity.  The subsequent slowdown to room and cryogenic temperatures were incorporated in the published version which just appeared during the final stages of this work, though with the differences noted above.\n\n\n\\acknowledgments\nWe would like to thank Richard Thompson for useful discussions regarding the particulars of the experimental data in ~\\cite{goodwin2016resolved}.  The work of DB and FSK was supported by the Cluster of Excellence ``Precision Physics, Fundamental Interactions, and Structure of Matter'' (PRISMA+ EXC 2118\/1) funded by the German Research Foundation (DFG) within the German Excellence Strategy (Project ID 39083149), by the European Research Council (ERC) under the European Union Horizon 2020 research and innovation program (project Dark-OST, grant agreement No 695405), and by the DFG Reinhart Koselleck project. SU acknowleges support by RIKEN and the Max Planck, RIKEN, PTB Center for Time, Constants and Fundamental Symmetries. \nCS acknowledges support by the ERC (project STEP, grant agreement No 852818) and the Institute of Physics in Mainz. \nPWG and HR acknowledge support from the Simons Investigator Award 824870, DOE Grant DE-SC0012012, NSF Grant PHY-2014215, DOE HEP QuantISED award \\#100495, and the Gordon and Betty Moore Foundation Grant GBMF7946.  This work was also supported by the U.S.~Department of Energy, Office of Science, National Quantum Information (NQI) Science Research Centers through the Fermilab SQMS NQI Center.\n\n\n\\onecolumngrid\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nHadronic scattering in the late stages of a heavy-ion collision may modify the chemical composition of the system up to the time of chemical freeze-out, defined as when all non-resonant particle ratios are fixed.  However, resonances may still be produced through inelastic hadronic scatterings, their production rates being dependent on the relatively poorly known hadronic cross-sections ($\\Xi^{-}+\\pi^{+}\\rightarrow\\Xi^{*}(1530)$, $\\Lambda+\\pi^{\\pm}\\rightarrow\\Sigma^{*}(1385)$, etc.) \\cite{Bass00,Rapp01,vanHees05,Adler02}.  Similarly, rescattering of the daughters of resonances that decay during the lifetime of the colliding system may destroy their correlations.  This prohibits the parent resonance reconstruction, leading to an underestimated primary yield.  The interplay of these two effects continues from the time of chemical freeze-out ($t_{chem}$) to the time of thermal freeze-out ($t_{therm}$, currently estimated to be $\\sim$8-10 fm\/$c$ \\cite{Retiere04}), the degree to which the primary resonance yields are modified depends on both the length of this time interval ($\\Delta t = t_{therm} - t_{chem}$) and the relative strengths of the two effects \\cite{Bleicher02, Bleicher03, Torrieri01, Rafelski01, Rafelski02}.  It has also been suggested that the \\textit{in vacuo} resonance masses and widths might be modified in the presence of the nuclear medium \\cite{STARRes}.\n\nThe STAR experiment \\cite{STAR} has measured several mesonic and baryonic resonance states \\cite{STARRes, STARK*, STARrho, STARphi}.  The variation in the composition and lifetimes of the measured states provides multiple constraints on $\\Delta t_{therm-chem}$ and the degree of late stage hadronic interactions.  The measurements presented herein add to the emerging picture of how the medium produced in heavy-ion collisions at RHIC affects resonance production.\n\n\\section{Analysis}\n\nThe results presented here come from an analysis of 7.6$\\times 10^{6}$, 5.4$\\times 10^{6}$, and 6.8$\\times 10^{6}$ Au+Au events in the 0-12\\%, 10-40\\%, and 40-80\\% centrality ranges respectively.  The greatest difficulty in performing an analysis of short-lived particles is acquiring an accurate description of the underlying combinatorial  background.  For this analysis, we use a narrow rotational method that is implemented as follows.  \n\nIn each event, the invariant mass is calculated for each topologically found $\\Xi^{-}$ candidate that passes a set of quality cuts combined with each $\\pi^{+}$ candidate that has passed a set of particle identification cuts.  These combinations comprise the ``signal''.  A second set of combinations is then made in which the transverse momentum ($\\mathrm p_{\\mathrm T}$) vectors of the $\\Xi^{-}$ candidates are rotated in the azimuthal plane by angles from 170$^{\\circ}$ to 188$^{\\circ}$ and the $\\pi^{+}$ candidates are rotated from -8$^{\\circ}$ to +8$^{\\circ}$, both in steps of 2$^{\\circ}$.  The invariant mass is then calculated for each iteration and each pair.  These combinations comprise the ``background''.  The process is repeated as a function of $\\mathrm p_{\\mathrm T}$ and the background is then subtracted from the signal in each bin.  Lastly, the yield is determined by summing the bin contents in a narrow region around the canonical mass.  Efficiency and acceptance corrections are applied to the results.  Lastly, a correction is applied to account for contribution to the overall yield coming from the neutral decay channel, which we do not measure here.  