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+{"text":"\\chapter[The topological types of length bounded multicurves]{The topological types of length bounded multicurves}\n\n\n\\author{Hugo Parlier}\n\\authormark{H. Parlier}\n\n\n\\keywords{hyperbolic surfaces, curves, Teichm\\\"uller spaces, moduli spaces}\n\\subjclass{Primary 32G15, 52K20; secondary 53C22, 30F60}\n\n\n\n\n\\begin{abstract}\nThis article discusses inequalities on lengths of curves on hyperbolic surfaces. In particular, a characterization is given of which topological types of curves and multicurves always have a representative that satisfies a length inequality that holds over all of moduli space.\n\\end{abstract}\n\n\n\\section{Introduction}\n\nLength inequalities for curves play an important role in the understanding of hyperbolic surfaces and their moduli spaces. A prime example of this is a theorem of Bers, which states that any closed hyperbolic surface admits a pants decomposition of length bounded by a constant which only depends on the topology of the surface, and not its geometry. Short pants decompositions have been very useful in the understanding of the underlying Teichm\\\"uller space, and its large to medium scale geometry. This result generalizes bounds on the length of the shortest non-trivial closed curve (the systole). These are both examples of families of curves or multicurves that admit upper bounds over Teichm\\\"uller or moduli space of a given topological surface. More specifically, these results tell you that any hyperbolic surface of given topology admits a curve, or a multicurve, taken among a family of topological types, and which satisfies a certain length bound.\n\nThe goal of this short note is to characterize which families of curves or multicurves admit such upper bounds on their lengths. By multicurve, we mean a finite union of curves, all of them considered up to free homotopy, and by length we mean the length of a minimum length, thus geodesic, representative. The bounds are only allowed to depend on the topology of the underlying surface. Because of the nature of the game, we are only interested in the topological type of the curves, or said differently, in its mapping class group orbit. In order to try and satisfy the length inequality, we are allowed to choose curves of minimal length in the orbit. \n\nThe result is stated in terms of length functions. To a multicurve, we can associate the function, which takes a hyperbolic surface in the Teichm\\\"uller $\\mathcal{T}(\\Sigma)$ of a finite-type surface orientable $\\Sigma$, and which associates to it the length of the (unique) geodesic in the free homotopy class of the multicurve. The existence of the upper bounds described depends on whether certain length functions are bounded over Teichm\\\"uller or moduli space.\n\n\\begin{theorem}\\label{thm:main}\nLet $\\Gamma$ be a multicurve on $\\Sigma$. Then the quantity\n$$\\max_{X\\in \\mathcal{T}(\\Sigma)} \\min_{\\phi \\in \\mathrm{Mod}(\\Sigma)} \\ell_{\\phi(\\Gamma)}(X)$$\nis finite if and only if, for every pants decomposition $P$, there exists $\\phi$ in the mapping class group  $\\mathrm{Mod}(\\Sigma)$ such that $i(P, \\phi(\\Gamma))) = 0$.\n\\end{theorem}\nFor simplicity, it is stated only for multicurves but more generally it holds for families of multicurves (see Theorem \\ref{thm:maingeneral}). In more colloquial terms, what the result says is that if you are given a type of multicurve, then the function that associates to any hyperbolic surface the multicurve of minimal length of that type, is a bounded function over moduli space if and only if there is a multicurve of that type disjoint from any pants decomposition. \n\nNote that, given a lower bound on the systole, by compactness of pinched moduli space, there is similar conditional length inequality statement (Proposition \\ref{prop:conditionallength}) that holds for any multicurve, but the implied constants depend on topology, curve type and the lower bound of the systole.\n\nNote that here we are only interested in the very first value in a subset of the length spectrum. This is obviously very different from recent results about the asymptotic growth of curves of a given type (see \\cite{Erlandsson-Souto} and references therein). This is also very different from the related problems of finding precise constants, and in particular, exploring surfaces that are extremal for different geometric quantities. The focus here is on understanding what type of inequalities are possible. This is of course inspired by the many uses that have been made of these, or related, inequalities in the study of the large or medium scale of Teichm\\\"uller spaces with different metrics \\cite{Brock,Cavendish-Parlier} for the Weil-Petersson metric, \\cite{Rafi,Rafi-Tao,Papadopoulos-Theret} for the Teichm\\\"uller and Thurston metrics. \n\nFinally, note that this note is about upper bounds of length functions. If one replaces the $\\max$ with a $\\min$ in the above theorem, the quantity is strictly positive if and only if the multicurve has intersection (coming from a non-simple closed curve or pairwise intersecting curves). This is a consequence of the collar lemma \\cite{Keen} and generalisations \\cite{BasmajianCollar}. Lower bounds that depend on curve type have been studied in some detail by Basmajian \\cite{BasmajianUniversal}.\\\\\n\n\\noindent{\\bf Organization.} The article is organized as follows. The second section is mainly notation and definitions, and includes Proposition \\ref{prop:conditionallength}. The third and final section is dedicated to Theorem \\ref{thm:maingeneral} and ends with a discussion of which previously known length inequalities fall within its framework.\n\n\\section{Preliminaries and setup}\nThroughout $\\Sigma$ will be an orientable finite-type surface with $\\chi(\\Sigma) <0$. $\\Sigma$ is entirely determined by its genus $g$ and number of ends $n$, and $\\chi(\\Sigma) = 2-2g -n$.The space of marked complete hyperbolic structures on $\\Sigma$, that is Teichm\\\"uller space, will be denoted $\\mathcal{T}(\\Sigma)$, and should be thought of as a continuous deformation space of hyperbolic metrics. For the purpose of simplicity, we ask the metrics to be geodesically complete, and so the ends of $X\\in \\mathcal{T}(\\Sigma)$ are realized as cusps. The underlying moduli space, that is the space of hyperbolic structures on $\\Sigma$ up to isometry, will be denoted $\\mathcal{M}(\\Sigma)$. $\\mathrm{Mod}(\\Sigma)$ will be the (full) mapping class group of $\\Sigma$, that is the group of self-homeomorphisms of $\\Sigma$ up to isotopy. This group acts on $\\mathcal{T}(\\Sigma)$ and its quotient is $\\mathcal{M}(\\Sigma)$. \n\nFormally a curve is the continuous image of a circle on $\\Sigma$, but we will only be interested in a curve up to free homotopy. In particular, we only consider essential curves, that is those non-homotopic to a point or a boundary. A multicurve is a (finite) collection of curves. Associated to a multicurve $\\Gamma$ is a function which associates to $X\\in \\mathcal{T}(\\Sigma)$ the quantity $\\ell_\\Gamma(X)$, the length of the unique geodesic representative of $\\Gamma$ on $X$. These length functions are continuous, analytic and in fact convex \\cite{Kerckhoff, Wolpert}. \n\nIntersection between curves is defined as minimal intersection among representatives, and is denoted $i(\\cdot,\\cdot)$. A curve is simple if it has no self-intersections. A pants decomposition is a maximal collection of disjoint and distinct simple closed curves, and decomposes the surface into three-holed spheres (pairs of pants). The boundary curves of a pair of pants are sometimes called cuffs.\n\nThe following result \\cite{Keen} is stated here in non-quantitative terms for simple closed geodesics. Note that a version also holds for non-simple closed geodesics as well \\cite{BasmajianCollar}, with the notable difference being that you cannot pinch a non-simple closed curve. \n\n\\begin{lemma}[Collar lemma]\\label{lem:collar}\nA simple closed geodesic of length $\\ell$ on a hyperbolic surface $X$ admits a cylindrical neighborhood (its collar) of positive width $w(\\ell)$ which only depends on $\\ell$ and such that $w(\\ell) \\to \\infty$ when $\\ell \\to 0$. \n\\end{lemma}\n\nOur main use of the above result will be to consider hyperbolic structures where curves of a pants decomposition have length tending towards $0$, and hence all curves that are not among the cuffs of the given pants decomposition have length that tend to infinity. \n\nGiven $X\\in \\mathcal{T}(\\Sigma)$, the length of its shortest essential curve is its systole and is denoted $\\mathrm{sys}(X)$. Unless $X$ is a pair of pants, the systole is realized by a simple closed geodesic. For $\\epsilon>0$, the $\\epsilon$-thick part of Teichm\\\"uller space is the subset $\\mathcal{T}^{\\epsilon}(\\Sigma)\\subset \\mathcal{T}(\\Sigma)$ consisting of surfaces with $\\mathrm{sys}(X)\\geq \\epsilon$. By Mahler's compactness criterion, the corresponding thick part of moduli space $\\mathcal{M}^{\\epsilon}(\\Sigma)$ is compact.\n\nIn this paper, we study length inequalities, which here will be upper bounds for lengths of curves or multicurves with given properties. Generally these inequalities will be about a topological type of curve or multicurve: two multicurves $\\Gamma$ and $\\Gamma'$ are of the same type if there exists $\\phi\\in \\mathrm{Mod}(\\Sigma)$ such that $\\phi(\\Gamma)=\\Gamma'$. Given a multicurve $\\Gamma$, we can consider its mapping class group orbit. These orbits divide the space of all multicurves into equivalence classes\n$$\n[\\Gamma]:= \\{ \\phi(\\Gamma) \\mid \\phi \\in \\mathrm{Mod}(\\Sigma)\\}\n$$\nsorted by type. Now given $X\\in \\mathcal{T}(\\Sigma)$, we can study the length of a minimal representative of an equivalence class:\n$$\n\\ell_{[\\Gamma]}(X) = \\min_{\\phi \\in \\mathrm{Mod}(\\Sigma)} \\big\\{ \\ell_X(\\phi(\\Gamma)\\big\\}\n$$\nThe existence of a minimum follows from the discreteness of the length spectrum. This function, due to its mapping class group invariance, descends nicely to $\\mathcal{M}(\\Sigma)$. Note that although the function $\\ell_{[\\Gamma]}(\\cdot)$ remains continuous over $\\mathcal{M}(\\Sigma)$, it is no longer smooth. This is due to possible changes of homotopy classes realizing the minimum length of in an equivalence class.\n\nThe following result is a non-explicit general bound that holds for any topological type of multicurve. The proof is a compactness argument. The constant $K$ in the statement depends on the topology of $\\Sigma$, the topological type of $\\Gamma$ and a lower bound on the systole.\n\n\\begin{proposition}\\label{prop:conditionallength}\nLet $[\\Gamma]$ be a type of multicurve on $\\Sigma$. For any $\\epsilon>0$, there exists a constant $K=K(\\epsilon, \\Sigma, [\\Gamma])$ such that for any $X\\in \\mathcal{T}^{\\epsilon}(\\Sigma)$, we have\n$$\n\\ell_{[\\Gamma]}(X) \\leq K.\n$$\n\\end{proposition}\n\n\\begin{proof}\nThe space $\\mathcal{M}^{\\epsilon}(\\Sigma)$ is compact, hence the continuous function $\\ell_{[\\Gamma]}(\\cdot)$ admits a maximum on $\\mathcal{M}^{\\epsilon}(\\Sigma)$. This maximum value is exactly $K$. \n\\end{proof}\n\nAs an explicit example of the above result, consider the following result due to Buser and S\\\"eppala \\cite{Buser-Seppala}. For a closed surface $\\Sigma$ of genus $g\\geq 2$, they consider {\\it canonical homology bases}, that is collections of $2g$ simple closed curves $\\{\\alpha_i,\\beta_i\\}_{1\\leq i \\leq g}$ that satisfy:\n\\begin{enumerate}[a)]\n\\item\n$i(\\alpha_i,\\beta_j)=\\delta_{ij}$ for all $i,j\\in \\{1,\\hdots,g\\}$ (where $\\delta_{ij}$ is the Kronecker delta),\n\\item\n$i(\\alpha_i,\\alpha_j)=i(\\beta_i,\\beta_j)=0$ for $i\\neq j$. \n\\end{enumerate}\nNote that such a system of curves automatically generate integer homology. See Figure \\ref{fig:homology} for an illustration in genus $2$.\n\n\\begin{figure}[htbp]\n\\leavevmode \\SetLabels\n\\L(.23*.5) $\\alpha_1$\\\\%\n\\L(.32*.27) $\\beta_1$\\\\%\n\\L(.74*.5) $\\alpha_2$\\\\%\n\\L(.66*.27) $\\beta_2$\\\\%\n\\endSetLabels\n\\begin{center}\n\\AffixLabels{\\centerline{\\includegraphics[width=8cm]{Figures\/N-homology.pdf}}}\n\\caption{A canonical homology basis in genus $2$}\n\\label{fig:homology}\n\\end{center}\n\\vspace{-0.8cm}\n\\end{figure}\n\nThey prove that any $X\\in \\mathcal{T}^{\\epsilon}(\\Sigma)$ admits a canonical homology basis with all curves of length at most\n$$\n(g-1)\\left(45+6\\,\\mathrm{arcsinh}\\frac{1}{\\epsilon}\\right).\n$$\nThis is an improvement over an earlier quantification in \\cite{Buser-Seppala2}. The above bound shows that the $K$ in this instance is bounded above by \n$$\n2g(g-1)\\left(45+6\\,\\mathrm{arcsinh}\\frac{1}{\\epsilon}\\right)\n$$\nbecause there are $2g$ curves in the family. Of course, for this to be an exact quantification of the constant $K$, it would have to be sharp (which it is not). Exact quantifications are rarely known but, as proved in \\cite{Buser-Seppala}, the real constant must depend on $\\epsilon$. Alternatively, this dependency on $\\epsilon$ can also be deduced from Theorem \\ref{thm:main}.\n\nNaturally, this leads to the existence of upper bounds which only depend on topology and curve type, but not on systole length. A multicurve type $[\\Gamma]$ is said to satisfy a strong length inequality if $\\ell_{[\\Gamma]}(\\cdot)$ is upper bounded over $\\mathcal{M}(\\Sigma)$. By continuity of the length function $\\ell_{[\\Gamma]}(\\cdot)$, this is equivalent to the existence of a surface $X_{\\max}\\in \\mathcal{M}(\\Sigma)$ such that \n$$\n\\max_{X\\in \\mathcal{M}(\\Sigma)} \\ell_{[\\Gamma]}(X) = \\ell_{[\\Gamma]}(X_{\\max}).\n$$\nTo see this, one must show that the supremum of the length function cannot be reached on the boundary of moduli space, that is on a noded surface. As it turns out, as a consequence of Lemma \\ref{lem:stretchpants} from the next section, a (finite) supremum is always reached in the ''thick\" part of moduli space, and so in particular the $\\sup$ is indeed a $\\max$.\n\n\\section{Stretching pants, Bers' theorem and consequences}\n\nThe following lemma is by now well-known, but a sketch proof is provided for completeness. The proof uses strip maps, introduced by Thurston \\cite{Thurston}, and used to great effect by many authors \\cite{Danciger-Gueritaud-Kassel, Gueritaud, Papadopoulos-Theret, Parlier} to study of deformations of hyperbolic structures. Recall that a hyperbolic pair of pants is uniquely determined by its cuff lengths. In order to allow pants with cusp boundary, we use the convention that a cusp is a cuff of $0$ length. \n\n\\begin{lemma}[Pants stretching lemma]\\label{lem:stretchpants}\nLet $Y_{x,y,z}$ be the unique hyperbolic pair of pants with cuff lengths $x,y,z \\geq 0$. Then, for any (non-boundary) homotopy class of closed curve $\\gamma$ on a pair of pants, and any $\\epsilon>0$, we have\n$$\n\\ell_\\gamma\\left(Y_{x,y,z}\\right) < \\ell_\\gamma\\left(Y_{x+\\epsilon,y,z}\\right)\n$$\n\\end{lemma}\n\\begin{proof}[Sketch proof]\nConsider on $Y=Y_{x+\\epsilon,y,z}$, the simple orthogeodesic $a$ with both endpoints on the boundary curve of length $x+\\epsilon$ (see Figure \\ref{fig:arc}). \n\n\\begin{figure}[htbp]\n\\leavevmode \\SetLabels\n\\L(.445*.5) $a$\\\\%\n\\endSetLabels\n\\begin{center}\n\\AffixLabels{\\centerline{\\includegraphics[width=5cm]{Figures\/N-arc.pdf}}}\n\\caption{The orthogeodesic $a$}\n\\label{fig:arc}\n\\end{center}\n\\vspace{-0.8cm}\n\\end{figure}\n\nNote the closed geodesic $\\gamma$ intersects $a$ at least once. Exactly like in the collar lemma, $a$ admits an embedded neighborhood, topologically a strip (see Figure \\ref{fig:strip}). The idea is to now remove at least part of this strip to reduce the length of the boundary curve. \n\n\\begin{figure}[htbp]\n\\leavevmode \\SetLabels\n\\L(.445*.5) $a$\\\\%\n\\endSetLabels\n\\begin{center}\n\\AffixLabels{\\centerline{\\includegraphics[width=5cm]{Figures\/N-strip.pdf}}}\n\\caption{A strip surrounding $a$}\n\\label{fig:strip}\n\\end{center}\n\\vspace{-0.8cm}\n\\end{figure}\n\nTo do this properly, it is more convenient to consider the complete pair of pants by adding funnels. The arc $a$ can now be extended into a complete simple and infinite length geodesics. In addition, because the boundary curve of length $x+\\epsilon$ is not $0$, there is a family of simple complete geodesics parallel to each other and to $a$. We can take any two of these, cut away the strip between them, and paste them together to obtain a (complete) hyperbolic metric, say $Y'$ (see Figure \\ref{fig:operation}). (The slightly sketchy part is here: in fact it is possible by a variational argument to show that this can be done such that the length of the new boundary component of $Y'$ is exactly $x$, but we won't dwell on this, the main point being that the boundary length has been reduced.)\n\n\\begin{figure}[H]\n\\leavevmode \\SetLabels\n\\endSetLabels\n\\begin{center}\n\\AffixLabels{\\centerline{\\includegraphics[width=10cm]{Figures\/N-construction.pdf}}}\n\\caption{Removing the strip neighborhood $\\mathcal{S}(a)$}\n\\label{fig:operation}\n\\end{center}\n\\vspace{-0.8cm}\n\\end{figure}\n\nWe denote the strip enclosed by these geodesics by $\\mathcal{S}(a)$.\n\nNow can analyse the result of this operation on the length of $\\gamma$. Note that, due to the topology of the strip neighborhood $\\mathcal{S}(a)$ of $a$, $\\gamma \\cap \\mathcal{S}(a)$ is a union of simple geodesic arcs. We look at each one. In order to find a representative of $\\gamma$ on $Y'$, we replace each simple geodesic arc with its projection to $a$ (see Figure \\ref{fig:project}). The point is that the projection strictly reduces lengths.\n\\begin{figure}[H]\n\\leavevmode \\SetLabels\n\\endSetLabels\n\\begin{center}\n\\AffixLabels{\\centerline{\\includegraphics[width=5cm]{Figures\/N-project.pdf}}}\n\\caption{A local picture of $\\gamma$ under the operation}\n\\label{fig:project}\n\\end{center}\n\\vspace{-0.8cm}\n\\end{figure}\nThus, this results in a curve on $Y'$, in the same homotopy class, and of length strictly smaller. The corresponding geodesic is thus of length strictly smaller than it was previously. Hence we have $\\ell_\\gamma\\left(Y\\right) > \\ell_\\gamma\\left(Y'\\right)$ as required. \n\\end{proof}\nTo make the proof fully rigorous \nWe will mainly need an immediate corollary of the above result.\n\\begin{corollary}\\label{cor:longpants}\nFor $L>0$, let $Y$ be a pair of pants with cuff lengths between $0$ and $L$. Let $Y_L$ be the pair of pants with all cuff lengths exactly $L$. Then, for any interior homotopy class of curve $\\gamma$ on the pair of pants, we have $\\ell_\\gamma(Y_L) \\geq \\ell_\\gamma(Y)$ with equality only occurring if $Y=Y_L$. \n\\end{corollary}\n\nPants decompositions play an essential role in this story. The following result, originally due to Bers \\cite{Bers1,Bers2}, has been since quantified by different authors \\cite{BuserBook,BPS,ParlierShort}.\n\n\\begin{theorem}[Bers' constants]\\label{thm:bers}\nThere exists a constant $B(\\Sigma)$ such that any $X\\in \\mathcal{T}(\\Sigma)$ admits a pants decomposition with all curves of length at most $B(\\Sigma)$.\n\\end{theorem}\n\nNote that this result does not fall in the framework of either Proposition \\ref{prop:conditionallength} or Theorem \\ref{thm:main}. Indeed, in order to find a short pants decomposition, we are allowed to choose any topological type of pants decomposition. The number of these grows with topology (for instance there are roughly $\\sim g^g$ different types when $\\Sigma$ is of genus $g$ and $g$ is large). So here we are not only minimizing among mapping class group orbits of a fixed multicurve, but we are minimizing among multicurves that belong to a family. The following result holds for all such families of multicurves. A family will be denoted $\\{\\Gamma_\\alpha\\}_{\\alpha\\in I}$ where $I$ is an index set (possibly infinite, but countable as there are only countably many topological types of finite multicurve on a finite type surface). \n\n\\begin{theorem}\\label{thm:maingeneral}\nLet $\\{\\Gamma_\\alpha\\}_{\\alpha\\in I}$ be a family of multicurves. Then the quantity \n$$\\max_{X\\in \\mathcal{M}(\\Sigma)} \\min_{\\alpha\\in I} \\ell_{[\\Gamma_\\alpha]}(X)$$\nis finite if and only if, for every pants decomposition $P$, there exists $\\alpha\\in I$ and $\\phi \\in \\mathrm{Mod}(\\Sigma)$ such that $i(P, \\phi(\\Gamma_\\alpha)) = 0$. \n\\end{theorem}\n\nThe statement might seem a little confusing at first, due to the fact that we are taking a maximum among hyperbolic structures of a double minimum (over a family and then over the mapping class group orbit). If the family is reduced to a single curve, the statement becomes Theorem \\ref{thm:main} from the introduction. The proofs of both statements are identical, so we prove the more general statement above.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:maingeneral}]\n\nWe begin with the more straightforward direction, showing that if there is a pants decomposition $P$ which is intersected by any mapping class group orbit of any multicurve in the family $\\{\\Gamma_\\alpha\\}_{\\alpha\\in I}$, then there is no upper bound on the length. This follows directly from Lemma \\ref{lem:collar} (the collar lemma). Indeed, by considering a sequence of hyperbolic structures with all curves in the pants decomposition of $P$ that converge to $0$, the length of any curve that intersects one of the pants curve necessarily goes to $+\\infty$. As, by hypothesis, there is at least one curve in every multicurve $\\phi(\\Gamma_\\alpha)$, for all $\\phi \\in \\mathrm{Mod}(\\Sigma)$ and all $\\alpha \\in I$, that intersects a curve in $P$, the result follows. \n\nThe other direction will follow from Bers' theorem (Theorem \\ref{thm:bers}) and Corollary \\ref{cor:longpants} above.