The corrected mid-rapidity spectra for three centralities are show in Figure \\ref{spectra}.\n\\begin{figure}\n \\centering\n \\includegraphics[bb= 0 0 567 547, scale=0.45]{figures\/figure1.eps}\n\n \\caption{Corrected $\\Xi^{0}(1530)$ $\\mathrm p_{\\mathrm T}$ spectra from Au+Au collisions at $\\sqrt{\\mathrm{s}_{\\mathrm NN}}=200$ GeV\/$c$ for three centrality bins.}\n \\label{spectra}\n\\end{figure}\n\n\\section{Results and Conclusions}\n\nInformation about medium effects on resonance production is gained by comparing the resonance yield to that of its non-resonant partner.  In Figure \\ref{ratios} we show this comparison, plotted as a ratio of the resonance to non-resonance yield, for the relevant species measured by STAR.  The mid-rapidity ($|y|<1.0$) $\\Xi^{0}(1530)$ $dN\/dy$ yields are compared with ($|y|<0.75$) $\\Xi$ $dN\/dy$ yields from reference \\cite{STARXi}.\n\\begin{figure*}[h]\n\\hspace*{-1.0cm}\n\\subfigure[Resonance to non-resonance ratios for several species in $p+p$, d+Au, and Au+Au collisions.]{\n\\hspace*{-1.0cm}\n \\label{ratios}\n \\includegraphics[bb= 6 5 561 542, scale=0.45]{figures\/figure2.eps}}\n\n\\hspace*{0.1cm}\n\\subfigure[Thermal model calculation.  Black dots are data, model fit and predictions are horizontal lines.  Data to the right side of the ``Fitted'' line were not included in the fit, but rather compared to predictions from the fit.]{\n \\label{thermus}\n \\includegraphics[bb= 0 0 567 547, scale=0.45]{figures\/figure3.eps}}\n\n\\end{figure*}\nThese comparisons are usually shown scaled by the $p+p$ value.  However, as there is currently no measured $p+p$ ratio for the $\\Xi^{0}(1530)$, the ratios have all been scaled by the most central Au+Au values.  This alternative scaling results in what is usually seen as a supression in central Au+Au appearing as an enhancement in peripheral (and $p+p$) collisions and an enhancement in central Au+Au appearing as a suppression in peripheral (and $p+p$) collisions.\n\nExamining the ratios we find the following: The shortest lived resonance measured by STAR, the K$^*$, shows a small suppression in central collisions relative to peripheral \\cite{STARK*ratio}, and the only slightly longer-lived $\\Sigma^{*}(1385)$ shows no measurable effect \\cite{STARRes}.  The $\\Xi^{0}(1530)$ ratios show an \\textit{enhancement} in the most central collisions relative to the peripheral collisions, the opposite of what is seen for the other reasonably long-lived baryonic resonance, the $\\Lambda^{*}(1520)$, which shows a strong suppression in central collisions relative to $p+p$.\n\nThe short-lived resonances would be expected to show suppression in more central events due to the rescattering of their decay products.  The fact that they are only weakly suppressed, if at all, suggests there must be significant regeneration contributing to the final state yields.  This conclusion is strengthened when one adds the $\\Xi^{0}(1530)$ measurement to the picture.  The $\\Xi^{0}(1530)$ should easily outlive the system (on average) once it's produced.  This means rescattering does not significantly affect the $\\Xi^{0}(1530)$ yield, (re-)generation plays a stronger role.  The observed $\\Xi^{0}(1530)$ enhancement in central collisions seems to indicate that significant interaction, and resonance production, occurs between $\\Xi$ and $\\pi$ particles in the (late) hadronic phase.  The $\\phi$ meson, like the $\\Xi^{0}(1530)$, will not on average decay before $t_{therm}$.  However, hadronic phase $\\phi$ production may be suppressed by the relatively low probability for K$^{+}$+K$^{-}$ scattering in the final stages of the collision \\cite{STARphi}.  The $\\Lambda^{*}(1520)$ is also reasonably long-lived.  However, alternative explanations have been put forward for the observed $\\Lambda^{*}(1520)$ suppression that do not involve rescattering and regeneration \\cite{Enyo,Kaskulov}.\n\nLastly, in Figure \\ref{thermus} we show the results of a thermal model fit to the published STAR data from central $\\sqrt{\\mathrm{s}_{\\mathrm NN}}=200$ Au+Au collisions.  The fit was performed using the THERMUS interface \\cite{thermus}.  The resulting fit parameters are T=0.169$\\pm$0.006 GeV, $\\mu_{\\mathrm B}$=0.04$\\pm$0.01 GeV, $\\mu_{\\mathrm S}$=0.016$\\pm$0.009 GeV, $\\mu_{\\mathrm Q}$=-0.01$\\pm$0.01 GeV, $\\gamma_{\\mathrm S}$=0.91$\\pm$0.06, and radius=7.5$\\pm$1.0 fm.  All of the resonance to non-resonance ratios are shown to the right of the ``Fitted'' line compared to their predicted values from the thermal fit.  The same pattern of suppression and enhancement as was seen in the ratios alone shows-up again.  The thermal model gives predictions for particle ratios only at $t_{chem}$, and therefore does not account for production that occurs during the hadronic phase, and so the observed $\\Xi^{0}(1530)$ enhancement could again be attributed to significant resonance production occurring during the hadronic phase.  This result strengthens the conclusion drawn from the resonance to non-resonance ratios alone.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}