\n\nTake any $X\\in \\mathcal{T}(\\Sigma)$. By Bers' theorem, there exists a pants decomposition of $X$, say $P$, with all curves of length at most $B(\\Sigma)$. Now, by hypothesis, there exists $\\Gamma_\\alpha$ such that $i(\\Gamma_\\alpha,P)= 0$. Note that $\\Gamma_\\alpha$ can contain curves from $P$, but none that intersect curves of $P$ transversally. \n\nBy hypothesis all curves that belong to both $\\Gamma_\\alpha$ and $P$ are upper bounded by $B(\\Sigma)$. We now the corollary to all remaining curves of $\\Gamma_\\alpha$. Indeed, any such a curve $\\gamma \\in \\Gamma_\\alpha$ is contained in a pair of pants. By hypothesis, the cuff lengths of this pair of pants are at most $B(\\Sigma)$. By Corollary \\ref{cor:longpants}, $\\ell_X(\\gamma)$ is at most the length when the pair of pants has all cuff lengths equal to $B(\\Sigma)$, which is some finite number that depends on $B(\\Sigma)$ and the topological type of $\\gamma$. As there are a finite number of such curves, the result follows.\n\\end{proof}\n\nThere are many instances of the above theorem, each obtained by changing the multicurve or family of multicurves. Note that only the topological type of a multicurve matters in the statement. Some of these results are in the positive direction: the theorem implies that there is an upper bound on a minimal length representative that only depends on the topology of $\\Sigma$. Others are in the negative direction, that is that certain multicurves, or families of multicurves, do not admit such upper bounds. We give a (non-exhaustive) list of results of this type.\n\n\\begin{enumerate}[I.]\n\\item Even though it was used in the proof, Bers' theorem is an example obtained by taking the family of multicurves to be the full set of pants decompositions (or simply one pants decomposition for each topological type). To put in the framework of the theorem, the length of a pants decomposition should be defined as the sum, and not the maximum value. \n\nThe quantification of the implied constants - for both the sum and the max - has attracted some attention over the years, but it does not seem to be an easy problem. In fact, even the rough growth in terms of genus is not known \\cite{BuserBook,ParlierShort,GPY}. On the positive side, the exact constant in genus $2$ is known by a result of Gendulphe \\cite{Gendulphe}, as is the rough growth in terms of the number of cusps \\cite{Balacheff-Parlier,BPS}. \n\n\\item One can also take the full set of all topological types of all multicurves: this boils down to the systolic inequality. Quantifying the exact constants that appear is an arduous task. For orientable closed surfaces, the only constant known is again in genus $2$ \\cite{Jenni}. Buser and Sarnak \\cite{Buser-Sarnak} showed that the constants must grow logarithmically in genus, and since then there have been multiple variations and refinements of this (see for instance \\cite{BFR,KSV,Fanoni-Parlier, SchmutzSchaller}), including generalizations in the world of variable curvature and higher dimensional manifolds \\cite{Gromov}.\n\n\\item A slight variation of the above is to look at the homological systole of a closed surface $\\Sigma$. That is, the shortest curve that is not only non-trivial in homotopy, but also in homology. By cut and paste arguments, the homological systole is always realized by a non-separating simple closed curve. Hence, it is always in the same mapping class orbit. As above, few optimal constants are known. However, for closed surfaces, it is known that the optimal constants are equal to those from the systolic inequality. \\cite{ParlierPapa} Said differently, the systole of a maximal surface is homologically non-trivial.\n\n\\item In addition to the results of Buser and Sepp\\\"al\\\"a mentioned previously, one can try and bound families of homologically independent curves, but that do not necessarily form (part of) a canonical basis. If one requests a full homology basis, by the theorem above, there is no upper bound on its length over moduli space. However, by Bers' theorem, and the observation that any pants decomposition contains $g$ homologically independent curves, one can find an upper bound on the length of up until $g$ curves by a function of topology. So what about $g+1$ curves?\n\nGromov observed \\cite[Section 5]{GromovFilling} that any minimal length homology basis consists of simple curves that pairwise intersect at most once. Now given $g+1$ homologically independent and simple curves, there must be at least a pair that intersect. And a pair of intersecting simple curves necessarily intersects {\\it all} pants decompositions. Hence by the theorem above, there is no upper bound for such a family and so strong length inequality stops at exactly $g$ homologically independent curves. \n\nMore precise quantifications of the constants have also attracted attention. In particular the Buser-Sarnak logarithmic bound can be extended to roughly $ag$ curves for any $a<1$ \\cite{BPS}. \n\nFinally note that Gromov's observation above (on the intersection properties of minimal bases) still forces one to consider multiple, although finite, topological types of multicurves.\n\n\\item In a somewhat opposite direction, consider $\\{\\Gamma\\}_\\alpha$ to be the set of separating simple closed curves on a closed surface $\\Sigma$ of genus $g$. Then, as there exists a pants decomposition consisting only of non-separating curves, it will essentially intersect any element of the mapping class group orbit of any element of $\\Gamma$. Hence, for all $\\alpha\\in I$, the function $\\ell_{[\\Gamma_\\alpha]}(\\cdot)$ admits no upper bound over $\\mathcal{M}(\\Sigma)$. If however one takes the larger set $\\{\\Gamma\\}_\\alpha$ of all homologically trivial curves (but not homotopically trivial), then it is a consequence of a theorem of Sabourau \\cite{Sabourau} that this admits an upper bound that only depends on genus. This subtle difference lead to a small gap in Mirzakhani's study of the expected value of the shortest {\\it simple} homologically trivial curve in \\cite{Mirzakhani}, but this was not the difficult part of her work and has since been cleared up \\cite{PWX}.\n\n\\item For multicurves containing more than one curve, many of these problems admit variations. Here we suppose the length of a multicurve is the sum of lengths of its components, but of course one could also consider other variations, such as the $\\max$ or the product. For instance, the product of lengths of homologically distinct curves was studied in \\cite{Balacheff-Karam-Parlier}. Replacing the sum of the lengths with the maximum will also satisfy the same boundedness condition of Theorem \\ref{thm:maingeneral}, but products - or other functions of the lengths of the components - need to be checked on a case-by-case basis. \n\\end{enumerate}\n\n\\ack{This work was supported by the Luxembourg National Research Fund OPEN grant O19\/13865598.}\n\n\n\n\\chapter[The topological types of length bounded multicurves]{The topological types of length bounded multicurves}\n\n\n\\author{Hugo Parlier}\n\\authormark{H. Parlier}\n\n\n\\keywords{hyperbolic surfaces, curves, Teichm\\\"uller spaces, moduli spaces}\n\\subjclass{Primary 32G15, 52K20; secondary 53C22, 30F60}\n\n\n\n\n\\begin{abstract}\nThis article discusses inequalities on lengths of curves on hyperbolic surfaces. In particular, a characterization is given of which topological types of curves and multicurves always have a representative that satisfies a length inequality that holds over all of moduli space.\n\\end{abstract}\n\n\n\\section{Introduction}\n\nLength inequalities for curves play an important role in the understanding of hyperbolic surfaces and their moduli spaces. A prime example of this is a theorem of Bers, which states that any closed hyperbolic surface admits a pants decomposition of length bounded by a constant which only depends on the topology of the surface, and not its geometry. Short pants decompositions have been very useful in the understanding of the underlying Teichm\\\"uller space, and its large to medium scale geometry. This result generalizes bounds on the length of the shortest non-trivial closed curve (the systole). These are both examples of families of curves or multicurves that admit upper bounds over Teichm\\\"uller or moduli space of a given topological surface. More specifically, these results tell you that any hyperbolic surface of given topology admits a curve, or a multicurve, taken among a family of topological types, and which satisfies a certain length bound.\n\nThe goal of this short note is to characterize which families of curves or multicurves admit such upper bounds on their lengths. By multicurve, we mean a finite union of curves, all of them considered up to free homotopy, and by length we mean the length of a minimum length, thus geodesic, representative. The bounds are only allowed to depend on the topology of the underlying surface. Because of the nature of the game, we are only interested in the topological type of the curves, or said differently, in its mapping class group orbit. In order to try and satisfy the length inequality, we are allowed to choose curves of minimal length in the orbit. \n\nThe result is stated in terms of length functions. To a multicurve, we can associate the function, which takes a hyperbolic surface in the Teichm\\\"uller $\\mathcal{T}(\\Sigma)$ of a finite-type surface orientable $\\Sigma$, and which associates to it the length of the (unique) geodesic in the free homotopy class of the multicurve. The existence of the upper bounds described depends on whether certain length functions are bounded over Teichm\\\"uller or moduli space.\n\n\\begin{theorem}\\label{thm:main}\nLet $\\Gamma$ be a multicurve on $\\Sigma$. Then the quantity\n$$\\max_{X\\in \\mathcal{T}(\\Sigma)} \\min_{\\phi \\in \\mathrm{Mod}(\\Sigma)} \\ell_{\\phi(\\Gamma)}(X)$$\nis finite if and only if, for every pants decomposition $P$, there exists $\\phi$ in the mapping class group  $\\mathrm{Mod}(\\Sigma)$ such that $i(P, \\phi(\\Gamma))) = 0$.\n\\end{theorem}\nFor simplicity, it is stated only for multicurves but more generally it holds for families of multicurves (see Theorem \\ref{thm:maingeneral}). In more colloquial terms, what the result says is that if you are given a type of multicurve, then the function that associates to any hyperbolic surface the multicurve of minimal length of that type, is a bounded function over moduli space if and only if there is a multicurve of that type disjoint from any pants decomposition. \n\nNote that, given a lower bound on the systole, by compactness of pinched moduli space, there is similar conditional length inequality statement (Proposition \\ref{prop:conditionallength}) that holds for any multicurve, but the implied constants depend on topology, curve type and the lower bound of the systole.\n\nNote that here we are only interested in the very first value in a subset of the length spectrum. This is obviously very different from recent results about the asymptotic growth of curves of a given type (see \\cite{Erlandsson-Souto} and references therein). This is also very different from the related problems of finding precise constants, and in particular, exploring surfaces that are extremal for different geometric quantities. The focus here is on understanding what type of inequalities are possible. This is of course inspired by the many uses that have been made of these, or related, inequalities in the study of the large or medium scale of Teichm\\\"uller spaces with different metrics \\cite{Brock,Cavendish-Parlier} for the Weil-Petersson metric, \\cite{Rafi,Rafi-Tao,Papadopoulos-Theret} for the Teichm\\\"uller and Thurston metrics. \n\nFinally, note that this note is about upper bounds of length functions. If one replaces the $\\max$ with a $\\min$ in the above theorem, the quantity is strictly positive if and only if the multicurve has intersection (coming from a non-simple closed curve or pairwise intersecting curves). This is a consequence of the collar lemma \\cite{Keen} and generalisations \\cite{BasmajianCollar}. Lower bounds that depend on curve type have been studied in some detail by Basmajian \\cite{BasmajianUniversal}.\\\\\n\n\\noindent{\\bf Organization.} The article is organized as follows. The second section is mainly notation and definitions, and includes Proposition \\ref{prop:conditionallength}. The third and final section is dedicated to Theorem \\ref{thm:maingeneral} and ends with a discussion of which previously known length inequalities fall within its framework.\n\n\\section{Preliminaries and setup}\nThroughout $\\Sigma$ will be an orientable finite-type surface with $\\chi(\\Sigma) <0$. $\\Sigma$ is entirely determined by its genus $g$ and number of ends $n$, and $\\chi(\\Sigma) = 2-2g -n$.The space of marked complete hyperbolic structures on $\\Sigma$, that is Teichm\\\"uller space, will be denoted $\\mathcal{T}(\\Sigma)$, and should be thought of as a continuous deformation space of hyperbolic metrics. For the purpose of simplicity, we ask the metrics to be geodesically complete, and so the ends of $X\\in \\mathcal{T}(\\Sigma)$ are realized as cusps. The underlying moduli space, that is the space of hyperbolic structures on $\\Sigma$ up to isometry, will be denoted $\\mathcal{M}(\\Sigma)$. $\\mathrm{Mod}(\\Sigma)$ will be the (full) mapping class group of $\\Sigma$, that is the group of self-homeomorphisms of $\\Sigma$ up to isotopy. This group acts on $\\mathcal{T}(\\Sigma)$ and its quotient is $\\mathcal{M}(\\Sigma)$. \n\nFormally a curve is the continuous image of a circle on $\\Sigma$, but we will only be interested in a curve up to free homotopy. In particular, we only consider essential curves, that is those non-homotopic to a point or a boundary. A multicurve is a (finite) collection of curves. Associated to a multicurve $\\Gamma$ is a function which associates to $X\\in \\mathcal{T}(\\Sigma)$ the quantity $\\ell_\\Gamma(X)$, the length of the unique geodesic representative of $\\Gamma$ on $X$. These length functions are continuous, analytic and in fact convex \\cite{Kerckhoff, Wolpert}. \n\nIntersection between curves is defined as minimal intersection among representatives, and is denoted $i(\\cdot,\\cdot)$. A curve is simple if it has no self-intersections. A pants decomposition is a maximal collection of disjoint and distinct simple closed curves, and decomposes the surface into three-holed spheres (pairs of pants). The boundary curves of a pair of pants are sometimes called cuffs.\n\nThe following result \\cite{Keen} is stated here in non-quantitative terms for simple closed geodesics. Note that a version also holds for non-simple closed geodesics as well \\cite{BasmajianCollar}, with the notable difference being that you cannot pinch a non-simple closed curve. \n\n\\begin{lemma}[Collar lemma]\\label{lem:collar}\nA simple closed geodesic of length $\\ell$ on a hyperbolic surface $X$ admits a cylindrical neighborhood (its collar) of positive width $w(\\ell)$ which only depends on $\\ell$ and such that $w(\\ell) \\to \\infty$ when $\\ell \\to 0$. \n\\end{lemma}\n\nOur main use of the above result will be to consider hyperbolic structures where curves of a pants decomposition have length tending towards $0$, and hence all curves that are not among the cuffs of the given pants decomposition have length that tend to infinity. \n\nGiven $X\\in \\mathcal{T}(\\Sigma)$, the length of its shortest essential curve is its systole and is denoted $\\mathrm{sys}(X)$. Unless $X$ is a pair of pants, the systole is realized by a simple closed geodesic. For $\\epsilon>0$, the $\\epsilon$-thick part of Teichm\\\"uller space is the subset $\\mathcal{T}^{\\epsilon}(\\Sigma)\\subset \\mathcal{T}(\\Sigma)$ consisting of surfaces with $\\mathrm{sys}(X)\\geq \\epsilon$. By Mahler's compactness criterion, the corresponding thick part of moduli space $\\mathcal{M}^{\\epsilon}(\\Sigma)$ is compact.\n\nIn this paper, we study length inequalities, which here will be upper bounds for lengths of curves or multicurves with given properties. Generally these inequalities will be about a topological type of curve or multicurve: two multicurves $\\Gamma$ and $\\Gamma'$ are of the same type if there exists $\\phi\\in \\mathrm{Mod}(\\Sigma)$ such that $\\phi(\\Gamma)=\\Gamma'$. Given a multicurve $\\Gamma$, we can consider its mapping class group orbit. These orbits divide the space of all multicurves into equivalence classes\n$$\n[\\Gamma]:= \\{ \\phi(\\Gamma) \\mid \\phi \\in \\mathrm{Mod}(\\Sigma)\\}\n$$\nsorted by type. Now given $X\\in \\mathcal{T}(\\Sigma)$, we can study the length of a minimal representative of an equivalence class:\n$$\n\\ell_{[\\Gamma]}(X) = \\min_{\\phi \\in \\mathrm{Mod}(\\Sigma)} \\big\\{ \\ell_X(\\phi(\\Gamma)\\big\\}\n$$\nThe existence of a minimum follows from the discreteness of the length spectrum. This function, due to its mapping class group invariance, descends nicely to $\\mathcal{M}(\\Sigma)$. Note that although the function $\\ell_{[\\Gamma]}(\\cdot)$ remains continuous over $\\mathcal{M}(\\Sigma)$, it is no longer smooth. This is due to possible changes of homotopy classes realizing the minimum length of in an equivalence class.\n\nThe following result is a non-explicit general bound that holds for any topological type of multicurve. The proof is a compactness argument. The constant $K$ in the statement depends on the topology of $\\Sigma$, the topological type of $\\Gamma$ and a lower bound on the systole.\n\n\\begin{proposition}\\label{prop:conditionallength}\nLet $[\\Gamma]$ be a type of multicurve on $\\Sigma$. For any $\\epsilon>0$, there exists a constant $K=K(\\epsilon, \\Sigma, [\\Gamma])$ such that for any $X\\in \\mathcal{T}^{\\epsilon}(\\Sigma)$, we have\n$$\n\\ell_{[\\Gamma]}(X) \\leq K.\n$$\n\\end{proposition}\n\n\\begin{proof}\nThe space $\\mathcal{M}^{\\epsilon}(\\Sigma)$ is compact, hence the continuous function $\\ell_{[\\Gamma]}(\\cdot)$ admits a maximum on $\\mathcal{M}^{\\epsilon}(\\Sigma)$. This maximum value is exactly $K$. \n\\end{proof}\n\nAs an explicit example of the above result, consider the following result due to Buser and S\\\"eppala \\cite{Buser-Seppala}. For a closed surface $\\Sigma$ of genus $g\\geq 2$, they consider {\\it canonical homology bases}, that is collections of $2g$ simple closed curves $\\{\\alpha_i,\\beta_i\\}_{1\\leq i \\leq g}$ that satisfy:\n\\begin{enumerate}[a)]\n\\item\n$i(\\alpha_i,\\beta_j)=\\delta_{ij}$ for all $i,j\\in \\{1,\\hdots,g\\}$ (where $\\delta_{ij}$ is the Kronecker delta),\n\\item\n$i(\\alpha_i,\\alpha_j)=i(\\beta_i,\\beta_j)=0$ for $i\\neq j$. \n\\end{enumerate}\nNote that such a system of curves automatically generate integer homology. See Figure \\ref{fig:homology} for an illustration in genus $2$.\n\n\\begin{figure}[htbp]\n\\leavevmode \\SetLabels\n\\L(.23*.5) $\\alpha_1$\\\\%\n\\L(.32*.27) $\\beta_1$\\\\%\n\\L(.74*.5) $\\alpha_2$\\\\%\n\\L(.66*.27) $\\beta_2$\\\\%\n\\endSetLabels\n\\begin{center}\n\\AffixLabels{\\centerline{\\includegraphics[width=8cm]{Figures\/N-homology.pdf}}}\n\\caption{A canonical homology basis in genus $2$}\n\\label{fig:homology}\n\\end{center}\n\\vspace{-0.8cm}\n\\end{figure}\n\nThey prove that any $X\\in \\mathcal{T}^{\\epsilon}(\\Sigma)$ admits a canonical homology basis with all curves of length at most\n$$\n(g-1)\\left(45+6\\,\\mathrm{arcsinh}\\frac{1}{\\epsilon}\\right).\n$$\nThis is an improvement over an earlier quantification in \\cite{Buser-Seppala2}. The above bound shows that the $K$ in this instance is bounded above by \n$$\n2g(g-1)\\left(45+6\\,\\mathrm{arcsinh}\\frac{1}{\\epsilon}\\right)\n$$\nbecause there are $2g$ curves in the family. Of course, for this to be an exact quantification of the constant $K$, it would have to be sharp (which it is not). Exact quantifications are rarely known but, as proved in \\cite{Buser-Seppala}, the real constant must depend on $\\epsilon$. Alternatively, this dependency on $\\epsilon$ can also be deduced from Theorem \\ref{thm:main}.\n\nNaturally, this leads to the existence of upper bounds which only depend on topology and curve type, but not on systole length. A multicurve type $[\\Gamma]$ is said to satisfy a strong length inequality if $\\ell_{[\\Gamma]}(\\cdot)$ is upper bounded over $\\mathcal{M}(\\Sigma)$. By continuity of the length function $\\ell_{[\\Gamma]}(\\cdot)$, this is equivalent to the existence of a surface $X_{\\max}\\in \\mathcal{M}(\\Sigma)$ such that \n$$\n\\max_{X\\in \\mathcal{M}(\\Sigma)} \\ell_{[\\Gamma]}(X) = \\ell_{[\\Gamma]}(X_{\\max}).\n$$\nTo see this, one must show that the supremum of the length function cannot be reached on the boundary of moduli space, that is on a noded surface. As it turns out, as a consequence of Lemma \\ref{lem:stretchpants} from the next section, a (finite) supremum is always reached in the ''thick\" part of moduli space, and so in particular the $\\sup$ is indeed a $\\max$.\n\n\\section{Stretching pants, Bers' theorem and consequences}\n\nThe following lemma is by now well-known, but a sketch proof is provided for completeness. The proof uses strip maps, introduced by Thurston \\cite{Thurston}, and used to great effect by many authors \\cite{Danciger-Gueritaud-Kassel, Gueritaud, Papadopoulos-Theret, Parlier} to study of deformations of hyperbolic structures. Recall that a hyperbolic pair of pants is uniquely determined by its cuff lengths. In order to allow pants with cusp boundary, we use the convention that a cusp is a cuff of $0$ length. \n\n\\begin{lemma}[Pants stretching lemma]\\label{lem:stretchpants}\nLet $Y_{x,y,z}$ be the unique hyperbolic pair of pants with cuff lengths $x,y,z \\geq 0$. Then, for any (non-boundary) homotopy class of closed curve $\\gamma$ on a pair of pants, and any $\\epsilon>0$, we have\n$$\n\\ell_\\gamma\\left(Y_{x,y,z}\\right) < \\ell_\\gamma\\left(Y_{x+\\epsilon,y,z}\\right)\n$$\n\\end{lemma}\n\\begin{proof}[Sketch proof]\nConsider on $Y=Y_{x+\\epsilon,y,z}$, the simple orthogeodesic $a$ with both endpoints on the boundary curve of length $x+\\epsilon$ (see Figure \\ref{fig:arc}). \n\n\\begin{figure}[htbp]\n\\leavevmode \\SetLabels\n\\L(.445*.5) $a$\\\\%\n\\endSetLabels\n\\begin{center}\n\\AffixLabels{\\centerline{\\includegraphics[width=5cm]{Figures\/N-arc.pdf}}}\n\\caption{The orthogeodesic $a$}\n\\label{fig:arc}\n\\end{center}\n\\vspace{-0.8cm}\n\\end{figure}\n\nNote the closed geodesic $\\gamma$ intersects $a$ at least once. Exactly like in the collar lemma, $a$ admits an embedded neighborhood, topologically a strip (see Figure \\ref{fig:strip}). The idea is to now remove at least part of this strip to reduce the length of the boundary curve. \n\n\\begin{figure}[htbp]\n\\leavevmode \\SetLabels\n\\L(.445*.5) $a$\\\\%\n\\endSetLabels\n\\begin{center}\n\\AffixLabels{\\centerline{\\includegraphics[width=5cm]{Figures\/N-strip.pdf}}}\n\\caption{A strip surrounding $a$}\n\\label{fig:strip}\n\\end{center}\n\\vspace{-0.8cm}\n\\end{figure}\n\nTo do this properly, it is more convenient to consider the complete pair of pants by adding funnels. The arc $a$ can now be extended into a complete simple and infinite length geodesics. In addition, because the boundary curve of length $x+\\epsilon$ is not $0$, there is a family of simple complete geodesics parallel to each other and to $a$. We can take any two of these, cut away the strip between them, and paste them together to obtain a (complete) hyperbolic metric, say $Y'$ (see Figure \\ref{fig:operation}). (The slightly sketchy part is here: in fact it is possible by a variational argument to show that this can be done such that the length of the new boundary component of $Y'$ is exactly $x$, but we won't dwell on this, the main point being that the boundary length has been reduced.)\n\n\\begin{figure}[H]\n\\leavevmode \\SetLabels\n\\endSetLabels\n\\begin{center}\n\\AffixLabels{\\centerline{\\includegraphics[width=10cm]{Figures\/N-construction.pdf}}}\n\\caption{Removing the strip neighborhood $\\mathcal{S}(a)$}\n\\label{fig:operation}\n\\end{center}\n\\vspace{-0.8cm}\n\\end{figure}\n\nWe denote the strip enclosed by these geodesics by $\\mathcal{S}(a)$.\n\nNow can analyse the result of this operation on the length of $\\gamma$. Note that, due to the topology of the strip neighborhood $\\mathcal{S}(a)$ of $a$, $\\gamma \\cap \\mathcal{S}(a)$ is a union of simple geodesic arcs. We look at each one. In order to find a representative of $\\gamma$ on $Y'$, we replace each simple geodesic arc with its projection to $a$ (see Figure \\ref{fig:project}). The point is that the projection strictly reduces lengths.\n\\begin{figure}[H]\n\\leavevmode \\SetLabels\n\\endSetLabels\n\\begin{center}\n\\AffixLabels{\\centerline{\\includegraphics[width=5cm]{Figures\/N-project.pdf}}}\n\\caption{A local picture of $\\gamma$ under the operation}\n\\label{fig:project}\n\\end{center}\n\\vspace{-0.8cm}\n\\end{figure}\nThus, this results in a curve on $Y'$, in the same homotopy class, and of length strictly smaller. The corresponding geodesic is thus of length strictly smaller than it was previously. Hence we have $\\ell_\\gamma\\left(Y\\right) > \\ell_\\gamma\\left(Y'\\right)$ as required. \n\\end{proof}\nTo make the proof fully rigorous \nWe will mainly need an immediate corollary of the above result.\n\\begin{corollary}\\label{cor:longpants}\nFor $L>0$, let $Y$ be a pair of pants with cuff lengths between $0$ and $L$. Let $Y_L$ be the pair of pants with all cuff lengths exactly $L$. Then, for any interior homotopy class of curve $\\gamma$ on the pair of pants, we have $\\ell_\\gamma(Y_L) \\geq \\ell_\\gamma(Y)$ with equality only occurring if $Y=Y_L$. \n\\end{corollary}\n\nPants decompositions play an essential role in this story. The following result, originally due to Bers \\cite{Bers1,Bers2}, has been since quantified by different authors \\cite{BuserBook,BPS,ParlierShort}.\n\n\\begin{theorem}[Bers' constants]\\label{thm:bers}\nThere exists a constant $B(\\Sigma)$ such that any $X\\in \\mathcal{T}(\\Sigma)$ admits a pants decomposition with all curves of length at most $B(\\Sigma)$.\n\\end{theorem}\n\nNote that this result does not fall in the framework of either Proposition \\ref{prop:conditionallength} or Theorem \\ref{thm:main}. Indeed, in order to find a short pants decomposition, we are allowed to choose any topological type of pants decomposition. The number of these grows with topology (for instance there are roughly $\\sim g^g$ different types when $\\Sigma$ is of genus $g$ and $g$ is large). So here we are not only minimizing among mapping class group orbits of a fixed multicurve, but we are minimizing among multicurves that belong to a family. The following result holds for all such families of multicurves. A family will be denoted $\\{\\Gamma_\\alpha\\}_{\\alpha\\in I}$ where $I$ is an index set (possibly infinite, but countable as there are only countably many topological types of finite multicurve on a finite type surface). \n\n\\begin{theorem}\\label{thm:maingeneral}\nLet $\\{\\Gamma_\\alpha\\}_{\\alpha\\in I}$ be a family of multicurves. Then the quantity \n$$\\max_{X\\in \\mathcal{M}(\\Sigma)} \\min_{\\alpha\\in I} \\ell_{[\\Gamma_\\alpha]}(X)$$\nis finite if and only if, for every pants decomposition $P$, there exists $\\alpha\\in I$ and $\\phi \\in \\mathrm{Mod}(\\Sigma)$ such that $i(P, \\phi(\\Gamma_\\alpha)) = 0$. \n\\end{theorem}\n\nThe statement might seem a little confusing at first, due to the fact that we are taking a maximum among hyperbolic structures of a double minimum (over a family and then over the mapping class group orbit). If the family is reduced to a single curve, the statement becomes Theorem \\ref{thm:main} from the introduction. The proofs of both statements are identical, so we prove the more general statement above.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:maingeneral}]\n\nWe begin with the more straightforward direction, showing that if there is a pants decomposition $P$ which is intersected by any mapping class group orbit of any multicurve in the family $\\{\\Gamma_\\alpha\\}_{\\alpha\\in I}$, then there is no upper bound on the length. This follows directly from Lemma \\ref{lem:collar} (the collar lemma). Indeed, by considering a sequence of hyperbolic structures with all curves in the pants decomposition of $P$ that converge to $0$, the length of any curve that intersects one of the pants curve necessarily goes to $+\\infty$. As, by hypothesis, there is at least one curve in every multicurve $\\phi(\\Gamma_\\alpha)$, for all $\\phi \\in \\mathrm{Mod}(\\Sigma)$ and all $\\alpha \\in I$, that intersects a curve in $P$, the result follows. \n\nThe other direction will follow from Bers' theorem (Theorem \\ref{thm:bers}) and Corollary \\ref{cor:longpants} above.\n\nTake any $X\\in \\mathcal{T}(\\Sigma)$. By Bers' theorem, there exists a pants decomposition of $X$, say $P$, with all curves of length at most $B(\\Sigma)$. Now, by hypothesis, there exists $\\Gamma_\\alpha$ such that $i(\\Gamma_\\alpha,P)= 0$. Note that $\\Gamma_\\alpha$ can contain curves from $P$, but none that intersect curves of $P$ transversally. \n\nBy hypothesis all curves that belong to both $\\Gamma_\\alpha$ and $P$ are upper bounded by $B(\\Sigma)$. We now the corollary to all remaining curves of $\\Gamma_\\alpha$. Indeed, any such a curve $\\gamma \\in \\Gamma_\\alpha$ is contained in a pair of pants. By hypothesis, the cuff lengths of this pair of pants are at most $B(\\Sigma)$. By Corollary \\ref{cor:longpants}, $\\ell_X(\\gamma)$ is at most the length when the pair of pants has all cuff lengths equal to $B(\\Sigma)$, which is some finite number that depends on $B(\\Sigma)$ and the topological type of $\\gamma$. As there are a finite number of such curves, the result follows.\n\\end{proof}\n\nThere are many instances of the above theorem, each obtained by changing the multicurve or family of multicurves. Note that only the topological type of a multicurve matters in the statement. Some of these results are in the positive direction: the theorem implies that there is an upper bound on a minimal length representative that only depends on the topology of $\\Sigma$. Others are in the negative direction, that is that certain multicurves, or families of multicurves, do not admit such upper bounds. We give a (non-exhaustive) list of results of this type.\n\n\\begin{enumerate}[I.]\n\\item Even though it was used in the proof, Bers' theorem is an example obtained by taking the family of multicurves to be the full set of pants decompositions (or simply one pants decomposition for each topological type). To put in the framework of the theorem, the length of a pants decomposition should be defined as the sum, and not the maximum value. \n\nThe quantification of the implied constants - for both the sum and the max - has attracted some attention over the years, but it does not seem to be an easy problem. In fact, even the rough growth in terms of genus is not known \\cite{BuserBook,ParlierShort,GPY}. On the positive side, the exact constant in genus $2$ is known by a result of Gendulphe \\cite{Gendulphe}, as is the rough growth in terms of the number of cusps \\cite{Balacheff-Parlier,BPS}. \n\n\\item One can also take the full set of all topological types of all multicurves: this boils down to the systolic inequality. Quantifying the exact constants that appear is an arduous task. For orientable closed surfaces, the only constant known is again in genus $2$ \\cite{Jenni}. Buser and Sarnak \\cite{Buser-Sarnak} showed that the constants must grow logarithmically in genus, and since then there have been multiple variations and refinements of this (see for instance \\cite{BFR,KSV,Fanoni-Parlier, SchmutzSchaller}), including generalizations in the world of variable curvature and higher dimensional manifolds \\cite{Gromov}.\n\n\\item A slight variation of the above is to look at the homological systole of a closed surface $\\Sigma$. That is, the shortest curve that is not only non-trivial in homotopy, but also in homology. By cut and paste arguments, the homological systole is always realized by a non-separating simple closed curve. Hence, it is always in the same mapping class orbit. As above, few optimal constants are known. However, for closed surfaces, it is known that the optimal constants are equal to those from the systolic inequality. \\cite{ParlierPapa} Said differently, the systole of a maximal surface is homologically non-trivial.\n\n\\item In addition to the results of Buser and Sepp\\\"al\\\"a mentioned previously, one can try and bound families of homologically independent curves, but that do not necessarily form (part of) a canonical basis. If one requests a full homology basis, by the theorem above, there is no upper bound on its length over moduli space. However, by Bers' theorem, and the observation that any pants decomposition contains $g$ homologically independent curves, one can find an upper bound on the length of up until $g$ curves by a function of topology. So what about $g+1$ curves?\n\nGromov observed \\cite[Section 5]{GromovFilling} that any minimal length homology basis consists of simple curves that pairwise intersect at most once. Now given $g+1$ homologically independent and simple curves, there must be at least a pair that intersect. And a pair of intersecting simple curves necessarily intersects {\\it all} pants decompositions. Hence by the theorem above, there is no upper bound for such a family and so strong length inequality stops at exactly $g$ homologically independent curves. \n\nMore precise quantifications of the constants have also attracted attention. In particular the Buser-Sarnak logarithmic bound can be extended to roughly $ag$ curves for any $a<1$ \\cite{BPS}. \n\nFinally note that Gromov's observation above (on the intersection properties of minimal bases) still forces one to consider multiple, although finite, topological types of multicurves.\n\n\\item In a somewhat opposite direction, consider $\\{\\Gamma\\}_\\alpha$ to be the set of separating simple closed curves on a closed surface $\\Sigma$ of genus $g$. Then, as there exists a pants decomposition consisting only of non-separating curves, it will essentially intersect any element of the mapping class group orbit of any element of $\\Gamma$. Hence, for all $\\alpha\\in I$, the function $\\ell_{[\\Gamma_\\alpha]}(\\cdot)$ admits no upper bound over $\\mathcal{M}(\\Sigma)$. If however one takes the larger set $\\{\\Gamma\\}_\\alpha$ of all homologically trivial curves (but not homotopically trivial), then it is a consequence of a theorem of Sabourau \\cite{Sabourau} that this admits an upper bound that only depends on genus. This subtle difference lead to a small gap in Mirzakhani's study of the expected value of the shortest {\\it simple} homologically trivial curve in \\cite{Mirzakhani}, but this was not the difficult part of her work and has since been cleared up \\cite{PWX}.\n\n\\item For multicurves containing more than one curve, many of these problems admit variations. Here we suppose the length of a multicurve is the sum of lengths of its components, but of course one could also consider other variations, such as the $\\max$ or the product. For instance, the product of lengths of homologically distinct curves was studied in \\cite{Balacheff-Karam-Parlier}. Replacing the sum of the lengths with the maximum will also satisfy the same boundedness condition of Theorem \\ref{thm:maingeneral}, but products - or other functions of the lengths of the components - need to be checked on a case-by-case basis. \n\\end{enumerate}\n\n\\ack{This work was supported by the Luxembourg National Research Fund OPEN grant O19\/13865598.}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\n Repulsive gravitational effects are\nan essential ingredient of the Standard Cosmological Model (SCM), where they are evoked to overcome the difficulties faced by the Friedmann model, such as the horizon, flatness, and origin of the primordial density fluctuations problems \\cite{GUTH_1981, LINDE_1982, ALBRECHT_STEINHARDT_1982, LINDE_1983, LINDE_2005}, as well as the observed late-time accelerated expansion of the universe \\cite{RIESS_1998, PERLMUTTER_1999}. According to General Relativity (GR), when the minimal coupling principle is enforced, repulsive gravitational effects can only be generated by fluids with negative pressure. Such fluids are invariably behind inflationary models (see \\cite{LINDE_2005} and references therein) and dynamical dark energy models \\cite{DE_PDG_2013, COPELAND_2006, FRIEMAN_TURNER_HUTERER_2008}.\n\n A series of attempts to overcome the cosmological singularity problem, arguably the most severe difficulty faced by the Friedmann model (see \\cite{NOVELLO_BERGLIAFFA_2008} and references therein), were proposed in the past in which the alternative concept of repulsive gravitational effects generated by fields non-minimally coupled to gravity was introduced \\cite{LINDE_1979,NOVELLO_SALIM_1979, NOVELLO_1980}. In \\cite{LINDE_1979}, in particular, a conformally coupled Higgs-like field with a potential $V(\\phi) = m^2\\phi^2 + \\sigma\\phi^4$ ($\\sigma >0$) was responsible for a reversal of the effective gravitational constant in the early universe, when the density reached a critical value $\\rho_c$. Such model was later ruled-out on stability grounds, since as $\\rho \\to \\rho_c$ the effective gravitational constant diverges, $\\kappa_{eff} \\to \\infty$ \\cite{STAROBINSKY_1981}. Despite that, this idea paved the way for inflationary models based on non-minimally coupled Higgs-like scalar fields \\cite{SALOPEK_BOND_BARDEEN_1989, FAKIR_UNRUH_1990, KAISER_1995, KOMATSU_FUTAMASE_1999, BEZRUKOV_SHAPOSHNIKOV_2008, BARVINSKY_KAMENSHCHIK_STAROBINSKY_2008} (see \\cite{LINDE_2005} and references therein), and was the source of other gravitational repulsion models (see \\cite{HOHMANN_WOHLFARTH_2010} and references therein). As we shall see, that criticism does not apply here.\n\n\n In the present work we revisit a model \\cite{NOVELLO_1980}\nin which ordinary matter (radiation) can generate repulsive gravitational effects by intervention of a conformally coupled scalar field. The main idea can be synthesized in the following steps:\n\n\\begin{itemize}\n \\item[(i)] We assume the existence of a scalar field $\\phi$ which has a quartic self interaction potential with the form $V(\\phi) = m^2\\phi^2 -\\sigma \\phi^4$ (we call attention to the relative sign);\n \\item[(ii)] This scalar field couples non-minimally to gravity;\n \\item[(iii)] We select among all possible candidates of such a coupling a conformal one;\n \\item[(iv)] Matter interacts with the scalar field only through gravity;\n \\item[(v)] The symmetry of the theory is broken by a state $\\phi = \\phi_0 = $ constant;\n \\item[(vi)] When the field is in the broken symmetry state, the net result\non the equation for the metric is to produce an effective gravitational constant which, under certain conditions, has the opposite sign of the bare Newton's\ngravitational constant, provided that the energy-momentum tensor for the matter fields is traceless ($T=0$).\n \\item[(vii)] It will be shown that a necessary consequence of this process is that a ordinary radiation ($T=0$) dominated universe exhibits a bounce. \n\\end{itemize}\n\n\n\n\n\\section{The model}\n\n\n \\subsection{Non-minimally coupled scalar field}\n\n\n\nIn what follows, we adopt the space-time metric with signature $(+---)$.\nThe theory considered here is defined by a Lagrangian with a scalar field non-minimally (conformally) coupled to gravity \n\\begin{equation}\nL = \\sqrt{-g}\\left[ \\frac{1}{\\kappa}R + g^{\\alpha\\beta}\\partial_{\\alpha}\\phi^{\\ast}\\partial_{\\beta}\\phi - V(\\phi^{\\ast}\\phi)\n- \\frac{1}{6}R\\phi^{\\ast}\\phi + \\mathcal{L}_m \\right],\n\\label{lagrangian}\n\\end{equation}\nwhere $g_{\\alpha\\beta}$ are the components of the metric field, $g = \\mbox{det}(g_{\\alpha\\beta})$, $R$ is the Ricci scalar,\n$\\kappa = 8\\pi G$ is the reduced gravitational constant ($c=\\hslash = 1$), $\\mathcal{L}_m$ is the matter Lagrangian density,\nand the self-interaction potential of the scalar field is given by \\cite{NOVELLO_1980}\n\\begin{equation}\nV(\\phi^{\\ast}\\phi) = m^2\\phi^{\\ast}\\phi - \\sigma(\\phi^{\\ast}\\phi)^2 - 2V_0.\n\\end{equation}\nWe call attention for the fact that this potential differs from the usual ones employed in \\cite{LINDE_1979} or in inflation models (see \\cite{LINDE_2005}) by the relative sign between the mass term, $m^2\\phi^{\\ast}\\phi$, and the quartic self-interaction term, $\\sigma(\\phi^{\\ast}\\phi)^2$. \n\nThe set of field equations obtained from the Lagrangian (\\ref{lagrangian}) are\n\\begin{widetext}\n\\begin{subequations}\n \\begin{align}\n   R_{\\alpha\\beta} - \\frac{1}{2}Rg_{\\alpha\\beta} = & -\\frac{1}{{\\frac{1}{\\kappa} - \\frac{1}{6}\\phi^2}}\\left[ \\tau_{\\alpha\\beta}(\\phi) + T_{\\alpha\\beta} \\right],\n\\label{grav_field_eq}\n \\end{align}\n\\begin{equation}\n \\Box\\phi + m^2\\phi - 2\\sigma\\phi^3 + \\frac{1}{6}R\\phi = 0, \\label{scalar_field_eq}\n\\end{equation}\n\\end{subequations}\n\\end{widetext}\nwhere $\\Box \\equiv g^{\\alpha\\beta}\\nabla_{\\alpha}\\nabla_{\\beta}$, $\\phi^2 \\equiv \\phi^{\\ast}\\phi$, $\\phi^3 = (\\phi^{\\ast}\\phi)\\phi$, $T_{\\alpha\\beta}$ is the energy-momentum tensor of the matter fields, and\n\\begin{widetext}\n\\begin{align}\n \\tau_{\\alpha\\beta}(\\phi) \\equiv \\partial_{\\alpha}\\phi^{\\ast}\\partial_{\\beta}\\phi\n- \\frac{1}{2}\\Big( \\partial_{\\mu}\\phi^{\\ast}\\partial^{\\mu}\\phi - m^2\\phi^2 + \\sigma\\phi^4 \\Big)g_{\\alpha\\beta}\n- \\frac{1}{6}\\Big( \\nabla_{\\alpha}\\nabla_{\\beta}\\phi^2 - \\Box\\phi^2g_{\\alpha\\beta} \\Big) - V_0 g_{\\alpha\\beta} \\label{em_scalar}\n\\end{align}\n\\end{widetext}\nis the ``improved\" energy-momentum tensor of the scalar field.\nTaking the trace of the field equation (\\ref{grav_field_eq}),\nand using equation (\\ref{scalar_field_eq}), it follows that the Ricci scalar can be expressed as follows\n\\begin{equation}\n R = m^2\\phi^2 - 4V_0 + T,\n\\end{equation}\nwhere $T\\equiv g^{\\alpha\\beta}T_{\\alpha\\beta}$. This enables us to rewrite equation (\\ref{scalar_field_eq}) in the form\n\\begin{equation}\n  \\Box\\phi + \\left( m^2 - \\frac{2}{3}V_0 + \\frac{1}{6} T\\right)\\phi + \\left(\\frac{1}{6} m^2 - 2\\sigma\\right)\\phi^3 = 0. \\label{scalar_field_eq_b}\n \\end{equation}\n\n\n\n\n \\subsection{Broken symmetry and gravitational repulsion from ordinary matter}\n\n\n\nIt is clear from equation (\\ref{grav_field_eq}) that in the case where the ground state of the scalar field potential vanishes, $V_0=0$,\nthe improved energy-momentum tensor of the scalar field $\\tau_{\\alpha\\beta}(\\phi)$\nis irremediably not conserved. Following Callan et al. \\cite{CALLAN_COLEMAN_JACKIW_1970}, however, we can define the conserved energy-momentum tensor\n\\begin{equation}\n E_{\\alpha\\beta}(\\phi) \\equiv \\frac{1}{{\\frac{1}{\\kappa} - \\frac{1}{6}\\phi^2}}\\tau_{\\alpha\\beta}(\\phi). \\label{em_complex}\n\\end{equation}\nWe now look for a constant solutions of equation (\\ref{scalar_field_eq_b}) \nwhich correspond to stable vacua of the scalar field. Clearly, equation (\\ref{scalar_field_eq_b}) only admits a constant solution in the special case where the trace of the energy momentum-tensor of matter fields is a constant. \nWe stress the fact that the symmetry breaking process is only possible in this case, \notherwise no nontrivial ground state solution of equation (\\ref{scalar_field_eq_b}) exists.\nLet us concentrate, from now on, on\nmatter fields described by a traceless energy-momentum tensor, $T=0$.\n\nAccording to expressions (\\ref{em_complex})-(\\ref{em_scalar}), for a constant solution $\\phi=\\phi_0$ the energy density $E_{00}(\\phi) \\equiv E(\\phi)$\ncorresponding to the energy-momentum tensor (\\ref{em_complex}) has the form\n\\begin{equation}\nE(\\phi_0) = \\frac{1}{{\\frac{1}{\\kappa} - \\frac{1}{6}\\phi_0^2}}(m^2\\phi_0^2 - \\sigma\\phi_0^4 - 2V_0). \\label{energy_func}\n\\end{equation}\nOn the other hand, for $T= 0$ equation (\\ref{scalar_field_eq_b}) admits, besides the trivial solution $\\phi_0 = 0$, two constant solutions\n\\begin{equation}\n \\phi_0^2 = \\frac{6m^2 - 4V_0}{12\\sigma - m^2}. \\label{relation}\n\\end{equation}\nThe constant nontrivial solutions which minimizes the energy density functional (\\ref{energy_func}), and satisfies relation (\\ref{relation}),\nare given by\n\\begin{equation}\n \\phi_0 = \\pm \\frac{2\\sqrt{V_0}}{m}, \\ \\ \\ \\  \\sigma = \\frac{m^4}{8V_0}. \\label{const_sol}\n\\end{equation}\nThe resulting behavior of the energy density $E(\\phi_0)$ exhibits three uncommunicating regions, two of them containing the stable local minima\nat $\\phi_0 = \\pm 2\\sqrt{V_0}\/m$, to which correspond $E(\\phi_0) = 0$, the other\ncontaining two unstable maxima and a metastable minimum at $\\phi_0 = 0$ (see Fig \\ref{energy_fig}).\n\n\n\n\nConsequently, when the energy-momentum tensor for the matter fields is traceless ($T=0$), and the field is in a non-trivial stable ground state (\\ref{const_sol}), the gravitational field equation (\\ref{grav_field_eq}) assumes the form\n\\begin{align}\n  R_{\\alpha\\beta} = -\\kappa_{eff}T_{\\alpha\\beta} \\label{grav_field_eq_2},\n \\end{align}\nwhere\n\\begin{equation}\n\\kappa_{eff} \\equiv \\frac{1}{\\frac{1}{\\kappa} - \\frac{1}{6}\\phi_0^2} =  \\left(\\frac{3m^2\\kappa}{3m^2 - 2\\kappa V_0}\\right)\n\\end{equation}\nis the renormalized gravitational constant.\nThe term multiplying the energy-momentum tensor of matter (ordinary radiation) in equation (\\ref{grav_field_eq_2}) can, thus,\nbe viewed as a ``renormalized'' gravitational constant. Clearly, in the regime where the relation\n\\begin{equation}\n\\kappa V_0 > \\frac{3}{2}m^2 > 0 \\label{condition_mass}\n\\end{equation}\nis satisfied ({\\it i.e.} either the mass $m$ of the scalar field has to be very small, or $V_0$ has to be very large), gravity is reversed, {\\it i.e.} $\\kappa_{eff} < 0$. \n\n \\begin{figure}[H]\n   \\centering\n \\vspace{15mm}\n  \\includegraphics[scale=\n0.8 \n]{ENplots.eps} \n   \\caption{Plot of the energy density $E(\\phi_0)$, for $\\kappa V_0>3m^2\/2>0$.\nThe figure shows the solutions $\\phi_0 = \\pm 2\\sqrt{V_0}\/m$ corresponding to the nontrivial stable vacua,\nand also two asymptotes at $\\phi_0 = \\pm \\sqrt{6}$ separating three uncommunicating regionsa are indicated (dashed lines).} \\label{energy_fig}\n   \\end{figure} \n \n \n\\section{Bouncing cosmological model} \n\n\n  \\subsection{Scalar field induced bounce}\n  \n\nWe now investigate what cosmological solutions are compatible with the model discussed above. We assume spatial homogeneity and isotropy, and adopt a Friedmann metric for the space-time\n\\begin{equation}\nds^2 = dt^2 - a^2(t)\\left[ \\frac{dr^2}{1- \\epsilon r^2} + r^2 \\big(d\\theta^2 + \\sin^2\\theta d\\varphi^2\\big) \\right], \\label{friedmann_metric}\n\\end{equation}\nwhere $a(t)$ is the scale factor and $\\epsilon$ determine the geometry of the spatial section. In order to be the source of the Friedmann metric (\\ref{friedmann_metric}) the scalar field must depend on the cosmic time only, $\\phi = \\phi(t)$. We consider a radiation-dominated universe, $\\rho = 3p$ ($T= 0$).\n In the regime $\\phi = \\phi_0$ the field equations reduce to\n\\begin{subequations}\\label{cosm_eq}\n\\begin{equation}\n\\left( \\frac{\\dot{a}}{a} \\right)^2 + \\frac{\\epsilon}{a^2} = \\frac{1}{3}\\kappa_{eff} \\rho, \n\\end{equation}\n\\begin{equation}\n\\frac{\\ddot{a}}{a} = -\\frac{1}{6}\\kappa_{eff}(\\rho + 3p), \n\\end{equation}\n\\end{subequations}\nwhere $\\rho(t)$ and $p(t)$ are the density and pressure of the radiation fluid. In this case, if condition (\\ref{condition_mass}) holds, gravity is reversed, $\\kappa_{eff}<0$, and the field equations (\\ref{cosm_eq}) only admit a solution for an open spatial section $\\epsilon = -1$. Form energy conservation we have $\\rho(t) \\propto \\rho_0 a^{-4}(t)$, where $\\rho_0$ is a constant, so that the system of equations (\\ref{cosm_eq}) yield\n\\begin{equation}\n\\dot{a} = \\sqrt{ 1 - \\frac{1}{3}|\\kappa_{eff}| \\rho_0 a^{-2}}.\n\\end{equation}\nThis equation can be readily integrated, and we obtain the following form for the scale factor\n\\begin{equation}\na(t) = \\sqrt{ t^2  + a_0^2}. \\label{scale_fac_bounce}\n\\end{equation}\nTherefore, the universe in this model exhibits a bounce around $t = 0$, the constant $a_0 = \\sqrt{|\\kappa_{eff}| \\rho_0\/3}$ being the minimum value attainable by the scale factor. \n\nInterestingly, the bouncing solution obtained above coincides with a model based on a non-minimally coupled vector field proposed by one of the authors \\cite{NOVELLO_SALIM_1979}, even though the two models differ considerably (see appendix A).\n\n\n\n\n\\section{Final Remarks}\n\n\nIn the present work we revisited a model \\cite{NOVELLO_1980} in which a scalar field conformally coupled to gravity can generate\nrepulsive gravitational effects when only ordinary matter with traceless energy-momentum tensor (radiation) is coupled to gravity. It was shown that, in a radiation-dominated universe, when the scalar field is in a non-trivial ground (broken symmetry) state the only solution admissible by the field equations is a bouncing universe. \n\n\n\n\n\n\n\n \\begin{appendix}\n  \n \\section{Bouncing model generated by a vector field non-minimally coupled to gravity \\label{app_ns}}\n \n \n We include here a short review of the bouncing cosmological model proposed in \\cite{NOVELLO_SALIM_1979} and the one presented above for an easy comparison between them. The Lagrangian describing a vector field non-minimally coupled to gravity employed in \\cite{NOVELLO_SALIM_1979} is\n \\begin{equation}\n L = \\sqrt{-g}\\left[ \\frac{1}{\\kappa}R + \\beta RA_{\\mu}A^{\\mu} - \\frac{1}{4}F_{\\mu\\nu}F^{\\mu\\nu} \\right],\n \\label{lagrangian_d}\n \\end{equation}\nwhere $\\beta$ is a constant, and $F_{\\mu\\nu} = \\nabla_{\\mu}A_{\\nu} - \\nabla_{\\nu}A_{\\mu}$. The field equation are\n\\begin{widetext}\n\\begin{subequations} \\label{field_eq_app}\n \\begin{align}\n   R_{\\alpha\\beta} - \\frac{1}{2}Rg_{\\alpha\\beta} = & -\\frac{1}{{\\frac{1}{\\kappa} + \\beta A_{\\mu}A^{\\mu}}}\\tau_{\\alpha\\beta}(A^{\\mu}) ,\n \\end{align}\n\\begin{equation}\n \\nabla_{\\nu}F^{\\mu\\nu} + \\beta A^{\\mu} = 0, \n\\end{equation}\n\\end{subequations}\n\\end{widetext}\nwhere $\\tau_{\\alpha\\beta}(A^{\\mu})$ is the improved energy-momentum tensor of the vector field. \n\nIn a Friedmann geometry determined by the metric (\\ref{friedmann_metric}), making the choice $A_{\\mu} = A(t)\\delta^0_{\\mu}$, and defining $\\Omega(t) = \\frac{1}{\\kappa} + \\beta A^2(t)$, the set of field equations (\\ref{field_eq_app}) assume the form\n\\begin{subequations}\n\\begin{equation}\n3\\frac{\\ddot{a}}{a} = -\\frac{\\ddot{\\Omega}}{\\Omega},\n\\end{equation}\n\\begin{equation}\n\\frac{\\ddot{a}}{a} + 2\\left( \\frac{\\dot{a}}{a} \\right)^2 + 2\\frac{\\epsilon}{a^2} = -\\frac{\\dot{a}}{a}\\frac{\\dot{\\Omega}}{\\Omega},\n\\end{equation}\n\\begin{equation}\n\\Box\\Omega = 0.\n\\end{equation}\n\\end{subequations}\nA particular solution for an open spatial section, $\\epsilon = -1$, furnishes a scale factor with the form\n\\begin{equation}\na(t) = \\sqrt{ t^2  + a_0^2},\n\\end{equation}\nwhere $a_0$ is the minimum value of the scale factor. Although this is the same solution (\\ref{scale_fac_bounce} ) obtained for the scalar field, the two models differs in crucial points. First, the quantity $\\Omega(t)$, which is analogous to the effective gravitational constant of the model discussed in the main text, is not a constant but a function of the cosmic time (although a model similar to the one considered in the main text would arise in the special case $A_{\\mu}A^{\\mu} =$ constant). Second, here the non-minimally coupled vector field alone can be the source of the geometry, while in the bouncing model induced by the scalar field, in the absence of matter fields and for the scalar field in the non-trivial (broken symmetry) ground state, the gravitational field equations reduce to the vacuum Einstein equations.\n\n\n\\end{appendix}\n\n\n\n\\vspace{5mm}\n\n\\section*{Acknowlegments}\n\nThe authors would like to thank the Brazilian National Council of Technological and Scientific Development (CNPq) and the Research Support Foundation of the State of Rio de Janeiro (FAPERJ) for a grant. \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Bar formation: nature vs. nurture}  \nThe observational proofs about the influence of the environment on bar\nformation and\/or  evolution are still rare.   In \\citet{aguerri09}, we\ninvestigate  a volume-limited  sample  of 2106  disc  galaxies in  the\nnearby universe to derive the bar  fraction as a function of the local\ngalaxy density.  The local  density was\ncalculated for  every sample galaxy using the  fifth nearest neighbor\nmethod,  obtaining that  80\\% of  our  galaxies were  located in  very\nlow-density  environments  ($\\Sigma_5   <$  1  Mpc$^{-2}$),  and  20\\%\n(corresponding to  more than 400  galaxies) covers mostly  the typical\nvalues  measured for  loose ($\\Sigma_5  >$ 1  Mpc$^{-2}$)  and compact\ngalaxy groups  ($\\Sigma_5 \\sim$  10 Mpc$^{-2}$). We  did not  find any\ndifference  between the local  galaxy density  of barred  and unbarred\ngalaxies in our  range of densities.  \n\nTo  extend this  conclusion to  higher density  environments,  we have\ninvestigated the fraction of barred galaxies present in the two nearby\n\\emph{benchmark}  clusters: Virgo  (Zarattini et  al., in  prep.)  and\nComa (M\\'endez-Abreu et al., in  prep.).  In both clusters, we found a\nbar fraction in their bright galaxy population consistent with that in\nthe  field. Similar  results have  been recently  found in  cluster at\nhigher   redshift  by  Marinova   et  al.    (2009)  and   Barazza  et\nal.  (2009). However,  the errors  in the  clusters bar  fractions are\nlarge  and  the  results  might   be  affected  by  the  small  number\nstatistics. To deal with this  problem, we are undergoing an ambitious\nproject  to  study the  bar  fraction in  a  large  sample of  cluster\ngalaxies drawn from the WINGS project.\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\nModels of complex scalar matter fields coupled to gauge fields have\nbeen much studied in condensed matter physics, since they are believed\nto describe several interesting systems, such as superconductors and\nsuperfluids, quantum Hall states, quantum SU($N$) antiferromagnets,\nunconventional quantum phase transitions, etc., see, {\\em e.g.},\nRefs.~\\cite{SSNHS-03,SSS-04,BSA-04,MV-04,SBSVF-04,KHI-11} and\nreferences therein.  Scalar electrodynamics, or Abelian-Higgs (AH)\nmodel, is a paradigmatic model, in which a $N$-component complex\nscalar field ${\\bm \\Phi}$ is minimally coupled to the electromagnetic\nfield $A_\\mu$. The corresponding continuum Lagrangian reads\n\\begin{equation}\n{\\cal L} = \n|D_\\mu{\\bm\\Phi}|^2  + r\\, |{\\bm \\Phi}|^2 + \n{1\\over 6} u \\,(|{\\bm \\Phi}|^2)^2\n+ {1\\over 4 g^2} \\,F_{\\mu\\nu}^2 \n\\,,\n\\label{abhim}\n\\end{equation}\nwhere $F_{\\mu\\nu}\\equiv \\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu$, and\n$D_\\mu \\equiv \\partial_\\mu + i A_\\mu$.  The renormalization-group (RG)\nanalysis of the continuum AH model~\\cite{HLM-74,MZ-03} should provide\ninformation on the nature of the finite-temperature phase transitions\noccurring in $d$-dimensional systems characterized by a global U($N$)\nsymmetry and a local U(1) gauge symmetry.\n\nIn this paper we consider the multicomponent AH model, in which the\nscalar field ${\\bm\\Phi}$ has $N\\ge 2$ components. Such a model has a\nlocal U(1) gauge invariance and a global U($N$) invariance. We assume\nthat the field belongs to the fundamental representation of the U(1)\ngroup, i.e., it has charge 1. The one-component AH model has been\nextensively discussed in the literature\n\\cite{KKLP-98,KNS-02,MHS-02,WBJSS-05,CIS-06}. In three dimensions,\nthese systems may undergo continuous transitions in the XY\nuniversality class.\n\nLattice formulations of the three-dimensional AH model are obtained by\nassociating complex $N$-component unit vectors ${\\bm z}_{\\bm x}$ with\nthe sites ${\\bm x}$ of a cubic lattice, and U(1) variables\n$\\lambda_{{\\bm x},\\mu}$ with each link connecting the site ${\\bm x}$\nwith ${\\bm x}+\\hat\\mu$ (where $\\hat\\mu=\\hat{1},\\hat{2},\\ldots$ are\nunit vectors along the lattice directions).  The partition function of\nthe system reads\n\\begin{equation}\nZ = \\sum_{\\{{\\bm z}\\},\\{\\lambda\\}} e^{-H}\\,,\n\\label{partfun}\n\\end{equation}\nwhere the Hamiltonian is~\\cite{footnoteH}\n\\begin{eqnarray}\nH &=& - \\beta N\n\\sum_{{\\bm x}, \\mu}\n\\left( \\bar{\\bm{z}}_{\\bm x} \n\\cdot \\lambda_{{\\bm x},\\mu}\\, {\\bm z}_{{\\bm x}+\\hat\\mu} \n+ {\\rm c.c.}\\right)\n\\label{gllf}\\\\\n&-& \\beta_g \\sum_{{\\bm x},\\mu\\neq\\nu} \n\\left(\n\\lambda_{{\\bm x},{\\mu}} \\,\\lambda_{{\\bm x}+\\hat{\\mu},{\\nu}} \n\\,\\bar{\\lambda}_{{\\bm x}+\\hat{\\nu},{\\mu}}  \n  \\,\\bar{\\lambda}_{{\\bm x},{\\nu}} \n+ {\\rm c.c.}\\right)\\,,\n\\nonumber\n\\end{eqnarray}\nwhere the first sum is over all links of the lattice, while the second\none is over all plaquettes.  \n\nFor $\\beta_g=0$ we recover a particular lattice formulation of the\nCP$^{N-1}$ model, which is quadratic with respect to the spin\nvariables and contains explicit gauge link variables. The CP$^{N-1}$\nmodel has been extensively studied.  In spite of several\nfield-theoretical and numerical studies for $N=2,3,4$ and\n$N\\to\\infty$, there are still some controversies on the nature of its\ntransition \\cite{MZ-03,KHI-11,NCSOS-11,NCSOS-13,DPV-15,PTV-17,PV-19}.\nFor $\\beta_g\\to\\infty$ the gauge link variables are all equal to 1\nmodulo gauge transformations and the AH model becomes equivalent to\nthe standard O($n$) vector model with $n=2N$, whose critical behavior\nis well understood \\cite{PV-02}.  We also mention that some numerical\nresults for the AH lattice model (\\ref{gllf}) have been reported in\nRefs.~\\cite{KHI-11,ODHIM-07,TIM-05}, but a definite picture has not\nbeen achieved yet~\\cite{footnoteold}.\n\nIt is important to stress that we consider the lattice compact version\nof electrodynamics (the so-called Wilson lattice formulation of gauge\ntheories).  In the absence of matter fields, its behavior \\cite{QED}\nis controlled by topological excitations, the monopoles, which are\ninstead suppressed in noncompact formulations. Therefore, the critical\nproperties of the AH lattice model that we consider might differ from\nthose of the model in which gauge fields are noncompact.\n\nIn this paper we investigate the phase diagram and the nature of the\nphase transitions of the three-dimensional AH model (\\ref{gllf}).  We\nconsider systems with $N=2$ and $N=4$, and investigate the nature of\nthe transition line by varying $\\beta$ at fixed gauge coupling\n$\\beta_g$, for some values of $\\beta_g$.  In both cases the phase\ndiagram of the AH lattice model (\\ref{gllf}) turns out to present two\nphases: for small $\\beta$ there is a disordered confined phase, while\nfor large values of $\\beta$ there is an ordered Higgs phase in which\ncorrelations of the gauge-invariant hermitian operator $Q_{\\bm x}^{ab}\n= \\bar{z}_{\\bm x}^a {z}_{\\bm x}^b - \\delta^{ab}\/N$ show long-range\norder.  In both phases, and also along the transition line, the\ncorrelations of gauge variables do not show critical behaviors. The\ngauge coupling $\\beta_g$ does not play any significant role: the\nfeatures of two phases are the same for any finite $\\beta_g$.  The two\nphases are separated by a single transition line, which connects the\nCP$^{N-1}$ transition point ($\\beta_g = 0$) to the O($2N$) transition\npoint ($\\beta_g = \\infty$) in the space of the two parameters $\\beta$\nand $\\beta_g$.  For $\\beta_g=0$ the transition is continuous for $N=2$\n(belonging to the Heisenberg universality class) and of first order\nfor $N=4$.  We conjecture that the nature of the transitions along the\nline separating the Higgs and confined phases does not change with\n$\\beta_g$.  Therefore, the transition is always continuous\n(discontinuous) for $N=2$ ($N=4$).  We also observe significant\ndeviations for $\\beta_g$ large ($\\beta_g\\gtrsim 1$), i.e., when gauge\nfluctuations are suppressed. They are interpreted as a crossover\nphenomenon due to the presence of a O($2N$) vector transition in the\nlimit $\\beta_g\\to\\infty$.\n\nThe paper is organized as follows. In Sec.~\\ref{FTanalysis} we review\nthe field-theoretical results for the AH model. In Sec.~\\ref{rgeps} we\nreview the $\\varepsilon$-expansion predictions obtained in the\ncontinuum AH model, and in Sec.~\\ref{GLW} we present instead the\nresults of the Landau-Ginzburg-Wilson (LGW) approach based on a\ngauge-invariant order parameter. The two approaches are critically\ncompared in Sec.~\\ref{compFT}.  The numerical results are presented in\nSec.~\\ref{numres}.  The definitions of the quantities we consider are\ngiven in Sec.~\\ref{nusim}, while Sec.~\\ref{N2res} and \\ref{N4res}\npresent our results for $N=2$ and 4, respectively, focusing on the\nbehavior of the gauge-invariant order parameter.  Results for vector\nand gauge observables are presented in Sec.~\\ref{Vector}.  In\nSec.~\\ref{Conclusions} we summarize and present our conclusions.  In\nApp.~\\ref{FSSON} we discuss the limit $\\beta_g\\to\\infty$. More details\non the behavior of the different observables in this limit are given\nin the supplementary material \\cite{suppl-mat}.\n\n\\section{Field theoretical approaches}\n\\label{FTanalysis}\n\nIn this section we outline some apparently alternative\nfield-theoretical approaches which can be employed to infer the nature\nof the phase transitions in systems characterized by a U($N$) global\nsymmetry and a local U(1) gauge symmetry, such as the AH lattice\nmodel.\n\n\\subsection{Renormalization-group flow \nin the AH model close to four dimensions}\n\\label{rgeps}\n\nWe now summarize the main features of the RG flow in the continuum AH\nmodel (\\ref{abhim}), which has been analyzed close to four dimensions\nin the $\\varepsilon\\equiv 4-d$ expansion\nframework~\\cite{HLM-74,FH-96,IZMHS-19}, and in the large-$N$\nlimit~\\cite{MZ-03}.\n\nClose to four dimensions, the RG flow in the space of the renormalized\ncouplings $u$ and $f\\equiv g^2$ (we rescale them as $u\\to u\/(24\\pi^2)$\nand $f\\to f\/(24\\pi^2)$ to simplify the equations) can be computed in\nperturbation theory. At one loop, the $\\beta$ functions\nread~\\cite{HLM-74}\n\\begin{eqnarray}\n&&\\beta_u \\equiv \\mu {\\partial u\\over \\partial \\mu}\n= - \\varepsilon u + \n(N+4) u^2 - 18 u f + 54 f^2\\,,\n\\nonumber \\\\\n&&\\beta_f \\equiv \\mu {\\partial f \\over \\partial \\mu}\n= - \\varepsilon f + N f^2\\,.\n\\label{betafunc}\n\\end{eqnarray} \nOne can easily verify that a stable fixed point exists only for $N >\nN_c(\\varepsilon)$,  with \n\\begin{equation}\nN_c(\\varepsilon) =  N_{4}\n+ O(\\varepsilon)\\,,\\quad N_{4} = 90 +\n24\\sqrt{15} \\approx 183\\,.\n\\label{nceps}\n\\end{equation}\nThe corresponding zero of the $\\beta$ functions is \n\\begin{eqnarray}\n&&f^* = {\\varepsilon \\over N}\\,,\\label{fstar}\\\\\n&&u^*  = \n{N+18 + \\sqrt{N^2-180N-540}\\over 2 N(N+4)}\\,\\varepsilon\n\\approx {\\varepsilon\\over N}\\,.\n\\label{ufstar}\n\\end{eqnarray}\nThe presence of a stable fixed point indicates that these systems may\nundergo a continuous transition if $N$ is large enough [$N>N_c(1)$ in\n  three dimensions], in agreement with the direct large-$N$\nanalysis~\\cite{MZ-03}.  The qualitative picture obtained in the\none-loop calculation is not changed by higher-order calculations.  The\nperturbative expansion has been recently extended to four loops\n\\cite{IZMHS-19}, obtaining $N_c(\\varepsilon)$ to $O(\\varepsilon^3)$,\n\\begin{equation}\nN_c(\\varepsilon) = N_{4}\\left[1 - 1.752 \\,\\varepsilon + 0.789\\,\n\\varepsilon^2 + 0.362 \\,\\varepsilon^3+ O(\\varepsilon^4)\\right] .\n\\end{equation}\nThe large coefficients make a reliable three-dimensional\n($\\varepsilon=1$) estimate quite problematic.  Nevertheless, by means\nof a resummation of the expansion, Ref.~\\cite{IZMHS-19} obtained the\nestimate $N_c = 12.2(3.9)$ in three dimensions, which confirms the\nabsence of a stable fixed point for small values of $N$.\n\nIn the limit $\\beta_g \\to\\infty$, the lattice AH model (\\ref{gllf}) is\nequivalent to the symmetric O$(2N)$ vector theory. Therefore, for\nlarge $\\beta_g$ one expects significant crossover effects, which\nincrease as $\\beta_g$ increases, due to the nearby O$(2N)$ critical\nbehavior.  In the continuum AH model, the crossover is controlled by\nthe RG flow in the vicinity of the O$(2N)$ fixed point\n\\begin{equation}\nu^*_{{\\rm O}(2N)}= {1\\over N+4}  \\varepsilon\\,,\\qquad\nf=0\\,.\n\\label{O2nfp}\n\\end{equation}\nThis fixed point exists for any $N$ and is always unstable.  The\nanalysis of the stability matrix $\\Omega_{ij} = \\partial\n\\beta_i\/\\partial g_j$ shows that it has a positive eigenvalue\n$\\lambda_u=\\omega$, where $\\omega>0$ is the exponent controlling the\nleading scaling corrections in O$(2N)$ vector models~\\cite{PV-02}, and\na negative eigenvalue, which gives the dimension of the operator that\ncontrols the crossover behavior,\n\\begin{equation}\n\\lambda_f = \\left. {\\partial \\beta_f \\over \\partial f} \\right|_{f=0,u=u^*}\n \\; .\n\\end{equation}\nSince the $\\beta$-function $\\beta_f(u,f)$ \nassociated with $f$ has the general form\n\\begin{equation}\n\\beta_f = -\\varepsilon f + f^2 F(u,f)\n\\label{betafform}\n\\end{equation}\n where $F(u,f)$ has a regular perturbative expansion (see, e.g., the\n four-loop expansion reported in Ref.~\\cite{IZMHS-19}), we obtain\n\\begin{equation}\n\\lambda_f = -\\varepsilon\n\\label{lambdafest}\n\\end{equation}\nto all orders in perturbation theory. Therefore, the crossover exponent \n$y_f=-\\lambda_f$ is 1 in\nthree dimensions.  Note that these crossover features related to the\nunstable O($2N$) fixed point are independent of the existence of the\nstable fixed point, which is only relevant to predict the eventual\nasymptotic behavior.\n\n\\subsection{Gauge-invariant Landau-Ginzburg-Wilson framework}\n\\label{GLW}\n\nAn alternative field-theoretical approach is the LGW\nframework~\\cite{Landau-book,WK-74,Fisher-75,Ma-book,PV-02,PV-19}, in\nwhich one assumes that the relevant critical modes are associated with\nthe gauge-invariant local composite site variable\n\\begin{equation}\nQ_{{\\bm x}}^{ab} = \\bar{z}_{\\bm x}^a z_{\\bm x}^b - {1\\over N}\n\\delta^{ab},\n\\label{qdef}\n\\end{equation}\nwhich is a hermitian and traceless $N\\times N$ matrix.  As discussed\nin Refs.~\\cite{PTV-17,PTV-18,PV-19}, this is a highly nontrivial\nassumption, as it postulates that gauge\nfields do not play an important role in the effective theory.  \nThe order-parameter field in\nthe corresponding LGW theory is therefore a traceless hermitian matrix\nfield $\\Psi^{ab}({\\bm x})$, which can be formally defined as the\naverage of $Q_{\\bm x}^{ab}$ over a large but finite lattice domain.\nThe LGW field theory is obtained by considering the most general\nfourth-order polynomial in $\\Psi$ consistent with the U($N$) global\nsymmetry:\n\\begin{eqnarray}\n{\\cal H}_{\\rm LGW} &=& {\\rm Tr} (\\partial_\\mu \\Psi)^2 \n+ r \\,{\\rm Tr} \\,\\Psi^2 \\label{hlg}\\\\\n&+&   w \\,{\\rm tr} \\,\\Psi^3 \n+  \\,u\\, ({\\rm Tr} \\,\\Psi^2)^2  + v\\, {\\rm Tr}\\, \\Psi^4 .\n\\nonumber\n\\end{eqnarray}\nAlso in this framework continuous transitions may only arise if the RG\nflow in the LGW theory has a stable fixed point.\n\nFor $N=2$, the cubic term in Eq.~(\\ref{hlg}) vanishes and the two\nquartic terms are equivalent.  Therefore, one recovers the\nO(3)-symmetric vector LGW theory, leading to the prediction that the\nphase transition may be continuous and, in this case, that it belongs to\nthe Heisenberg universality class. For $N\\ge 3$, the cubic term is\ngenerally expected to be present.  Its presence is usually taken as an\nindication that phase transitions occurring in this class of systems\nare generally of first order.  Indeed, a straightforward mean-field\nanalysis shows that the transition is of first order in four\ndimensions where mean field applies. If statistical fluctuations are\nsmall---this is the basic assumption---the transition should be of\nfirst order also in three dimensions.  In this scenario, continuous\ntransitions may still occur, but they require a fine tuning of the\nmicroscopic parameters leading to the effective cancellation of the\ncubic term.  These arguments were originally \\cite{DPV-15,PV-19}\napplied to predict the behavior of CP$^{N-1}$ models. However, as they\nare only based on symmetry considerations, they can be extended to AH\nlattice models, as well.\n\n\\subsection{Comparison of the alternative field-theoretical approaches}\n\\label{compFT}\n\nThe two field-theoretical approaches outlined above give inconsistent\npredictions both for small and large values of $N$.  The contradiction\nis quite striking for the two-component $N=2$ case.  For this value of\n$N$, the continuum AH model predicts the absence of continuous\ntransitions, due to the absence of a stable fixed point. On the other\nhand, a stable fixed point---it is the usual Heisenberg O(3) fixed\npoint---exists in the effective LGW theory based on a gauge-invariant\norder parameter, leaving open the possibility of observing continuous\ntransitions (first-order transitions are never excluded as the\nstatistical model may be outside the attraction domain of the fixed\npoint). The numerical results for the CP$^1$ lattice\nmodels~\\cite{NCSOS-11,PV-19}, as well as the AH lattice results we\nshall present below, confirm the existence of continuous transitions\nin models with $N=2$: the LGW theory provides the correct description\nof the large-scale behavior of these systems. There are at least two\npossible explanations for the failure of the continuum AH model. A\nfirst possibility is that it does not encode the relevant degrees of\nfreedom at the transition. A second possibility is that the problem is\nnot in the continuum AH model, but rather in the perturbative\ntreatment around four dimensions. The three-dimensional fixed point\nmay not be analytically related to a four-dimensional fixed point, and\ntherefore it escapes any perturbative analysis in powers of\n$\\varepsilon$.\n\nWe also recall that the perturbative AH approach of Sec.~\\ref{rgeps}\nalso fails for $N=1$. Although no stable fixed point is identified in\nthe $\\varepsilon$ expansion, see Sec.~\\ref{rgeps}, these models may\nundergo continuous transitions in the XY universality class\n\\cite{KKLP-98,KNS-02,MHS-02}.  It is worth mentioning that there are\nalso other systems in which the $\\varepsilon$ expansion fails to\nprovide the correct physical picture in three dimensions. We mention\nthe $\\phi^4$ theories describing frustrated spin models with\nnoncollinear order~\\cite{CPPV-04,NO-14} and the $^3$He superfluid\ntransition from the normal to the planar phase~\\cite{DPV-04}.\n\nFor large values of $N$, the continuum AH theory and the effective LGW\napproach give again contradictory results. Indeed, the former approach\nindicates that continuous transitions are possible, a prediction which\nis supported by the large-$N$ analysis of lattice models, see, e.g.,\nRef.~\\cite{PV-19}.  If one trusts the argument based on the relevance\nof the cubic term, the LGW approach predicts instead a first-order\ntransition, unless a fine tuning of the microscopic parameters is\nperformed.  Again, there are two possible explanations for the\ndifferent conclusions obtained in the LGW approach.  A first\npossibility is that the critical modes at the transition are not\nexclusively associated with the gauge-invariant order parameter $Q$\ndefined in Eq.~(\\ref{qdef}). Other features, for instance the gauge\ndegrees of freedom, may become relevant, requiring an effective\ndescription different from that of the LGW theory (\\ref{hlg}). If this\ninterpretation is correct, the continuum AH model would be the correct\ntheory as it includes the gauge fields explicitly.  A second\npossibility is that the presence of a cubic term in the LGW\nHamiltonian does not necessarily imply the absence of continuous\ntransitions in three dimensions, as it is usually assumed. It might be\nthat statistical fluctuations soften the transition as one moves from\nfour to three dimensions; see, e.g., Refs.~\\cite{NCSOS-11,NCSOS-13} for\na discussion of this issue.\n\nWhile the two field-theoretical approaches give different predictions\nfor $N=1,2$ and $N$ large (more precisely, for $N>N_c$, see\nSec.~\\ref{rgeps}), for $3 \\le N < N_c$ they both predict that all\nmodels undergo a first-order transition. For $N=3$ simulation results\ndo not presently confirm it.  Indeed, while numerical results for the\nlattice model with Hamiltonian (\\ref{gllf}) and $\\beta_g = 0$ show a\nrobust indication that the transition is of first order~\\cite{PV-19},\nthe results for the loop model considered in\nRefs.~\\cite{NCSOS-11,NCSOS-13} apparently favor a continuous\ntransition.  The available numerical results for lattice CP$^3$\nmodels, i.e., for $N=4$, are generally consistent with first-order\ntransitions~\\cite{KHI-11,NCSOS-13,PV-19}.  We also mention that\nRef.~\\cite{KHI-11} claims that the AH lattice model (\\ref{gllf})\nundergoes a continuous transition for $\\beta_g=1$ and $N=4$, a result\nwhich is at odds with the above arguments. However, as we shall show,\nthe numerical results that we present later do not confirm their\nconclusions, but are instead consistent with a relatively weak\nfirst-order transition.\n\n\n\\section{Numerical results}\n\\label{numres}\n\n\\subsection{Numerical simulations and observables}\n\\label{nusim}\n\n\\begin{figure}[tbp]\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4}]{betac.eps}\n\\caption{ The phase diagram of the AH lattice model in the space of\n  parameters $\\beta$ and $\\beta_g$, for $N=2$ and $N=4$. The points\n  are the MC estimates of the critical points, the dotted lines that\n  connect them are only meant to guide the eye.  The horizontal lines\n  indicate the limiting values of $\\beta_c$ for $\\beta_g\\to \\infty$\n  for $N=2$ ($\\beta_c\\approx 0.23396$, dashed) and $N=4$\n  ($\\beta_c\\approx 0.24084$, dot-dashed): they correspond to the\n  critical $\\beta_c$ for the standard three-dimensional O$(4)$ and\n  O$(8)$ vector models, respectively. For $\\beta_g = 0$, we have \\cite{PV-19}\n  $\\beta_c = 0.7102(1)$ ($N=2$) and $\\beta_c = 0.5636(1)$ ($N=4$).\n}\n\\label{betac}\n\\end{figure}\n\n\nIn this section we present a finite-size scaling (FSS) analysis of\nnumerical results of Monte Carlo (MC) simulations for $N=2$ and $N=4$.\nFor this purpose we consider cubic lattices of linear size $L$ with\nperiodic boundary conditions. We study the behavior of the system as a\nfunction of $\\beta$ at fixed $\\beta_g$.\n\nThe linearity of Hamiltonian (\\ref{gllf}) with respect to each lattice\nvariable allows us to employ an overrelaxed algorithm for the updating\nof the lattice configurations.  It consists in a stochastic mixing of\nmicrocanonical and standard Metropolis updates of the lattice\nvariables~\\cite{CRV-92,DMV-04,Hasenbusch-17}.  To update each lattice\nvariable, we randomly choose either a standard Metropolis update,\nwhich ensures ergodicity, or a microcanonical move, which is more\nefficient than the Metropolis one but does not change the energy. On\naverage, we perform three\/four microcanonical updates for every\nMetropolis proposal.  In the Metropolis update, changes are tuned so\nthat the acceptance is 1\/3.\n\nWe compute the energy density and the specific heat, defined as\n\\begin{eqnarray}\nE = {1\\over N V} \\langle H \\rangle,\\qquad\nC ={1\\over N^2 V}\n\\left( \\langle H^2 \\rangle \n- \\langle H  \\rangle^2\\right),\n\\label{ecvdef}\n\\end{eqnarray}\nwhere $V=L^3$.  We consider correlations of the hermitean gauge\ninvariant operator (\\ref{qdef}). Its two-point correlation function is\ndefined as\n\\begin{equation}\nG({\\bm x}-{\\bm y}) = \\langle {\\rm Tr}\\, Q_{\\bm x} Q_{\\bm y} \\rangle,  \n\\label{gxyp}\n\\end{equation}\nwhere the translation invariance of the system has been taken into\naccount.  The susceptibility and the correlation length are defined as\n$\\chi=\\sum_{\\bm x} G({\\bm x})$ and\n\\begin{eqnarray}\n\\xi^2 \\equiv  {1\\over 4 \\sin^2 (\\pi\/L)}\n{\\widetilde{G}({\\bm 0}) - \\widetilde{G}({\\bm p}_m)\\over \n\\widetilde{G}({\\bm p}_m)},\n\\label{xidefpb}\n\\end{eqnarray}\nwhere $\\widetilde{G}({\\bm p})=\\sum_{{\\bm x}} e^{i{\\bm p}\\cdot {\\bm x}}\nG({\\bm x})$ is the Fourier transform of $G({\\bm x})$, and ${\\bm p}_m =\n(2\\pi\/L,0,0)$ is the minimum nonzero lattice momentum.  We also\nconsider the Binder parameter\n\\begin{equation}\nU = {\\langle \\mu_2^2\\rangle \\over \\langle \\mu_2 \\rangle^2} \\,, \\qquad\n\\mu_2 = \n\\sum_{{\\bm x},{\\bm y}} {\\rm Tr}\\,Q_{\\bm x} Q_{\\bm y}\\,.\n\\label{binderdef}\n\\end{equation}\nWe consider correlations of the fundamental variable ${\\bm z}_x$.\nTo obtain a gauge-invariant quantity, we consider correlations with\n$\\lambda$ strings, i.e., averages like\n\\begin{equation}\n \\hbox{Re } \\left\\langle  \n  \\bar{\\bm z}_{\\bm x}\\cdot {\\bm z}_{\\bm y} \\prod_{\\ell \\in \\cal C} \\lambda_\\ell\n   \\right\\rangle,\n\\end{equation}\nwhere the product extends over the link variables that belong to a\nlattice path $\\cal C$ connecting points $\\bm x$ and $\\bm y$. To\ndefine quantities that have the correct FSS, the path ${\\cal C}$ must\nbe chosen appropriately, as discussed in Ref.~\\cite{APV-08}. Here, to\nsimplify the calculations, we only consider correlations between\npoints that belong to lattice straight lines.  We define\n\\begin{eqnarray}\nG_V(d,L) ={1\\over V} \n   \\sum_{{\\bm x}} \n  \\hbox{Re}\\left\\langle\n  \\bar{\\bm z}_{\\bm x}\\cdot {\\bm z}_{{\\bm x}+d \\hat{\\mu}}  \n   \\prod_{n=0}^{d-1} \\lambda_{{\\bm x}+n \\hat{\\mu},\\mu}\n   \\right\\rangle,\n   \\label{Gd}\n\\end{eqnarray}\nwhere all coordinates should be taken modulo $L$ because of the\nperiodic boundary conditions.  Note that in the definition of $G_V$ we\naverage over all lattice sites ${\\bm x}$ exploiting the translation\ninvariance of systems with periodic boundary conditions, and select a generic\nlattice direction $\\hat{\\mu}$ (in our MC simulations we also average\nover the three equivalent directions).  Note also that $G_V(0,L) = 1$\nand that $G_V(L,L)$ is the average value $P(L)$ of the Polyakov loop,\n\\begin{equation}\n   P(L) = {1\\over V} \\sum_{\\bm x} \n   \\hbox{Re } \\left\\langle \\prod_{n=0}^{L-1} \\lambda_{{\\bm x}+n \\hat{\\mu},\\mu}\n   \\right\\rangle\\,.\n   \\label{Polyakov}\n\\end{equation}\nFinally, we consider the so-called Wilson loop\ndefined as\n\\begin{equation}\n    W(m,L) = \\hbox{Re } \\left\\langle \\prod_{\\ell\\in \\cal C}\n    \\lambda_\\ell \\right\\rangle, \n    \\label{Wilson}\n\\end{equation}\nwhere the path $\\cal C$ is a square of linear size $m$.\n\nIn the following we present a FSS analysis of the above observables,\nfor $N=2$ and $N=4$ and some values of $\\beta_g>0$. In\nFig.~\\ref{betac} we anticipate the resulting phase diagrams.  For both\n$N=2$ and $4$, $\\beta_c$ decreases as $\\beta_g$ increases and,\neventually, it converges to the value appropriate for the $n$-vector\nmodel with $n=4$ and 8, $\\beta_c=0.233965(2)$~\\cite{BFMM-96,CPRV-96}\nand $\\beta_c = 0.24084(1)$~\\cite{DPV-15}.\n\nAs we shall discuss, our numerical data are consistent with a simple\nscenario in which the nature of the transitions along the line\nseparating the confined and Higgs phases is unchanged for any finite\n$\\beta_g\\ge 0$.  Therefore, for $N=2$ the phase transitions are continuous and\nbelong to the Heisenberg universality class as it occurs in the CP$^1$\nmodel.  The O(4) critical behavior occurs only for $\\beta_g$ strictly\nequal to $\\infty$. For $N=4$ instead, transitions are of first order,\nexcept for $\\beta_g=\\infty$, where the system develops an O(8)\nvector critical behavior.\n\n\n\n\\subsection{Continuous transitions for $N=2$}\n\\label{N2res}\n\n\nAs already mentioned, lattice versions of the three-dimensional CP$^1$\nmodel undergo continuous transitions belonging to the Heisenberg\nuniversality class, i.e., that of the standard $N=3$ vector model.\nThis has been also shown~\\cite{PV-19} for model\n(\\ref{gllf}) with $\\beta_g=0$ [$\\beta_c = 0.7102(1)$ in this case].\nOn the other hand, for $\\beta_g=\\infty$ the model is equivalent \nto the standard O(4) vector model that has a continuous transition for \n\\cite{BFMM-96,CPRV-96} $\\beta_c=0.233965(2)$. \nAs already inferred\nfrom the RG flow of the AH continuum theory, the $\\beta_g=\\infty$ O(4)\ncritical behavior is expected to be unstable against perturbations\nassociated with nonzero values of $\\beta_g^{-1}$. Therefore, the most\nnatural hypothesis is that the all transitions for finite $\\beta_g\\ge\n0$ belong to the Heisenberg universality class.  However, a\nsubstantial crossover from the O(4) to the O(3) behavior is expected\nto characterize the transition for relatively large values of\n$\\beta_g$, $\\beta_g\\gtrsim 1$ say.\n\n\\begin{figure}[tbp]\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4}]{rxi-n2-c1.eps}\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4}]{rxi-n2-c1o2.eps}\n\\caption{ $R_\\xi$ versus $\\beta$ for the $N=2$ AH lattice model, for\n  $\\beta_g=0.5$ (bottom) and $\\beta_g=1$ (top).  In both cases the\n  data for different values of $L$ show a crossing point, whose\n  position provides an estimate of the critical point:\n  $\\beta_c=0.4145(5)$ and $\\beta_c=0.276(1)$ for $\\beta_g=0.5$ and\n  $\\beta_g=1$, respectively.  }\n\\label{rxi-n2}\n\\end{figure}\n\nTo provide evidence of this scenario, we perform MC simulations for\n$\\beta_g=0.5$ and 1. As in our previous work \\cite{PV-19}, we study\nthe FSS behavior of the Binder parameter $U$ and of $R_\\xi = \\xi\/L$.\nAt continuous transitions the FSS limit is obtained by taking\n$\\beta\\to \\beta_c$ and $L\\to\\infty$ keeping \n\\begin{equation}\nX \\equiv (\\beta-\\beta_c)L^{1\/\\nu}\n\\label{Xdef}\n\\end{equation}\nfixed. Any RG invariant quantity $R$, such\nas $R_\\xi\\equiv \\xi\/L$ and $U$, is expected to asymptotically behave\nas\n\\begin{eqnarray}\nR(\\beta,L) = f_R(X) + O(L^{-\\omega})\\,,\n\\label{rsca}\n\\end{eqnarray}\nwhere $\\omega>0$ is the leading scaling correction\nexponent~\\cite{PV-02}, and $f_R(X)$ is universal apart from a\nnormalization of its argument.  The function $f_R(X)$ only depends on\nthe shape of the lattice and on the boundary conditions.  In the case\nof the Heisenberg universality we\nhave~\\cite{GZ-98,PV-02,CHPRV-02,HV-11} $\\nu=0.7117(5)$ and\n$\\omega=0.78(1)$.  As $R_\\xi$ is monotonically increasing as a\nfunction of $X$, Eq.~(\\ref{rsca}) implies that\n\\begin{equation}\nU = F(R_\\xi) + O(L^{-\\omega})\\,,\n\\label{r12sca}\n\\end{equation}\nwhere $F(x)$ is a universal scaling function.  As in our previous work\n\\cite{PV-19}, we will use Eq.~(\\ref{r12sca}) to perform a direct check\nof universality, because no model-dependent normalizations enter: If\ntwo models belong to the same universality class, the data for both of\nthem should collapse onto the same curve as $L$ increases.  The only\ndifficulty in the approach is that one should be careful in\nidentifying corresponding operators in the two models.\n\nTo identify the correct operators, one may reason as follows. In the AH\nlattice model the basic quantity that we consider is the local\noperator (\\ref{qdef}).  To identify the corresponding operator in the\nHeisenberg model, we use the explicit relation between the CP$^1$ and\nthe O(3) vector model. Under the mapping, the parameter $U$ and $\\xi$\ncorrespond to the usual O(3) vector Binder parameter and correlation\nlength (i.e., computed from correlations of the fundamental spin\nvariable ${\\bm s}_{\\bm x}$). The mapping of the large-$\\beta_g$ limit\nof the AH lattice model into the O(4) vector model is instead more\ncomplex and is discussed in detail in App.~\\ref{FSSON} and in the\nsupplementary material \\cite{suppl-mat}. The correspondence is not\ntrivial and $U$ is identified with a combination of suitably defined\nO(4) tensor Binder parameters.\n\n\\begin{figure}[H]\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4}]{bi-rxi-n2-c0.eps}\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4}]{bi-rxi-n2-c1o2.eps}\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4}]{bi-rxi-n2-c1.eps}\n\\caption{ The Binder parameter $U$ versus $R_\\xi$ for the $N=2$ AH\n  lattice model, and for $\\beta_g=0$ (top, data from\n  Ref.~\\cite{PV-19}), $\\beta_g=0.5$ (middle) and $\\beta_g=1$ (bottom).\n  In all panels the dashed line is the Heisenberg curve, as obtained\n  from MC simulations of the O(3) vector model.  The dot-dashed line\n  in the lower panel is the limiting curve for\n  $\\beta_g\\to\\infty$ (see App.~\\ref{FSSON}).  The inset enlarges the\n  region $R_\\xi<0.25$, showing that the nonmotonic behavior that\n  characterizes the small-size data (similar to the one of the O(4)\n  curve) for $R_\\xi\\approx 0.1$ disappears with increasing $L$.  The\n  horizontal dashed line shows the asymptotic value $U(R_\\xi\\to 0) =\n  5\/3$.  }\n  \\label{bi-rxi-n2}\n\\end{figure}\n\n\nIn Fig.~\\ref{rxi-n2} we plot $R_\\xi$ versus\n$\\beta$, for several values of $L$.  The data for different values of\nthe size $L$ show crossing points, which provide estimates of the\ncritical point: $\\beta_c = 0.4145(5)$ and $\\beta_c=0.276(1)$ for\n$\\beta_g=0.5$ and $\\beta_g=1$, respectively.  Data are consistent with\na continuous transition. \n\nWe now argue that the transitions are consistent with the expected\nasymptotic Heisenberg behavior.  The best evidence is provided by the\nplots of $U$ versus $R_\\xi$, see Fig.~\\ref{bi-rxi-n2}. In all panels\nwe report the data and the corresponding O(3) curve. If our simple\nscenario is correct, the data for all values of $\\beta_g$ must\napproach the O(3) curve with increasing $L$.  For $\\beta_g = 0$ we\nobserve very good agreement, as already discussed in\nRef.~\\cite{PV-19}. For $\\beta_g = 0.5$ convergence is slower,\nindicating that scaling corrections increase with increasing\n$\\beta_g$.  For $R_\\xi \\lesssim 0.25$ we observe a good collapse of\nthe data, while in the opposite case, we observe a clear upward trend,\nconsistent with an asymptotic O(3) behavior.  For $\\beta_g = 1$, for\nsmall values of $L$ we observe significant differences between data\nand O(3) curve.  These discrepancies can be explained by scaling\ncorrections. For $R_\\xi\\lesssim 0.25$, the results for $L=64$ fall on\ntop of the O(3) scaling curve, as predicted. For larger values of\n$R_\\xi$, crossover effects are stronger, but the trend of the data is\nthe expected one.\n\nTo be more quantitative, let us note that, for large values of $L$,\nthe Binder parameter $U$ should behave as~\\cite{PV-02}\n\\begin{equation}\nU(\\beta,\\beta_g,L) = F(R_\\xi) + a(\\beta_g)\\ G(R_\\xi)\\ L^{-\\omega} + \\ldots\n\\label{corrections}\n\\end{equation}\nwhere $F(R_\\xi)$ is the O(3) scaling function, $G(R_\\xi)$ is a\nuniversal function, and $a(\\beta_g)$ is a constant that encodes the\n$\\beta_g$-dependent size of the leading scaling corrections decaying\nas $L^{-\\omega}$.  We have verified that our data for $\\beta_g = 0.5$\nand 1 are consistent with Eq.~(\\ref{corrections}), if we take $\\omega\n= 0.78$ (the leading correction-to-scaling exponent in Heisenberg\nsystems \\cite{GZ-98,PV-02,CHPRV-02,HV-11}) and $a(1)\/a(0.5) \\approx\n5$.  This can be checked from Fig.~\\ref{differenze}, where we report\n\\begin{equation}\n\\Delta(\\beta,\\beta_g,L) = \n    {1\\over a(\\beta_g)} L^\\omega [U(\\beta,\\beta_g,L) - F(R_\\xi)]\\,,\n\\label{deltadef}    \n\\end{equation}\nwhere $F(R_\\xi)$ has been determined in the O(3) vector model, $\\omega\n= 0.78$, $a(1) = 5$, and $a(0.5) = 1$. All data reported in the figure\nare consistent with a single scaling curve that would be identified\nwith the function $G(R_\\xi)$ in Eq.~(\\ref{corrections}).  The\nexistence of similar crossover effects for $\\beta_g = 0.5$ and 1 is\nanother demonstration of universality.\n\n\\begin{figure}[!ht]\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4}]{scalcorr-n2.eps}\n\\caption{The quantity $\\Delta(\\beta,\\beta_g,L)$ defined in \nEq.~(\\ref{deltadef}) versus $R_\\xi$. We report data for $\\beta_g=1$ \nand 0.5 and several values of $L$.}\n\\label{differenze}\n\\end{figure}\n\nIt is interesting to note that the behavior of the data for small $L$\nat $\\beta_g=1$ can be interpreted as due to the presence of O(4) fixed\npoint that controls the critical behavior for $\\beta_g\\to\\infty$.  In\nthe lower panel of Fig.~\\ref{bi-rxi-n2}, we also plot the O(4) scaling\ncurve.  The data for $\\beta_g=1$ apparently follow the O(4) curve for\nsmall lattice sizes, and then move toward the O(3) curve with\nincreasing $L$.  In particular, note the nonmotonic behavior of the\ndata for small lattice sizes and $R_\\xi\\approx 0.1$, similar to the\none that characterizes the O(4) curve.  Such a behavior disappears\nwith increasing $L$ (see the inset in the lower panel of\nFig.~\\ref{bi-rxi-n2}).\n\nOn the basis of the above numerical results we argue that the\nfinite-temperature transition is continuous for any finite $\\beta_g\\ge\n0$ and belongs to the Heisenberg universality class.  However, for\nrelatively large values of $\\beta_g$, say $\\beta_g\\gtrsim 1$, notable\ncrossover effects emerge.  They are apparently related to the presence\nof the O(4) fixed point, which is the relevant one $\\beta_g\\to\\infty$.\nFor large values of $\\beta_g$, such effects may hide the asymptotic\nHeisenberg behavior. For intermediate sizes, data are expected to show\nan effective O(4) critical behavior, converging to the Heisenberg\nbehavior only for very large lattices.\n\n\n\n\\subsection{First-order transitions for $N=4$}\n\\label{N4res}\n\n\\begin{figure}[tbp]\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4}]{u-n4-c0.eps}\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4}]{u-n4-c1o2.eps}\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4}]{u-n4-c1.eps}\n\\caption{Plot of the Binder parameter $U$ versus $\\beta$, for\n  $\\beta_g=0$ (top, from Ref.~\\cite{PV-19}), $\\beta_g=1\/2$ (middle),\n  and $\\beta_g=1$ (bottom), for $N=4$.  The vertical lines correspond\n  to the estimates of the transition points.  The horizontal dashed\n  lines show the values $U(\\beta \\to 0) = 17\/15$ and\n  $U(\\beta\\to \\infty) = 1$.  } \n\\label{U-N4}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4}]{bi-rxi-n4-c0.eps}\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4}]{bi-rxi-n4-c1o2.eps}\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4}]{bi-rxi-n4-c1.eps}\n\\caption{ The Binder parameter $U$ versus $R_\\xi$ versus $\\beta$, for\n  $\\beta_g=0$ (top, from Ref.~\\cite{PV-19}), $\\beta_g=1\/2$ (middle),\n  and $\\beta_g=1$ (bottom) for $N=4$.  The dashed line in the lower\n  panel is the O(8) limiting scaling curve, see App.~\\ref{FSSON}.  The\n  horizontal dashed lines show the asymptotic values $U(R_\\xi\\to 0) =\n  17\/15$ and $U(R_\\xi\\to \\infty) = 1$.  }\n\\label{UvsR-N4}\n\\end{figure}\n\n\nWe now discuss the behavior of the $N=4$ AH lattice model, providing\nevidence that the transitions along the line separating the Higgs and\nconfined phases are of first order for any finite $\\beta_g$. Only when\n$\\beta_g$ is strictly infinity is the transition continuous: it\nbelongs to the O(8) vector universality class.\n\nAs shown in Ref.~\\cite{PV-19}, the transition is of first order for\n$\\beta_g=0$.  To show that the nature of the transition is unchanged\nfor $\\beta_g>0$, we first consider the specific heat and the Binder\nparameter $U$. Both of them are expected to increase as the volume at\na first-order transition.  Indeed, according to the standard\nphenomenological theory~\\cite{CLB-86}, for a lattice of size $L$ there\nexists a value $\\beta_{{\\rm max},C}(L)$ of $\\beta$ where $C$ takes its\nmaximum value $C_{\\rm max}(L)$, which asymptotically increases as\n\\begin{eqnarray}\n&&C_{\\rm max}(L) = V\\left[ {1\\over 4} \\Delta_h^2 + O(1\/V)\\right]\\,,\n\\label{cmaxsc}\\\\\n&&\\beta_{{\\rm max},C}(L)-\\beta_c\\approx c\\,V^{-1}\\,, \\label{betamax}\n\\end{eqnarray}\nwhere $V=L^d$ and $\\Delta_h$ is the latent heat\n[defined as $\\Delta_h = E(\\beta\\to\\beta_c^+) - E(\\beta\\to\\beta_c^-)$].\nAnalogously, the\nbehavior of the Binder parameter $U(\\beta,L)$ is expected to show a\nmaximum $U_{\\rm max}(L)$ at fixed $L$ (for sufficiently large $L$) at\n$\\beta = \\beta_{{\\rm max},U}(L) < \\beta_c$\nwith~\\cite{VRSB-93,CPPV-04,PV-19}\n\\begin{eqnarray}\n&&U_{\\rm max} \\sim a\\,V + O(1)\\,,\\label{umaxsca}\\\\ &&\\beta_{{\\rm\n    max},U}(L) - \\beta_c \\approx b \\,V^{-1}\\,.\n\\label{bpeakU}\n\\end{eqnarray}\nThe previous relations are valid in the asymptotic limit and, for weak\ntransitions, require data on large lattices.  As we discussed in\nRef.~\\cite{PV-19}, one can identify first-order transitions on\nsignificantly smaller lattices from the analysis of the behavior of\nthe Binder parameter $U$. In the presence of a first-order transition,\none observes large violations of the scaling relation (\\ref{r12sca})\nfor values of $L$ that are significantly smaller than those at which\nrelations (\\ref{cmaxsc}) and (\\ref{bpeakU}) hold. We will follow this\napproach here, considering again two values of $\\beta_g$, 0.5 and 1.\n\nIn Fig.~\\ref{U-N4} we report numerical estimates of $U$ at $\\beta_g=0$\n(taken from Ref.~\\cite{PV-19}), $\\beta_g=0.5$, and\n$\\beta_g=1$. Clearly, the maximum $U_{\\rm max}$ increases with\nincreasing $L$, as expected for a first-order transition. However,\nwith increasing $\\beta_g$, the rate of increase becomes smaller,\nindicating that the transition becomes weaker. The specific heat\nbehaves analogously.\n\nTo obtain a better evidence that the finite-size behavior is not\ncompatible with a continuous transition, we plot $U$ versus $R_\\xi$,\nsee Fig.~\\ref{UvsR-N4}. Data do not show any scaling behavior, as\nexpected at a first-order transition.\n\n\\begin{figure}[tbp]\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4},angle=0]{bi-rxi-n4-c4.eps}\n\\caption{ The Binder parameter $U$ versus $R_\\xi$ for $\\beta_g = 4$\n  and $N=4$.  The full line connects the data with $L=16$.  The dashed\n  line is the O(8) limiting scaling curve, see App.~\\ref{FSSON}.  }\n\\label{U-N4-c4}\n\\end{figure}\n\n\nNote that for $\\beta_g = 1$ the small size data show an apparent\nscaling behavior for small values of $R_\\xi$ and small $L$, which may\nlead to erroneous conclusions when limiting the FSS analysis to small\nlattices (as in Ref.~\\cite{KHI-11}).  To clarify the origin of the\ntransient effects, and understand whether they can be interpreted as\ndue to the O(8) fixed point that controls the behavior for $\\beta_g =\n\\infty$, we have performed MC simulations for $\\beta_g =\n4$. This value is so large that, for our range of values of $L$, we do\nnot expect to observe effects related to the first-order nature of the\ntransition and therefore all data should be in the crossover\nregion. The analysis of $U$ as a function of $\\beta$ allows us to\nestimate $\\beta_c = 0.2484(2)$, which is close to the O(8) value,\n$\\beta_c \\approx 0.2408$. At the transition, gauge fields are\nsignificantly ordered and indeed, the average value of the product of\nthe gauge fields along an elementary plaquette (a Wilson loop of size\n1) is 0.95780(5) (for comparison, such a product is equal to 0.8235(5)\nat the transition for $\\beta_g = 1$). In Fig.~\\ref{U-N4-c4} we report\n$U$ versus $R_\\xi$ and compare it with the O(8) curve. The numerical\ndata with $16\\le L \\le 48$ apparently fall onto a single scaling\ncurve, while the data corresponding to $L=64$ and 96 begin to show the\ndrift that characterizes the results at $\\beta_g \\le 1$ and which is\nrelated to the asymptotic first-order nature of the transition.  The\napparent scaling curve for small values of $L$ is different from the\nO(8) one, indicating that for $\\beta_g = 4$ we are observing a sizable\ncontribution due the relevant operator that destabilizes the O(8)\nfixed point for finite $\\beta_g$. We can also infer from the\nsubstantial stability of the results for $L\\le 48$ that it has a very\nsmall (positive) scaling dimensions $y$.  Indeed, close to the O(8)\nfixed point, we expect\n\\begin{equation}\n  U(\\beta,\\beta_g) = F(R_\\xi,b(\\beta_g,\\beta) L^y) \\,,\n\\end{equation}\nwhere $b(\\beta_g,\\beta)$ is a nonuniversal amplitude, which vanishes\nfor $\\beta_g \\to \\infty$. For each $\\beta_g$ the crossover region is\nthe one in which $b(\\beta_g,\\beta_c) L^y \\ll 1$. If this condition\nholds, $U$ can be written as\n\\begin{equation}\n  U(\\beta,\\beta_g) = F(R_\\xi,0) + b(\\beta_g,\\beta) L^y G(R_\\xi)\\,,\n\\end{equation}\nwhere the first term is the O(8) scaling function.  This equation\nwould imply that the deviations from the O(8) behavior scale, at least\nfor $\\beta_g$ very large, as $L^y$. Our results therefore imply that\n$y$ should be small enough, so that $L^y$ does not change\nsignificantly as $L$ varies from 16 to 48.\n\nIn conclusion, the numerical results favor a phase diagram based on a\nfirst-order transition line for $\\beta_g>0$, starting from the\nfirst-order transition of the CP$^3$ models, corresponding to\n$\\beta_g=0$.  With increasing $\\beta_g$ the first-order transition\nbecomes weaker.  We observe substantial crossover phenomena for\n$\\beta_g\\gtrsim 1$.  They may be explained in terms of the O(8) fixed\npoint controlling the behavior for $\\beta_g\\to\\infty$, perturbed by a\nrelevant operator with a relatively small scaling dimension.\n\n\n\n\\subsection{Vector and gauge observables} \\label{Vector}\n\nIn the previous sections we discussed the behavior of quantities\ndefined in terms of the gauge-invariant order parameter $Q_{\\bm\n  x}^{ab}$. Here we discuss instead the vector correlation function\n(\\ref{Gd}) and the gauge observables (\\ref{Polyakov}) and\n(\\ref{Wilson}). We focus on $N=4$.\n\n\\begin{figure}[tbp]\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4},angle=0]{gdcrit-beta0p8.eps}\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4},angle=0]{xiz-beta0p8.eps}\n\\caption{Top: Vector correlation function $G_V(x,L)$ versus $x$ for\n  $\\beta=0.8$ and several values fo $\\beta_g$. Bottom: Vector\n  correlation length $\\xi_z$ as a function of $\\beta_g$ for $\\beta =\n  0.8$. The line shows that $\\xi_z$ scales as $\\beta_g$ for large\n  $\\beta_g$ (the parameters have been determined by performing a\n  linear fit of the data with $\\beta_g \\ge 1.6$).  For $x=L$, $G_V(L,L)$ \n  corresponds to the average of the Polyakov loop.\n}\n\\label{Gdbeta0p8}\n\\end{figure}\n\n\\begin{figure}[tbp]\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4},angle=0]{gdcrit-c0p5.eps}\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4},angle=0]{gdcrit-c1.eps}\n\\includegraphics*[scale=\\twocolumn@sw{0.3}{0.4},angle=0]{gdcrit-c4.eps}\n\\caption{The vector correlation function $G_V(x,L)$ in the critical\n  region for $\\beta_g = 0.5$ (top), 1 (middle) and 4 (bottom). For\n  $\\beta_g = 0.5$ we report the estimate for $\\beta = 0.365$\n  (high-temperature phase) and for $\\beta = 0.370$ (low-temperature\n  phase). For $\\beta_g = 1$ and 4, we report an effective estimate of\n  the correlation function in the coexistence region (see text for a\n  discussion), computed at $\\beta = 0.279$ for $\\beta_g = 1$, and at\n  $\\beta = 0.248$ for $\\beta_g = 4$.  }\n\\label{Gdcrit}\n\\end{figure}\n\nLet us first discuss their behavior in the two different phases. In\nthe high-temperature phase $\\beta < \\beta_c$, we find that the\ncorrelation function can be approximated as ($x > 0$)\n\\begin{equation}\n  G_V(x,L) = A e^{-x\/\\xi_z}\\,,\n\\label{Gd-exp}\n\\end{equation}\nas soon as $x$ is 2 or 3. Moreover, for small $\\beta$, $\\xi_z$ is very\nlittle dependent on $\\beta_g$.  For instance, for $\\beta = 0.1$, the\nstrong-coupling behavior $G_V(x,L) \\sim (N\\beta)^x$ holds quite precisely\nfor all values of $\\beta_g$. Wilson loops behave in a very similar\nfashion. We find $W(m,L) \\approx B \\exp(-4 m\/\\xi_w)$ with $\\xi_w\n\\approx \\xi_z$ as long as $m\\gtrsim 2$.  Clearly, in the\nhigh-temperature phase a single gauge mode controls the behavior of\nall observables that involve gauge degrees of freedom.\n\nThe behavior in the low-temperature phase is analogous.  The\ncorrelation function $G_V(x,L)$ behaves as in Eq.~(\\ref{Gd-exp}); see\nthe upper panel of Fig.~\\ref{Gdbeta0p8} for results at $\\beta = 0.8$.\nMoreover, Polyakov and Wilson loops satisfy\n\\begin{equation}\nP(L) = A e^{-L\/\\xi_z}\\,, \\qquad W(m) = B e^{-4 m \/\\xi_z}\\,,\n\\label{P-exp}\n\\end{equation}\nwith the same correlation length and $A,B\\approx 1$.\nFig.~\\ref{Gdbeta0p8} also shows that $G_V(x,L)$ has a very precise\nexponential decay even when $\\xi_z \\gtrsim L$. Clearly, it couples to\na single isolated mode and hence there are no corrections to the\nleading exponential behavior.  In this phase the correlation length\nincreases with $\\beta_g$ (see the lower panel of\nFig.~\\ref{Gdbeta0p8}): in agreement with perturbation theory, it\nscales linearly with $\\beta_g$ in the limit $\\beta_g \\to \\infty$. Note\nthe $\\xi_z$ is also expected to diverge in the limit $\\beta\\to \\infty$\nat fixed $\\beta_g$. Indeed, for $\\beta\\to \\infty$, the relevant\nconfigurations are those that minimize the Hamiltonian term that\ndepends on the fields ${\\bm z}$. If we perform a local minimization on\neach link, we find the constraint\n\\begin{equation}\n    z_{{\\bm x} + \\hat{\\mu}} = \\bar\\lambda_{{\\bm x},\\mu} z_{\\bm x}\\,.\n\\end{equation}\nThis constraint can be satisfied simultaneously on the four links\nbelonging to a plaquette only if the product of the gauge fields along\nthe plaquette is 1. Analogously, the constraint is satisfied on the\nlinks that belong to a loop that wraps around the lattice only if the\nPolyakov operator is 1. It follows that gauge configurations are\ntrivial---$\\lambda_{{\\bm x},\\mu}$ is 1 on all links modulo gauge\ntransformations---and $\\xi_z$ is infinite in this limit.\n\nThese results for the gauge observables indicate that gauge and vector\nobservables are noncritical in both phases and that their behavior is\nanalogous for small and large values of $\\beta$.  Only the limit\n$\\beta_g\\to \\infty$ distinguishes the two sectors. If\n$\\beta_{c,\\infty}$ is the transition point for $\\beta_g\\to \\infty$\n[therefore in the O($2N$) theory], for $\\beta_g\\to \\infty$, the\ncorrelation length $\\xi_z$ is finite for $\\beta < \\beta_{c,\\infty}$\nand infinite in the opposite case. This guarantees that vector\ncorrelations are critical in the O($2N$) theory with a finite\nlow-temperature magnetization.  But this only occurs for $\\beta_g$\nstrictly equal to infinity. For finite $\\beta_g$, only $Q$\ncorrelations display criticality.\n\nFinally, let us discuss the behavior of vector and gauge quantities\nalong the transition line.  For $N=4$, as we are dealing with\nfirst-order transitions, we expect $G_V(x,L)$ to depend on the phase\none considers. In the CP$^3$ model ($\\beta_g = 0$) the transition is\nstrong and therefore the high-temperature (HT) and low-temperature\n(LT) correlation functions can be easily computed by fixing $\\beta$ in\nthe coexistence region and starting the simulation from a random or an\nordered configuration.  We find that in both cases the correlation\nfunction decays very rapidly and estimate $\\xi_z \\approx 1.9$ and\n$\\xi_z \\approx 1.7$ in the LT and HT phase, respectively.  Clearly,\nvector modes are not critical. Similar results hold for the CP$^1$ and\nCP$^2$ models. In the first case, we obtain $\\xi_z \\approx 2.2 $.  For\n$N=3$ the transition is so weak that we cannot identify the two phases\nand we are only able to compute an effective correlation function,\nwhich is a linear combination of those appropriate for the two phases.\nThis quantity still allows us to compute the largest of the two\ncorrelation lengths, i.e., $\\xi_z$ in the LT phase, obtaining $\\xi_z\n\\approx 2.2$.  Results for finite $\\beta_g$ are reported in\nFig.~\\ref{Gdcrit}. The first distinctive feature is that, for the\ncases we consider, the correlation function does not behave as a\nsingle exponential, although an exponential behavior sets when $x \\gg\n\\xi_z$.  Clearly, at the critical point several modes are playing an\nimportant role and an exponential behavior is only observed when\n$\\xi_z$ is significantly less than $L$.  Second, the correlation\nlength $\\xi_z$ increases with increasing $\\beta_g$ along the\ntransition line. For $\\beta_g = 0.5$ we obtain $\\xi_z \\approx 3.8$,\n3.6, 3.1 for $L=32$, 48, and 64 in the LT phase (runs at $\\beta =\n0.370$ with ordered start).  In the HT phase (runs at $\\beta = 0.365$)\nwe obtain $\\xi_z \\approx 2.2$ with a small $L$ dependence.  Although\n$\\xi_z$ is small, it is larger than the value it takes in the CP$^3$\nmodel, i.e. the AH model with $\\beta_g=0$.  For $\\beta_g = 1$, we are\nnot able to distinguish the two phases and, therefore, we only compute\nthe LT estimate of $\\xi_z$. Results for $L=32,48,64$ essentially agree\nand give $\\xi_z \\approx 6.9$. This estimate is confirmed by the\nanalysis of the Polyakov loop.  A fit to Eq.~(\\ref{P-exp}) gives\n$\\xi_z = 6.9(1)$, in very good agreement with the results obtained\nfrom $G_V(x)$.  For $\\beta_g = 4$, even for $L = 96$ we are not yet in\nthe regime in which one can reliably identify a range of distances in\nwhich the correlation function decays exponentially. If we fit the\ncorrelation function to Eq.~(\\ref{Gd-exp}) in the range $L\/3 \\le 2\nL\/3$, we obtain $\\xi_z = 17.1(1)$ and 17.6(1) for $L=64$ and 96,\nrespectively.  The analysis of the Polyakov loop gives a somewhat\nlarger value $\\xi_z = 20.9(3)$. Whatever the exact asymptotic result is,\ndata confirm that, for $\\beta_g = 4$, we are deep in the crossover\nregion, where vector and gauge excitations compete with\ngauge-invariant excitations associated with $Q_{\\bm x}$ (for\ncomparison note that $\\xi = 20.3(3)$ for $L=96$).  These results\nprovide us a physical explanation of the crossover effects we\nobserve. The asymptotic first-order behavior is only observed when the\ncorrelation length $\\xi(L)$ at the transition point is significantly\nlarger than $\\xi_z$. When $\\xi(L) \\sim \\xi_z$ we observe an apparent\nscaling behavior in which both the (gauge-field independent) degrees\nof freedom associated with $Q$ and the (gauge-field dependent) ones,\nthat are encoded in the gauge observables and in the vector\ncorrelations, are both relevant.\n\n\\section{Conclusions} \\label{Conclusions}\n\nWe have studied the phase diagram and critical behavior of\nmuticomponent AH lattice models, in which an $N$-component complex\nfield ${\\bm z}_{\\bm x}$ is coupled to quantum electrodynamics.  We\nconsider the compact Wilson formulation of Abelian lattice gauge\ntheories in which the fundamental gauge fields are complex numbers of\nunit modulus, see Eq.~(\\ref{gllf}).  For the scalar fields, we\nconsider the unit-length limit and fix $|{\\bm z}_{\\bm x}|^2 =\n1$. Finally, we fix $q=1$ for the charge of the matter fields. We\nfocus on systems with a small number of components, considering $N=2$\nand $N=4$.\n\n\nWe investigate the phase diagram of the model as a function of the\ncouplings $\\beta$ and $\\beta_g$. The phase diagram is characterized by\ntwo phases: a low-temperature phase (large $\\beta$) in which the order\nparameter $Q^{ab}$ condenses, and a high-temperature disordered phase\n(small $\\beta$).  The gauge coupling does not play any particular role\nin the two phases: gauge observables and vector observables do not\nshow long-range correlations for any finite $\\beta$ and $\\beta_g$. The\ntwo phases are separated by a transition line that connects the\nCP$^{N-1}$ transition point ($\\beta_g = 0$) with the O($2N$)\ntransition point ($\\beta_g = \\infty$).  Concerning the nature of the\ntransition line, our numerical data are consistent with a simple\nscenario, in which the nature of the transition line is independent of\n$\\beta_g$.  Therefore, we predict Heisenberg critical behaviors along\nthe whole line for $N=2$, and first-order transitions for $N=4$. Note\nthat, for $\\beta_g\\to \\infty$ the model becomes equivalent to the\nO($2N$) vector model, and therefore one expects strong crossover\neffects controlled by the O($2N$) fixed point.  These crossover\neffects are related to the presence of a second length scale $\\xi_z$\nassociated with the vector correlations, which is finite for any\n$\\beta_g$ and diverges in the limit $\\beta_g\\to\\infty$ in the whole\nlow-temperature phase. \n\nThe scenario supported by our numerical data is fully consistent with\nthe LGW approach that assumes a gauge-invariant order parameter. On\nthe other hand, at least for $N=2$, it disagrees with the\n$\\varepsilon$-expansion predictions obtained using the standard\ncontinuum AH model: for $N=2$ this approach does not predict a\ncontinuous transition.  Numerical results allow us to understand why\nthe LGW approach is more appropriate than the continuum AH model for\nthese values of $N$.  At the transition (for both $N=2$ and $N=4$)\nonly correlations of the gauge-invariant operator $Q^{ab}$ display\nlong-range order. Gauge modes represent a background that gives only\nrise to crossover effects and indeed, the asymptotic behavior sets in\nonly when the correlation length of the gauge fluctuations is\nnegligible compared to that of the $Q$ correlations.  It is important\nto note that a LGW approach based on a gauge-invariant order parameter\nhas also been applied to the study of phase transitions in the\npresence of nonabelian gauge symmetries, and, in particular, to the\nstudy of the finite-temperature transition of hadronic matter as\ndescribed by the theory of strong interactions, quantum chromodynamics\n\\cite{PW-84,BPV-03,PV-13}. Our results for the AH lattice model lend\nsupport to the correctness of the approach and of the predictions\nobtained.\n\nWe expect that AH lattice models with higher (but not too large)\nvalues of $N$ have a phase diagram similar to the one obtained for\n$N=4$, with a first-order transition line separating the ordered and\ndisordered phases. The phase diagram may change for large values of\n$N$. In this regime, the system may undergo continuous transitions\ncontrolled by the stable fixed point of the continuum AH model.  This\nissue requires additional investigations.\n\nIt is important to stress that we have considered here a compact\nversion of electrodynamics. Other models of interest in\ncondensed-matter physics consider complex fields (spinons) coupled to\nnoncompact electrodynamics \\cite{SVBSF-04,SBSVF-05,SBSVF-04}. Such a\nmodel may have a different critical behavior due the suppression of\nmonopoles \\cite{MHSF-08,BMK-13,SP-15,DMPS-15}.  Numerical studies have\nidentified the transition, but at present there is no consensus on its\norder.  The same is true for loop models which supposedly belong to\nthe same universality class (if it exists), see, e.g.,\nRefs.~\\cite{MV-04,KMPST-08,NCSOS-11,NCSOS-13,NCSOS-15}.  Clearly,\nadditional work is needed to settle the question.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe classic Poisson formula naively says that a harmonic function on the unit disk in the complex plane, that is a function whose Laplacian vanishes, can be represented as an integral transform of its values on the  boundary of the disk. The integral transform is with respect to a kernel known as the Poisson kernel.  Probabilistically, the Poisson kernel is the distribution of the position where Brownian motion exits the open disk. The idea of representing a harmonic function as an integral transform of its boundary values extends beyond Laplacian to other contexts. In particular, in the theory of Markov chains, the study of notion of Poisson formula  goes back to the works of  Blackwell \\cite{Blackwell1955} and Feller \\cite{Feller1956}. \n\nLet $\\EuScript{X}$ be an infinite, countable set. This will serve as our state space. Let $p:\\EuScript{X} \\times \\EuScript{X} \\to [0,1]$  be a transition function, that is $\\sum_y p(x,y)=1$ for all $x\\in \\EuScript{X}$.   A function $f:\\EuScript{X} \\to {\\mathbb R}$ is $p$-harmonic if $\\sum_y p(x,y) |f(y)|<\\infty$ for all $x\\in \\EuScript{X}$ and it also satisfies the  mean value property\n$f(x)=\\sum_y p(x,y)f(y)$ for all $x\\in\\EuScript{X}$.  Note that the constant functions are bounded and  $p$-harmonic, and that the set of $p$-harmonic functions is a linear space. It is easy to see that the space of bounded $p$-harmonic functions is a Banach space with respect to the  sup-norm.  By Rohlin's theory of measurable partitions  \\cite{Rohlin52} and Doob's theory of martingales \\cite{Doob53}, there exists a probability space called the Poisson boundary such that its $L^\\infty$ as a Banach space is isometrically isomorphic to the space of bounded $p$-harmonic functions, see \\cite{Kaimanovich1996}. \n\n\nThere are  extensive developments of the theory of Poisson boundaries whenever the state space $\\EuScript{X}$ has some special structure, e.g. a group, a Riemannian manifold,  see \\cite{Furstenberg1963}, \\cite{Kaimanovich2002}, \\cite{Kaimanovich-Woess2002} and the references therein. \n\nFurstenberg \\cite{Furstenberg1973} showed that the Poisson boundary of a random walk on a group is isomorphic to the induced random walk to a recurrent subgroup. Later, Kaimanovich \\cite{Kaimanovich83}, Muchnik \\cite{Muchnik2006}, Willis \\cite{Willis1990} provided more constructions on a random walk on group that preserve the Poisson boundary. The most general method up to date to construct random walks on groups with a common  Poisson boundary was recently introduced  by  Kaimanovich and the second author in \\cite{Forghani-Kaimanovich2016}, \\cite{Behrang2016}.  Their method is based on applying a randomized stopping time to the space of sample paths to obtain a new random walk with identical  Poisson boundary.  A crucial step in the proof is that each countable group can be viewed as a quotient space of some free semigroup. Hence the proofs are  applicable only  to random walks on countable groups. This approach was employed to study the space of positive harmonic functions on a countable group with respect to a stopping time \\cite{Forghani-Mallahi2016}.\n\nIn this paper, we consider the case when the state space has \\emph{no additional structure}.  Given a Markov chain $\\boldsymbol{x} = (x_0,x_1,\\dots)$ on a countable state space  $\\EuScript{X}$ with transition function $p$ and a stopping time $\\tau$ which is almost surely finite (see equation \\eqref{eq:finiteness}), \nwe consider the process obtained by looking at the Markov chain corresponding to $p$ along iterations of the stopping time, a sequence we denote by $\\langle \\tau (\\boldsymbol{x}) \\rangle$.   Through the strong Markov property, see \\cite{Revuz1984}, this  process is a Markov chain on $\\EuScript{X}$ which we call the transformed chain, and whose transition function we denote by $p^\\tau$, given by $p^\\tau (x,y) = \\mbox{\\bf P}_x (x_\\tau = y)$.  We say  $\\tau$ can be asymptotically recovered (Assumption~\\ref{as:asymp}) when there exists a positive integer-valued map $\\rho$ on the space of sample paths such that  \n$$\n\\displaystyle\\limsup_{n\\to\\infty}  \\inf_{x} \\mbox{\\bf P}_x \\bigg(n+\\rho (x_n,x_{n+1},\\cdots) \\in \\langle \\tau(\\boldsymbol{x}) \\rangle\\bigg)=1.\n$$\n\nA simple example for a stopping time that can be asymptotically recovered is that of a hitting time. More examples are given in  Section \\ref{sec:examples}.\n\n\nWe investigate how the Poisson boundary (bounded harmonic functions) of the original and transformed Markov chains are related. We do this by showing the intuitively clear fact that  if the stopping time can be asymptotically recovered  (Assumption~\\ref{as:asymp}), then the space of bounded harmonic functions for the transformed process is  embedded in the space of space-time harmonic functions for the original chain.  This is the statement  of our main result,  Theorem \\ref{th:newmain}:\n\\begin{theo}\n Let  $p$ be  a transient  transition function on $\\EuScript{X}$, and suppose that $o\\in \\EuScript{X}$ is such that all $x\\in \\EuScript{X}-\\{o\\}$ are accessible from $o$.  Let $\\tau$ be a stopping time for which is finite a.s. under any initial distribution and can  be   asymptotically recovered (Assumption~\\ref{as:asymp}).\nThen for any positive bounded $p^\\tau$--harmonic function $u$ there exists an extension $\\bar{u}$ to   $\\EuScript{X}\\times {\\mathbb Z}_+$,  ${\\bar u}(x,0)=u(x)$, such that\n  \\begin{enumerate} \n \\item  $\\bar{u}$ is a positive bounded $p$-harmonic sequence. \n \\item $\\|{\\bar u} \\|_\\infty =\\|u\\|_\\infty$. \n \\end{enumerate} \n \\end{theo} \n We note that in general, and unlike the case of random walks on groups, the Poisson boundaries of the original and transformed  chains may be fundamentally  different, see the example in Section \\ref{splitting}.   In Theorem \\ref{th:martin} we extend the scope to positive harmonic functions under some additional conditions. \n\nOur proof of the theorem is based on the construction of the \\emph{Martin boundary}, which is one the main qualitative space in boundary theory and potential theory associated to Markov chains. The Martin boundary of a Markov chain is the  topological counterpart of the Poisson boundary which is responsible for representation of positive harmonic functions. If the Martin boundary as a Borel space equipped with an appropriate probability measure, then it is isomorphic (as a measure space) with the Poisson boundary, see \\cite{Dynkin69}. \n\nThe organization of the paper is as follows.  In Section \\ref{sec:preliminaries} we recall the theory of  Poisson, tail, and Martin boundaries. Section 3 devoted to constructing Markov chains via transformation.  In  Section 4 we show how the tools from the theory of Martin boundary can be applied to the transformed Markov chains via stopping times. \nOur main result is  proved in Section \\ref{sec:main}, and in Section \\ref{sec:examples} we present a number of examples. Two standard approximation results used in the proof of Theorem \\ref{th:martin} are  proved in the Appendix.\n\n\\subsection*{Acknowledgements}\nWe would like to thank the anonymous referee for reading  the manual very carefully and suggesting improvements.\n\n\\section{Preliminaries}\\label{sec:preliminaries} \n\\subsection{Markov chains}\n\\label{sec:MC_defn} \nLet $\\EuScript{X}$ be a countable set. The set $\\EuScript{X}$ and its power set form a measurable space.  Let $\\EuScript{X}^{{\\mathbb Z}_+}=\\{\\boldsymbol{x}=(x_0,x_1,\\dots):x_n \\in \\EuScript{X},n\\in{\\mathbb Z}_+\\}$, the set of $\\EuScript{X}$-valued sequences indexed by ${\\mathbb Z}_+$. For every $n\\in{\\mathbb Z}_+$ define the coordinate function $\\omega_n(\\boldsymbol{x})=x_n$. Denote by $\\cal{F}_k^\\infty(\\EuScript{X})=\\sigma(\\omega_k,\\omega_{k+1},\\cdots)$ the sigma-algebra generated by the coordinate functions $\\omega_i,~i\\ge k$. Let ${\\cal F^\\infty}=\\cal F^\\infty_0(\\EuScript{X})$.\nThe measurable  space  $(\\s^{\\mathbb{Z}_+},{\\cal F^\\infty})$ is called the  space of sample paths. For our work, we will also need to define an auxiliary process  $\\boldsymbol{z}=(z_n:n\\in {\\mathbb Z}_+)$, as follows. Given $\\boldsymbol{x}$ and $t \\in{\\mathbb Z}_+$, we let $ z_n = (x_n,t+n)$. That is, $\\boldsymbol{z}$ also keeps track of time. \n\nLet $p$ be a transition function on $\\EuScript{X}$. That is $p:\\EuScript{X} \\times \\EuScript{X} \\to [0,1]$ and $\\sum_y p(x,y) =1$ for all $x\\in \\EuScript{X}$. Let $m$ be a probability measure on $\\EuScript{X}$.  For $n\\in {\\mathbb Z}_+$, define the $n$-th iteration of $p$, denoted by $p^n$, through \n\\begin{equation}\n\\label{eq:p_iterated} \np^0(x,y) = \\delta_{x}(y), \\mbox{ and }p^{n+1} (x,y) = \\sum_z p(x,z) p^n (z,y)\n\\end{equation}\n(note that $p^1=p$, and that $p^n$ is also a transition function). By Kolmogorv's extension theorem, there exists a unique probability measure  $\\mbox{\\bf P}_m$ on the space of sample paths satisfying\n$$\n\\mbox{\\bf P}_m(a_0,a_1,\\cdots,a_n)=m(a_0)p(a_0,a_1)\\cdots p(a_{n-1},a_n)\n$$ \nwhere $(a_0,a_1,\\cdots,a_n)=\\{\\boldsymbol{x}\\in\\s^{\\mathbb{Z}_+}\\ :\\ \\omega_i(\\boldsymbol{x})=a_i,\\ i=0,\\cdots,n\\}$. The probability measure \n$m$ is usually referred to as the initial distribution under $\\mbox{\\bf P}_m$. As usual, we write $\\mbox{\\bf E}_m$ for the expectation operator associated with $\\mbox{\\bf P}_m$, also for $x\\in \\EuScript{X}$ we abbreviate and write $\\mbox{\\bf P}_x,\\mbox{\\bf E}_x$ instead of $\\mbox{\\bf P}_{\\delta_x},\\mbox{\\bf E}_{\\delta_x}$. Note then that \n$$\n\\mbox{\\bf P}_m=\\sum_x m(x)\\mbox{\\bf P}_x,\\ \\ \\ \\  \\mbox{\\bf E}_m=\\sum_xm(x)\\mbox{\\bf E}_x.\n$$\nThe triple $(\\EuScript{X},p,m)$ is called a Markov chain on the state space $\\EuScript{X}$ with the transition function $p$ and the initial distribution $m$.\n\\subsection{Harmonic functions}\nSuppose that $f:\\EuScript{X} \\to {\\mathbb R}$ satisfies $\\sum_y p(x,y) |f(y)|< \\infty$ for all $x\\in \\EuScript{X}$. Then we can define a function $pf:\\EuScript{X}\\to {\\mathbb R}$ through \n$$pf(x)=\\sum_yp(x,y)f(y).$$ \nIf $pf =f$, then $f$ is called $p$--harmonic.  We denote the set of all bounded $p$--harmonic functions and the set of all positive $p$--harmonic functions by $H^{\\infty}(\\EuScript{X},p)$ and $H_{+}(\\EuScript{X},p)$, respectively.  We also write $H^\\infty_+(\\EuScript{X},p)$ for the convex cone of bounded positive harmonic functions.  \n\\subsection{Harmonic sequences}\\label{Z}\nA sequence of functions $(f_n:n\\in {\\mathbb Z}_+)$, where  $f_n: \\EuScript{X}\\to{\\mathbb R}$, is called a $p$--harmonic sequence whenever \n$$pf_{n+1}=f_n.$$\nThe space $p$--harmonic sequence is denoted by $S(\\EuScript{X},p)$, while the subspace of  bounded $p$--harmonic sequences, and nonnegative $p$--harmonic  sequences are denoted by  $S^{\\infty}(\\EuScript{X},p)$ and  $S_+(\\EuScript{X},p)$, respectively.  If $f$ is a nonnegative $p$--harmonic, then  $(f,f,\\dots)$ is a nonnegative $p$--harmonic sequence.  Harmonic sequences sometimes are called space--time harmonic functions. Indeed, given a $p$--harmonic sequence $(f_n:n\\in{\\mathbb Z}_+)$, define a function $f:\\EuScript{X}\\times {\\mathbb Z}_+\\to {\\mathbb R}$ by letting $f(x,n)=f_n(x)$.  Now if one defines a transition function  $p^+$ on $\\EuScript{X}\\times {\\mathbb Z}_+$ by letting\n$$p^+((x,n),(y,n+1))=p(x,y),$$\nthen $p^+ f = f$. Conversely, if $f$ is $p^+$--harmonic, then $(f(x,0),f(x,1),\\dots)$ is a $p$-harmonic sequence. In terms of notation, ${\\cal S}(\\EuScript{X},p)=H(\\EuScript{X}\\times {\\mathbb Z}_+,p^+)$. For $(x,t)\\in \\EuScript{X}\\times{\\mathbb Z}_+$, write  $\\mbox{\\bf P}_{x,t}$ for the probability measure on the space of sample paths on $\\EuScript{X}\\times {\\mathbb Z}_+$ induced by  $p^+$ with initial distribution $\\delta_{x,t}$.  A sample path in that space will be written as  $\\boldsymbol z=(z_0,z_1,\\cdots)$.\n\\subsection{Poisson boundary}\\label{sec:Poisson}\nLet $S:\\s^{\\mathbb{Z}_+}\\to\\s^{\\mathbb{Z}_+}$ be the time-shift, that is \n$$\nS(x_0,x_1,\\cdots)=(x_1,x_2,\\cdots),\n$$\nand for $k\\ge 1$ write $S^k$ for the $k$-th iteration of $S$, i.e, $S^k (x_0,x_2,\\cdots) =(x_k,x_{k+1},\\cdots)$.\nThe sigma-algebra  $\\mathcal{I}=\\{A\\in\\mathcal F^\\infty\\ :\\ S^{-1}A=A\\}$ is called the  invariant sigma-algebra. Let $\\theta$ be a probability measure on $\\EuScript{X}$ with full support, that is $\\theta(x)>0$ for all $x\\in \\EuScript{X}$. Let $\\overline{\\mathcal{I}}(\\EuScript{X},p)$ denote the completion of ${\\cal I}$ with respect to the probability measure $\\mbox{\\bf P}_{\\theta}$. As can be easily seen, this completion is  equivalent to requiring that whenever $A\\subseteq \\s^{\\mathbb{Z}_+}$ is such that $A\\subseteq B$ for some $ B \\in{\\cal F^\\infty}$, satisfying $\\mbox{\\bf P}_x (B) = 0$ for all $x\\in \\EuScript{X}$, then $A$ is measurable with respect to the completion. Therefore $\\overline{\\cal I}(\\EuScript{X},p)$ is independent of the particular choice of $\\theta$.   The Poisson boundary is defined as the  restriction of $(\\s^{\\mathbb{Z}_+},\\cal F^\\infty,\\mbox{\\bf P}_{\\theta})$ to  $\\overline{\\mathcal{I}}(\\EuScript{X},p)$.  Denote the Poisson boundary with respect to the transition function $p$ as $\\mathcal{PB}(\\EuScript{X},p):=(\\EuScript{X}^{{\\mathbb Z}_+}, \\overline{\\mathcal{I}}(\\EuScript{X},p), \\mbox{\\bf P}_\\theta)$.  As an ergodic theoretic object, the Poisson boundary is identified with  the space of ergodic components of the time-shift on the space of sample paths \\cite{Kaimanovich91}.  \n\nThe Poisson boundary identifies the space of bounded harmonic functions. More precisely, let $f$ be a bounded  $p$--harmonic function, define \n\n\\begin{equation}\\label{eq:martin-approach}\n\\begin{array}{rcl}\nH^\\infty(\\EuScript{X}, p) &\\longrightarrow &L^\\infty(\\mathcal{PB}(\\EuScript{X},p))\\\\\nf  &\\longmapsto&  \\phi_f(\\boldsymbol{x}):= \\displaystyle\\lim_{n\\to\\infty} f(x_n)  \\ \\mbox{ a.e.} \n\\end{array}\n\\end{equation}\n\nBecause $f$ is a bounded $p$--harmonic function, the sequence $(f(x_n))_{n}$ is $\\mbox{\\bf P}_\\theta$--martingale, therefore $\\phi_f(x)$ almost surely exists. Because $\\overline{\\mathcal{I}}(\\EuScript{X},p)$ is a complete sigma--algebra, $\\phi_f$ is measurable with respect to the probability space $\\mathcal{PB}(\\EuScript{X},p)$.  We can define the inverse as follows.\n\\begin{equation}\\label{eq: bounded-infinity}\n\\begin{array}{rcl}\nL^\\infty(\\mathcal{PB}(\\EuScript{X},p)) &\\longrightarrow &H^\\infty(\\EuScript{X}, p) \\\\\n\\phi  &\\longmapsto&  f_\\phi(x):= \\int \\phi(\\boldsymbol{x}) \\ d\\mbox{\\bf P}_x(\\boldsymbol{x})\n\\end{array}\n\\end{equation}\nMoreover the definitions of $\\phi_f$ and $f_\\phi$ imply that $\\|f\\|_\\infty=\\|\\phi_f\\|_\\infty$, hence $H^\\infty(\\EuScript{X},p)$\nis isometrically isomorphic to $L^\\infty(\\mathcal{PB}(\\EuScript{X},p))$, see also Corollary~\\ref{cor:isometry}.\n\n\n\\subsection{Tail boundary}\nConsider the  tail sigma--algebra $\\mathcal{T}=\\displaystyle\\bigcap_{k=0}^{\\infty}S^{-k}(\\cal F^\\infty)$  on the space of sample paths. The completion of $\\mathcal{T}$ with respect to $\\mbox{\\bf P}_\\theta$, where $\\theta$ is as in the last paragraph, is denoted by $\\overline\\mathcal{T}(\\EuScript{X},p)$. The tail boundary is the  restriction of $(\\s^{\\mathbb{Z}_+},\\cal F,\\mbox{\\bf P}_{\\theta})$  to the sigma-algebra  $\\overline\\mathcal{T}(\\EuScript{X},p)$. The tail boundary associated with the transition function $p$ is denoted by $\\mathcal{TB}(\\EuScript{X},p):=(\\EuScript{X}^{{\\mathbb Z}_+}, \\overline{\\mathcal{T}}(\\EuScript{X},p), \\mbox{\\bf P}_\\theta))$. Similarly to the Poisson boundary, the space of bounded  $p$--harmonic sequences $S^\\infty(\\EuScript{X},p)$ is isometrically isomorphic to $L^\\infty(\\mathcal{TB}(\\EuScript{X},p))$:\n$$\n\\begin{array}{rcl}\nS^\\infty(\\EuScript{X}, p) &\\longrightarrow &L^\\infty(\\mathcal{TB}(\\EuScript{X},p))\\\\\nF=(f_n)_n  &\\longmapsto&  \\psi_F(\\boldsymbol{x}):= \\displaystyle\\lim_{n\\to\\infty} f_n(x_n)\\ a.e\n\\end{array}\n$$\nUsing the fact that the space of bounded $p^+$--harmonic functions can be viewed as the space of bounded $p$--harmonic functions implies $H^\\infty(\\EuScript{X}\\times {\\mathbb Z}_+,p^+)$ is isometrically isomorphic to $L^\\infty(\\mathcal{TB}(\\EuScript{X},p))$. On the other hand, $H^\\infty(\\EuScript{X}\\times {\\mathbb Z}_+,p^+)$ is isometrically isomorphic to \n$L^\\infty(\\mathcal{PB}(\\EuScript{X}\\times{\\mathbb Z}_+,p^+))$. Summarizing: we have $L^\\infty(\\mathcal{TB}(\\EuScript{X},p))$  is isometrically isomorphic to $L^\\infty(\\mathcal{PB}(\\EuScript{X}\\times{\\mathbb Z}_+,p^+))$. Therefore,  the tail boundary associated with $p$ is isomorphic (as a probability space) to the Poisson boundary associated with $p^+$, for more details see \\cite{Ka92,Kaimanovich1996}\n\n$$ \n \\begin{tikzcd}\nH^\\infty(\\EuScript{X}\\times{\\mathbb Z}_+,p^+)\\arrow[r, \"\\cong\"]  & S^\\infty(\\EuScript{X},p)  \\dar{\\cong} \\\\  \nL^\\infty(\\mathcal{PB}(\\EuScript{X}\\times{\\mathbb Z}_+,p^+)) \\arrow[u, \"\\cong\"]   & L^\\infty(\\mathcal{TB}(\\EuScript{X},p))\n\\end{tikzcd}\n$$\n\n\n Viewing the Poisson boundary as representing the  bounded harmonic functions,  the tail boundary as representing the  bounded harmonic sequences, and using the fact that  every harmonic function uniquely extends to a harmonic sequence, we can think of the Poisson boundary as a subset on the tail boundary. We will not get into any details here.   However, we have the following:\n\n$$ \n \\begin{tikzcd}\nH^\\infty(\\EuScript{X},p)\\arrow[r, hook]  & S^\\infty(\\EuScript{X},p)  \\dar{\\cong} \\\\  \nL^\\infty(\\mathcal{PB}(\\EuScript{X},p)) \\arrow[u, \"\\cong\"]   & L^\\infty(\\mathcal{TB}(\\EuScript{X},p))\n\\end{tikzcd}\n$$\n \n A transition function of a Markov chain is called steady whenever the tail boundary  coincides mod $\\mbox{\\bf P}_x$ with  the Poisson boundary  for any $x$ in $\\EuScript{X}$, and, in particular, all bounded harmonic sequences are bounded harmonic functions.  The ``0-2'' law  determines whether a transition function is steady, see  \\cite{Ka92} for more details. Here is one sufficient condition: \n \n\\begin{example}[\\cite{Ka92}]\nFor $x\\in \\EuScript{X}$, let  $g_x = \\inf \\{n\\ge 1: p^n(x,x)>0\\}$. Then $p$ is steady if the greatest common divisor of $\\{g_x>0:x\\in \\EuScript{X}\\}$ is $1$.  \n\\end{example}\n\n\\subsection{Martin boundary}\n\\label{sec:martin_boundary}\nThe relation between the Poisson and the tail boundaries to the invariant and tail sigma-algebras allow  to characterize the space of bounded harmonic functions and bounded harmonic sequences, respectively.  We now introduce the Martin boundary, a topological boundary also used to characterize positive harmonic functions (Theorem \\ref{th:doob59} below), and which is more suitable for our purposes.  We comment that the Poisson boundary can be identified as a subset of the Martin topology equipped with an appropriate probability measure \\cite{Kaimanovich1996}, \\cite{Sawyer1997}, and \\cite{Woess09}, also see Corollary \\ref{cor:isometry} below.  One can refer to \\cite{Derriennic1976}, \\cite{Kaimanovich1996}, \\cite{Sawyer1997} and  \\cite{Woess09} for the construction of Martin boundary.  In this section, we remind the reader about the definition of the Martin boundary and related results which will be used  later.\n\nLet $p$ be a transition function on $\\EuScript{X}$.  The  transition function is called transient whenever the Green's function $G^p(x,y)=\\sum_{n\\geq0}p^n(x,y)$ is finite for all $x,y\\in \\EuScript{X}$, $p^n$ is the $n$-th iteration of $p$, defined in \\eqref{eq:p_iterated}. \n\nWe always make the following assumption on $p$: \n\\begin{assum}~\n\\label{as:required} \n\\begin{enumerate}\n\\item $p$ is transient. \n\\item There exists a state $o$ such that all $y \\in \\EuScript{X}-\\{o\\}$ are accessible from $o$:  \n$$G^p(o,y)>0\\mbox{ for all }y\\in \\EuScript{X}.$$  \n\\end{enumerate}\n\\end{assum} \n\nWe also define  the {\\it Martin kernel} on $\\EuScript{X}\\times \\EuScript{X}$\n$$K^p(x,y)=\\begin{cases} \\frac{G^p(x,y)}{G^p(o,y)} & G^p(o,y)>0 \\\\ 0 & \\mbox{otherwise}\\end{cases}$$ \n The  {\\it Martin compactification} of $\\EuScript{X}$ is the topological space $M(\\EuScript{X},p)$\n\n satisfying the following requirements: \n \\begin{enumerate}\n \\item Every singleton $\\{x\\},~x\\in \\EuScript{X}$, is open. \n \\item $\\EuScript{X}$ is dense. \n\\item For  $x\\in \\EuScript{X}$, the function $K^p(x,\\cdot)$ extends to a continuous function on $M(\\EuScript{X},p) $, and the  set of extensions separate points in $M(\\EuScript{X},p) \\backslash \\EuScript{X}$. \n\\end{enumerate}\nThese  requirements uniquely determine a compact topological space (up to homeomorphism). Furthermore, the resulting space is metrizable \\cite{Woess09}. The compact topological space $\\partial(\\EuScript{X},p)=M(\\EuScript{X},p) \\backslash \\EuScript{X}$ is called the \\emph{Martin boundary} of the Markov chain with respect to the transition function $p$. \n\nThe \\emph{minimal Martin boundary} is the Borel subset $\\partial_m(\\EuScript{X},p)$ of $\\partial(\\EuScript{X},p)$  consisting of all $\\xi$ satisfying \n\\begin{enumerate} \n\\item $K^p(\\cdot, \\xi) \\in H_+(\\EuScript{X},p)$. \n\\item $K^p(\\cdot,\\xi)$ is {\\it minimal harmonic}: if $u\\in H_+(\\EuScript{X},p)$ and  $u \\le K^p(\\cdot,\\xi)$, then  $u = c K^p(\\cdot, \\xi)$ for some $c\\le 1$. \n\\end{enumerate} \n\\begin{theorem}\\cite{Doob59}\\label{th:doob59}\nLet $u\\in H_+(\\EuScript{X},p)$. Then  there exists a unique finite measure $\\mu_u$ on the Borel sigma-algebra on $\\partial_m(\\EuScript{X},p)$ such that \n$$\nu(x)=\\int_{\\partial_m(\\EuScript{X},p)}K^p(x,\\xi)d\\mu_u(\\xi).\n$$\n\\end{theorem}\nThe measure $\\mu_u$ is called the representation of $u$. Since    $K^p(o,\\xi)=1$ for all $\\xi\\in \\partial_m (\\EuScript{X},p)$, we have that  $u(o) = \\mu_u (\\partial_m(\\EuScript{X},p))$. Note that we can consider $\\mu_u$ as a finite measure on the compact metric space $\\partial (\\EuScript{X},p)$, by letting $\\mu_u(\\partial (\\EuScript{X},p)-\\partial_m(\\EuScript{X},p))=0$. \n A special role is reserved for $\\mu_{\\bf 1}$ the representation of the constant function ${\\bf 1}$. This is due to the following two results: \n\\begin{cor}\\label{cor:isometry}\nThe mapping  $T$ given by \n\\begin{equation}\\label{eq:mapping}  (T f)(x) = \\int_{\\partial_m(\\EuScript{X},p)} K^p(x,\\xi) f (\\xi) d\\mu_{\\bf 1}.\n\\end{equation}  \ndefines a linear isometry from $L^\\infty(\\mu_1)$ onto $H^\\infty (\\EuScript{X},p)$ and also $L^\\infty(\\mathcal{PB}(\\EuScript{X},p))$.\n\\end{cor} \n\\begin{proof}\nThe right-hand side of \\eqref{eq:mapping} defines a  linear mapping from $L^\\infty(\\mu_{\\bf 1})$ to the linear space of bounded real-valued functions on $\\EuScript{X}$ equipped with the $\\sup$-norm. Also,  \n $$|Tf(x) | \\le \\|f\\|_\\infty \\int_{\\partial_m(\\EuScript{X},p)} K^p(x,\\xi) d \\mu_{\\bf 1}(\\xi) = \\|f\\|_\\infty.$$ \n Therefore $\\|Tf \\|_\\infty \\le \\|f\\|_\\infty$. \n  By dominated convergence, \n$$ p (T f)(x) =  \\int_{\\partial_m(\\EuScript{X},p)}\\sum_y p(x,y)  K^p(y,\\xi) f (\\xi) d \\mu_{\\bf 1}(\\xi)=  \\int_{\\partial_m(\\EuScript{X},p)} K^p(x,\\xi) f(\\xi) d \\mu_{\\bf 1} (\\xi)=(Tf)(x),$$ \ntherefore $Tf \\in H^\\infty (\\EuScript{X},p)$. Next we show that $T$ is an isometry. Suppose first that $f\\in L^\\infty(\\mu_{\\bf 1})$ is nonnegative, and let $u = Tf$. Then since $\\|u\\|_{\\infty}{\\bf 1}-u$ and $u$ are both in $H_+(\\EuScript{X},p)$, it follows from the uniqueness assertion in Theorem \\ref{th:doob59}, that $\\|u\\|_\\infty \\mu_{\\bf 1} =\\mu_{ \\|u\\|_{\\infty}{\\bf 1}-u}+ \\mu_u$, the sum of two positive measures. Therefore, not only $\\mu_u \\ll \\mu_{\\bf 1}$, but also, $0\\le  \\frac{ d \\mu_u}{d \\mu_{\\bf 1}} \\le \\|u\\|_\\infty$. Since $\\frac{d \\mu_u}{d\\mu_{\\bf 1} } =f$, we have $\\|f\\|_\\infty \\le \\|u\\|_\\infty$. In the case of signed $f$, this means \n$$\n\\| \\left .  \\|f\\|_\\infty{\\bf 1} \\pm f\\right.  \\|_\\infty \\le \\| \\left. \\|f\\|_\\infty {\\bf 1} \\pm  Tf \\right.\\|_\\infty.\n$$ \nThe righthand side is bounded above by $\\|f\\|_\\infty + \\|Tf \\|_\\infty$. As for the left hand side, we can choose the sign so that the norm is equal to $2\\|f\\|_\\infty$. Therefore, $\\|Tf \\|_\\infty \\ge \\|f\\|_\\infty$, and since the reverse inequality is already established,  $T$ is an isometry. Finally, we show that $T$ is onto. If  $v\\in  H^\\infty_+(\\EuScript{X},p)$, then as already seen,  $\\mu_v \\ll \\mu_{\\bf 1}$, and $\\frac{d\\mu_v}{d\\mu_{\\bf 1}} \\in L^\\infty(\\mu_{\\bf 1})$, so that $v$ is in the range of $T$.  If $u \\in H^\\infty(\\EuScript{X},p)$, then we can write it as a difference of two elements in $H^\\infty_+(\\EuScript{X},p)$, i.e, $u=(\\|u\\|_\\infty{\\bf 1}  + u) - \\|u\\|_{\\infty}$.  Therefore, $u$ is also in the range of $T$. \n\\end{proof} \n\\begin{theorem}\\cite{Dynkin69,Woess09}\n\\label{th:limits}\n\\begin{enumerate} \n\\item There exists a $\\partial(\\EuScript{X},p)$-valued random variable $x_\\infty$ such that for $\\mbox{\\bf P}_x$-a.s. sample path $\\lim_{n\\to\\infty} x_n=x_\\infty $ is in the Martin topology for all $x$ in $\\EuScript{X}$. \n\\item  The random variable $x_\\infty$ is supported on $\\partial_m(\\EuScript{X},p)$ and for every measurable set $A$ in $\\partial_m(\\EuScript{X},p)$,   \n$$\n\\mbox{\\bf P}_x (x_\\infty \\in A) = \\int_A K^p(x,\\xi) d \\mu_{\\bf 1} (\\xi).\n$$\n\\end{enumerate} \n\\end{theorem} \n\n\n\\section{Transformed Markov chains}\\label{sec:transformed}\n\\subsection{Stopping time}\n\\label{sec:stopped_process}\nA measurable function $\\tau:\\EuScript{X}^{{\\mathbb Z}_+}\\to {\\mathbb Z}_+\\cup\\{\\infty\\}$ is called stopping time, if for every $k\\in {\\mathbb Z}_+$, the set $\\{ \\boldsymbol{x}: \\tau(\\boldsymbol{x})=k\\}$ is a measurable set in the sigma-algebra generated by the first $k+1$ coordinate function $\\sigma(\\omega_0,\\omega_1,\\cdots\\omega_k)$. In what follows, we will assume that \n\\begin{equation} \n\\label{eq:finiteness} \n\\mbox{\\bf P}_x(\\tau < \\infty)=1 \\mbox{  for all }x\\in \\EuScript{X}.\n\\end{equation}\nGiven a stopping time $\\tau$ satisfying \\eqref{eq:finiteness}, a nondecreasing sequence is induced by iteration:  \n$$\n\\tau_0 =0,\\ \\ ~\\tau_1=\\tau,\\ \\  \\tau_{n+1} = \\begin{cases} \\tau_n + \\tau \\circ  S^{\\tau_n} & \\tau_n <\\infty; \\\\ \\infty & \\mbox{otherwise.} \\end{cases}$$ \nWith this sequence, we obtain a transformed process $\\boldsymbol{x}^{\\tau}=(y_n(\\boldsymbol{x}):n\\in{\\mathbb Z}_+)$ given by $y_n = x_{\\tau_n}$. \nBy the strong Markov property, see \\cite{Revuz1984}, \n$$ \\mbox{\\bf P}( x_{\\tau_{n+1}}=z | \\sigma(x_{\\tau_1}\\cdots,x_{\\tau_n} )) = \\mbox{\\bf P}_{y_n} (x_{\\tau} = z).$$\nTherefore $\\boldsymbol{x}^{\\tau}$ is a Markov chain with  the transition function \n$$\np^{\\tau}(x,y)=\\mbox{\\bf P}_x (x_{\\tau} = y).\n$$\n\n\n\nNote that Doob's optional stopping theorem implies that for any stopping time $\\tau$, we can write\n$$\nH^{\\infty}(\\EuScript{X},p)\\subseteq H^{\\infty}(\\EuScript{X},p^\\tau).$$\nSimilarly,   \n$$H^{\\infty}(\\EuScript{X}\\times{\\mathbb Z}_+,p^+)\\subseteq H^{\\infty}(\\EuScript{X}\\times{\\mathbb Z}_+,(p^{+})^\\tau).\n$$\n\nLet $\\mbox{\\bf P}^\\tau$ denote the probability measure on the space of sample paths with respect to the transition function $p^\\tau$. We could map almost every sample path with respect to $p$ to  a sample path with respect to $p^\\tau$:\n$$\n\\begin{array}{rcl}\n(\\EuScript{X}^{{\\mathbb Z}_+},\\mbox{\\bf P}_\\theta)&\\longrightarrow &(\\EuScript{X}^{{\\mathbb Z}_+},\\mbox{\\bf P}^\\tau_\\theta)\\\\\n\\boldsymbol{x}=(x_n)_n &\\longmapsto & \\boldsymbol{x}_\\tau:=(x_{\\tau_n})_n,\n\\end{array}\n$$\nwhich implies $L^\\infty(\\mathcal{PB}(\\EuScript{X},p))$ is isomorphic to a subspace of $L^\\infty(\\mathcal{PB}(\\EuScript{X},p^\\tau))$.\n\nConsider the following:   \n\\begin{assum} \\label{as:asymp} \nThere exists a mapping $\\rho:{\\EuScript{X}}^{{\\mathbb Z}_+}\\to {\\mathbb Z}_+$ such that\n$$\n\\displaystyle\\limsup_{n\\to\\infty}  \\inf_{x} \\mbox{\\bf P}_x (\\rho_n (\\boldsymbol{x}) \\in \\langle \\tau(\\boldsymbol{x}) \\rangle)=1,\n$$\nwhere $\\rho_n({\\boldsymbol{x}}) = n + \\rho \\circ S^n({\\boldsymbol{x}})$ and  $\\langle \\tau(\\boldsymbol{x}) \\rangle=(\\tau_0(\\boldsymbol{x}),\\tau_1(\\boldsymbol{x}),\\cdots)$. \n\\end{assum} \nIn order to be able to employ the tools from the last section, we need to insure that $p^\\tau$ satisfies the conditions of Assumption~\\ref{as:required}. \nWe observe that if $\\boldsymbol{x}$ (equivalently, $p$) is transient then so is $\\boldsymbol{x}^\\tau$ (equivalently $p^\\tau$). \n\n\n\t\n\\begin{lemma}\nIf $p$ is transient, then so is  $p^\\tau$.  Furthermore, for all $x$ and $y\\in \\EuScript{X}$, we have $G^{p^\\tau}(x,y) \\le G^p(x,y)$. \n\\end{lemma}\n\\begin{proof}\n\tTransience of $p^\\tau$ is equivalent to $\\boldsymbol{x}^\\tau$ visiting each state finitely often under $\\mbox{\\bf P}_x$ for all $x\\in \\EuScript{X}$. Since this holds for $\\boldsymbol{x}$, and the paths of $\\boldsymbol{x}^\\tau$ are subsequences of $\\boldsymbol{x}$ both statements hold.  \n\\end{proof}\t\n\nNote that $p^\\tau$ may, in general, not satisfy the second condition in Assumption \\ref{as:required}. For example let $\\EuScript{X}={\\mathbb Z}_+$, the set of nonnegative integers and let  $p(n,n+1)=1$, then $G^p(0,n)=1>0$ for any natural number $n$. If we consider the stopping time $\\tau=2$, then for any $n\\in{\\mathbb Z}_+$, $G^{p^{\\tau}}(m,n)>0$ if and only if $n-m \\in 2{\\mathbb Z}_+$. Therefore there does not exist $m\\in{\\mathbb Z}_+$ such that $G^{p^{\\tau}}(m,n)>0$ for all $n\\in{\\mathbb Z}_+$.  In Section~\\ref{sec : extension}, we will remedy this by expanding the state space.  \n\n\\section{The extension} \\label{sec :  extension}\nAssumption \\ref{as:asymp} does not warrant that  $p^\\tau$ satisfies the second condition of Assumption \\ref{as:required} which is required for defining the Martin boundary. If it does, we need not do anything. Otherwise, we need to introduce the following completion. \n\nOur starting point is a transition function $p$ on $\\EuScript{X}$ satisfying Assumption~\\ref{as:required},  and a stopping time $\\tau$ for $p$ satisfying Assumption~\\ref{as:asymp}.\n\nThe first step is to append a state to $\\EuScript{X}$, and extend $p$ to the new resulting extended state space. Let $\\EuScript{X}^* = \\EuScript{X} \\cup\\{*\\}$, where $*$ is a state not in $\\EuScript{X}$. Let $\\theta$ be any probability measure on $\\EuScript{X}^*$ with $\\theta(x)>0$ for all $x\\in \\EuScript{X}$ and $\\theta(*)=0$.  Extend $p$ to $\\EuScript{X}^*$ by letting\n$$  p^*(x,y) = \\begin{cases} p(x,y)& ~x,y\\in \\EuScript{X}\\\\ \n\\theta(y)& x=* \\\\ \n0 & \\mbox{otherwise}. \n\\end{cases}$$\nWe write  $\\boldsymbol{x}^*$ for the corresponding Makrov chain. \nClearly $p^*$ satisfies both conditions in Assumption~\\ref{as:required} with $\\EuScript{X}$ replaced by $\\EuScript{X}^*$ and $o=*$. We also write $\\mbox{\\bf P}^*_x $ and $\\mbox{\\bf E}^*_x$ for the distribution and corresponding expectation associated with $\\boldsymbol{x}^*$, starting from $x$ in $\\EuScript{X}^*$. Note that for any $x\\in \\EuScript{X}$, we have $\\mbox{\\bf P}^*_x$ is supported on $\\EuScript{X}$-valued sequences and coincides with $\\mbox{\\bf P}_x$. It could be therefore viewed as an extension of $\\mbox{\\bf P}_x$. \n\nThis extension preserves the space of bounded harmonic functions: \n\\begin{lemma}\nLet $f$ be a bounded  function on $\\EuScript{X}$. Then, $f$ is $p$--harmonic if and only there exists a unique bounded function $f^*$ on $\\EuScript{X}^*$ such that \n\\begin{enumerate} \n\\item $f^*$ is an extension of $f$, that is $f^*(x)=f(x)$ for all $x$ in $\\EuScript{X}$,\n\\item $f^*$ is $p^*$--harmonic. \n\\end{enumerate} \n\\end{lemma}\n\\begin{proof}\nIt is enough to define $f^*(*) = \\sum_{y\\in \\EuScript{X}} \\theta(y) f(y)$ and $f^*(x)=f(x)$ for any $x$ in $\\EuScript{X}$. Them $f^*$ is an extension of $f$ and a bounded $p^*$--harmonic. The reverse direction is clear.\n\\end{proof}\nNext we extend the stopping time $\\tau$ to   $\\EuScript{X}^*$-valued sequences by setting \n \\[ \\tau^*(\\boldsymbol{x}^*) = \\begin{cases} \\tau(\\boldsymbol{x}^*)&\\mbox{if } x_0^*,x_1^*,\\dots \\in \\EuScript{X} \\\\\n  1+ \\tau(x_1^*,x_2^*,\\dots)&\\mbox{if }  x_0^*=*,x_1^*,x_2^*,\\dots \\in \\EuScript{X}\\\\\n  \\min\\{j:x_j^*= *\\}&  \\mbox{ otherwise} \n \\end{cases} \n \\]\n\nIt immediately follows that  $\\tau^*$ is a stopping time for $\\boldsymbol{x}^*$. Note that under $\\mbox{\\bf P}_x^*$ for  $x\\in \\EuScript{X}$, we have $\\tau^*=\\tau$ a.s. Furthermore,   if $x\\in \\EuScript{X}^*$,  then $\\tau^*$ is finite a.s. $\\mbox{\\bf P}^*_x$. As a result, and similarly to the definition of $p^\\tau$, we have an induced transition function $(p^*)^{\\tau^*}$, defined as follows: \n$$ \n(p^*)^{\\tau^*} (x,y) = \\mbox{\\bf P}_x^* (\\boldsymbol{x}^*_{\\tau^*}=y) = \\begin{cases} p^{\\tau}(x,y) & x,y \\in \\EuScript{X} \\\\ \\sum_{x'}\\theta(x') p^\\tau(x',y) & x =*,y \\in \\EuScript{X}\\\\\n0 & \\mbox{otherwise} \\end{cases}\n$$\n\nWe will write $\\boldsymbol{y}^*$ for the induced chain, that is $y^*_n = \\boldsymbol{x}^*_{\\tau^*_n},~n\\in{\\mathbb Z}_+,$ \nwhere, \n$$\n\\tau^*_0=0\\ \\mbox{ and }\\ \\tau^*_{n+1} = \\tau^*_n+\\tau^*\\circ S^{\\tau^*_n}$$ \nFinally, we restrict $(p^*)^{\\tau^*}$ to $\\widehat \\EuScript{X}$, the subset of  all states which can be reached from any state under $(p^*)^{\\tau^*}$ and the state $*$. More precisely, $\\widehat \\EuScript{X}$, defined as follows: \n$$\\widehat\\EuScript{X}  = \\{y \\in \\EuScript{X}^*: (p^*)^{\\tau^*}(x,y) >0\\mbox{ for some }x\\in \\EuScript{X}^* \\}\\cup\\{*\\}.$$\nEquivalently, \n\\begin{align*} \\widehat \\EuScript{X} \n & = \\{y\\in \\EuScript{X} : p^{\\tau}(x,y)>0\\mbox{ for some }x \\in \\EuScript{X}\\}\\cup \\{*\\}.\n \\end{align*} \n \n Denote the restriction of $(p^*)^{\\tau^*}$ to $\\widehat \\EuScript{X}$ by $\\widehat{p^\\tau}$.  The sample paths with respect to the transition function $\\widehat{p^\\tau}$ will be denoted by $\\widehat\\boldsymbol{y}=(\\widehat y_0,\\widehat y_1,\\widehat y_3,\\cdots)$. By construction, the Markov chain associated with the transition function $\\widehat{p^\\tau}$ on $\\widehat \\EuScript{X}$ satisfies Assumption~\\ref{as:required}.\n \n\n \\section{Main Result}\\label{sec:main}\n We are ready to state our main results. \n \\begin{theorem}\n \\label{th:newmain}\n  Suppose that $p$ is a transition function on $\\EuScript{X}$ satisfying Assumption \\ref{as:required} and that $\\tau$ is a stopping time on $\\EuScript{X}$-valued sequences satisfying Assumption \\ref{as:asymp}. \n  Then for any $0\\le u \\in H^\\infty (\\EuScript{X},p^\\tau)$, there exists a function  $\\bar{u}$ on   $\\EuScript{X}\\times {\\mathbb Z}_+$ such that\n  \\begin{enumerate} \n  \t\\item ${\\bar u}(x,0)=u(x),~x\\in \\EuScript{X}$. \n  \\item  $0\\le {\\bar u}\\in S^\\infty (\\EuScript{X},p)$. \n  \\item $\\|{\\bar u} \\|_\\infty =\\|u\\|_\\infty$. \n  \\end{enumerate} \n \\end{theorem} \n Note that if $p$ is steady, then $u(x)= {\\bar u}(x,t)$ for all $t\\in{\\mathbb Z}_+$  hence the embedding in the theorem gives $u\\in H^\\infty(\\EuScript{X},p)$. \n\nTo introduce the next result, recall that that the support of a Borel measure $\\mu$ on a metric space $(M,d)$, $\\mbox{Supp}(\\mu)$,  is defined as \n$$ \\mbox{Supp}(\\mu) = \\{y\\in M: \\mu (U) >0 \\mbox{ if } U\\mbox{ is open and }y\\in U\\}.$$ \nBy definition, the support is closed and its complement is a $\\mu$-null set.\n\n   We say that a transition function $p$ on $\\EuScript{X}$ has a locally finite range if for every $x\\in\\EuScript{X}$, the set $\\{y\\in\\EuScript{X}:p(x,y)>0\\}$ is finite.  \n \\begin{theorem}\n \t\\label{th:martin}\n Suppose that $p$ is a transition function on $\\EuScript{X}$ with locally finite range satisfying  Assumption \\ref{as:required}, and  that \n  $\\tau$ is  a stopping time on $\\EuScript{X}$-valued sequences satisfying Assumption \\ref{as:asymp}. \n  \n  Let  $u\\in H_+(\\widehat{\\EuScript{X}},{\\widehat {p^\\tau}})$ be such that  $\\mbox{Supp}(\\mu_u^{\\widehat{p^\\tau}})\\subseteq \\mbox{Supp}(\\mu_{\\bf 1}^{\\widehat{p^\\tau}})$. \n\nThen there exists a function ${\\bar u}$ on $\\EuScript{X}\\times {\\mathbb Z}_+$ such that \n\\begin{enumerate}\n\t\\item ${\\bar u}(x,0)=u(x),~x \\in \\EuScript{X}$. \n\t\\item  ${\\bar u}\\in S_+(\\EuScript{X},p)$. \n\t\\end{enumerate}\n \\end{theorem}\n The assumption on the support of $u$ is needed to ensure that one can approximate $u$ through bounded harmonic functions. We  need the local finite range assumption  to show that pointwise  limits of the approximating sequence are indeed harmonic sequences, avoiding a strict inequality in Fatou's lemma.  As for the assumption on the support of the measures, it is known that for random walks on regular trees the support of any harmonic function coincides with the minimal Martin boundary (\\cite[Section 8]{Sawyer1997}). Nevertheless, the  assumption on the support of a positive harmonic function does not hold in general, even for transition functions with locally finite range  (\\cite[Sections 6,7]{Sawyer1997}).\n\t\n\nWe now prove  two lemmas we will use to prove Theorem \\ref{th:newmain}. The lemmas will be followed by the proof of the Theorem~ \\ref{th:newmain} and the proof of Theorem~\\ref{th:martin}.\n\n  \\begin{lemma}\n\\label{lem:inv_to_tail}\nUnder the conditions of Theorem \\ref{th:newmain}, for  $A \\in \\partial({\\widehat \\EuScript{X}}, \\widehat {p^\\tau})$ there exists $I_A\\in {\\cal T} (\\EuScript{X}^*,p^*)$ such that \n\\begin{equation} \n\\label{eq:inv_in_tail} \\{\\lim_{n\\to\\infty} y^*_n \\in  A\\} = I_A,~\\mbox{\\bf P}_x-a.s. \\mbox{ for all }x\\in {\\widehat \\EuScript{X}}-\\{*\\}.\n\\end{equation}  \nFurthermore, if $A$ and $A'$ are disjoint,  $I_A$ and $I_{A'}$ are disjoint. \n\\end{lemma} \n\\begin{proof} \nWe split the proof of the lemma into two parts. In the first part, we show that \\eqref{eq:inv_in_tail} holds for all $A$ which are intersection of $\\partial( \\widehat\\EuScript{X}, \\widehat{p^\\tau})$ with an open ball (in the Martin topology on $\\widehat \\EuScript{X}$ relative to the transition function $\\widehat{p^\\tau}$),  centered at a point in $\\partial( \\widehat\\EuScript{X}, \\widehat{p^\\tau})$. Once this is proved, we show how the lemma extends to all Borel (in the subspace topology)  subsets of $\\partial(\\widehat \\EuScript{X}, \\widehat{p^\\tau})$. \n\nWe begin with the first part. Let $B_\\epsilon(\\zeta)$ be a neighborhood of $\\zeta$ in  $\\partial( \\widehat\\EuScript{X}, \\widehat{p^\\tau})$\n with radius $\\epsilon$. Since the topology on $\\partial(\\widehat \\EuScript{X}, \\widehat{p^\\tau})$ is the induced topology from the compact metric space $\\widehat\\EuScript{X}\\cup \\partial( \\widehat\\EuScript{X}, \\widehat{p^\\tau})$, a basis for the topology on $\\partial( \\widehat\\EuScript{X}, \\widehat{p^\\tau}) $ is the collection of sets of the form\n\n\\begin{equation}\n\\label{eq:special_As} \nA = B_\\epsilon(\\zeta)\\cap \\partial( \\widehat\\EuScript{X}, \\widehat{p^\\tau})\\mbox{ for some } \\epsilon>0\\mbox{  and }\\zeta \\in \\partial( \\widehat\\EuScript{X}, \\widehat{p^\\tau}).\n\\end{equation} \n \n Fix  $\\zeta$ in $\\partial( \\widehat\\EuScript{X}, \\widehat{p^\\tau})\n$ and $\\epsilon>0$. Let  $A = B_\\epsilon(\\zeta)  \\cap \\partial( \\widehat\\EuScript{X}, \\widehat{p^\\tau})$ and let  $B = B_\\epsilon(\\zeta) \\cap \\widehat \\EuScript{X}$.  Clearly, \n$$\\{ \\widehat y_\\infty  \\in A\\}=\\{x^*_{\\tau_n}\\in B \\mbox{ eventually}\\}$$ because by definition, $x^*_{\\tau_n} = \\widehat y_n$ for $n\\in{\\mathbb Z}_+$. Denote the event on the right hand side by $B_\\infty$. Therefore instead of dealing with the event on the left hand side, we will work with $B_\\infty$. Let \n$$\nK_n = \\left\\{\\boldsymbol{x} : \\rho_n(\\boldsymbol{x}) \\not\\in \\langle \\tau(\\boldsymbol{x})\\rangle\\right\\}.\n$$\n By assumption,  there exists a subsequence $(n_i:i\\in{\\mathbb N})$ such that \n$$ \\sum_{i} \\mbox{\\bf P}_x(K_{n_i})<\\infty,~x\\in \\EuScript{X}.$$ \nTherefore the event $\\Gamma=\\{K_{n_i} \\mbox{ finitely often}\\}$ has $\\mbox{\\bf P}_x (\\Gamma)=1$ for all $x\\in \\EuScript{X}$. Write $\\Gamma_x = \\Gamma \\cap \\{x_0=x\\}$. For each $i$, let $\\rho_{i,0} = \\rho_{n_i}$, and continue inductively, \n$$\n\\rho_{i,j+1} = \\rho_{i,j} + \\tau^* \\circ S^{\\rho_{i,j}}.\n$$\n Observe then that $\\rho_{i,j}$ are all ${\\cal F}_{n_i}^\\infty(\\EuScript{X}^*)$-measurable. Let \n$$\nC_{i} = \\bigcap_{j\\in{\\mathbb Z}_+} \\left\\{x^*_{\\rho_{i,j}} \\in B\\right\\}\n$$\nand \n$C=\\limsup C_i =  \\cap_{k=1}^\\infty \\cup_{i\\ge k} C_i.$\n Therefore $\\cup_{i\\ge k} C_i \\in {\\cal F}_{n_i}^\\infty(\\EuScript{X}^*)$, and since this union is decreasing in $k$, it follows that $C$ belongs to ${\\cal F}_{n_k}^\\infty(\\EuScript{X}^*)=S^{-n_k} ({\\cal F}^\\infty(\\EuScript{X}^*))$ for all $k\\in{\\mathbb N}$, that is $C\\in {\\cal T} (\\EuScript{X}^*)$. Clearly, on $\\Gamma_x$, $C$ implies $B_\\infty$, and conversely, on $\\Gamma_x$, $B_\\infty$ implies $C$. Therefore, \n$$ \nB_\\infty = C,~\\mbox{\\bf P}_x-\\mbox{ a.s.,}~x \\in\\widehat\\EuScript{X}-\\{*\\}.\n$$ \nThis proves \\eqref{eq:inv_in_tail} for the particular choice of $A$, with $I_A = C$. Note that by construction, if $A$ and $A'$ are disjoint, so are the corresponding $C$ and $C'$. \n \nWe continue to the second part. Under the subspace topology,  $\\partial(\\widehat \\EuScript{X},\\widehat{p^\\tau})$ is a compact metric space. Therefore  it is separable and  every open set is a countable union of such sets from this basis, and every compact subset is a complement of such a countable union.  \nIn particular, it follows from the first stage that for any compact set $K\\in \\partial(\\widehat \\EuScript{X},\\widehat{p^\\tau})$, there exists $I_K\\in \\overline\\mathcal{T}(\\EuScript{X}^*,{p^*})$, such that \n$$  \\{\\widehat y_\\infty \\in K\\}= I_K,~\\mbox{\\bf P}_x-\\mbox{a.s., for all }x\\in \\widehat\\EuScript{X}-\\{*\\}.$$  \n\nNote that  $\\mu_{\\bf 1}^{\\widehat{p^\\tau}}$ is a probability measure on a compact metric space (due to extension explained below Theorem \\ref{th:doob59}),  it is regular. So for a fixed  Borel set $A \\subseteq\\partial(\\widehat \\EuScript{X},\\widehat{p^\\tau})$, there exists an increasing sequence $(Q_j:j\\ge 1)$ such that for any $j$ the set $Q_j$ is a compact subsets of  $\\partial( \\widehat \\EuScript{X}, \\widehat{p^\\tau})$ and $\\mu_{\\bf 1}^{\\widehat{p^\\tau}} (A-Q_j) \\to 0$. Now \n$ \\{\\widehat y_\\infty \\in \\cup Q_j \\}=\\cup \\{\\widehat y_\\infty \\in Q_j\\}$, and therefore, there exists $I_{H} \\in {\\cal T} (\\EuScript{X}^*,p^*)$ such that \n$$ \\{\\widehat y_\\infty \\in \\cup Q_j\\} = I_H,\\quad \\mbox{\\bf P}_x-\\mbox{a.s.}\\quad x \\in \\widehat \\EuScript{X} - \\{*\\}.$$ \nWe also observe that \n $$\\mbox{\\bf P}_x ( \\widehat y_\\infty \\in A -\\cup Q_j) = \\int_{A-\\cup Q_j} K^{\\widehat{p^\\tau}}(x,\\zeta)d \\mu_{\\bf 1}^{\\widehat{p^\\tau}} (\\zeta)=0,$$ \n for   $x \\in \\widehat\\EuScript{X}$. \nNote that for $x=*$ or $x\\in \\EuScript{X} - \\widehat \\EuScript{X}$, then \n$$\\mbox{\\bf P}_x (y^*_\\infty \\in A - \\cup Q_j) = \\sum_{y\\in \\widehat\\EuScript{X}-\\{*\\}} {\\widehat{p^\\tau}}(x,y)\\mbox{\\bf P}_y(\\widehat y_\\infty \\in A -\\cup Q_j) =0.$$ \nBy completeness, it follows that the event $\\{y^*_\\infty \\in A - \\cup Q_j\\}$ is in $\\overline{{\\cal T}}(\\EuScript{X}^*,p^*)$. From this we conclude that \n$$ \\{\\widehat y_\\infty \\in A\\}= I_H\\quad \\mbox{\\bf P}_x \\mbox{-a.s.}\\quad x \\in \\widehat\\EuScript{X}-\\{*\\}.$$ \nThis completes the proof of the second part, and of the lemma. \n\\end{proof} \n\\begin{lemma}\n\\label{lem:supremum}\nFor $x$ in $\\EuScript{X}$ and $t$ in ${\\mathbb Z}_+$, let $v(x,t) = \\sum_{i=1}^N c_i \\mbox{\\bf P}_{x,t} (I_{A_i})$,  \nwhere $c_i\\ge 0$ and $A_i\\in \\partial( \\widehat \\EuScript{X},\\widehat{p^\\tau})$ are disjoint and  $I_{A_i}$ is as in Lemma \\ref{lem:inv_to_tail}. Then \n$$\\|v\\|_\\infty = \\sup_{x\\in \\widehat \\EuScript{X}-\\{*\\}}\\|v(x,0)\\|_{\\infty}.$$ \n\\end{lemma}\n\\begin{proof}\n  Clearly, $\\|v \\|_\\infty \\ge \\|v(\\cdot,0)\\|_\\infty$. \n  On the other hand,\n  $$ v(x,0) = \\sum c_i \\mbox{\\bf P}_x (y^*_\\infty \\in A_i),$$ therefore by Theorem \\ref{th:limits}, $\\sup_{x \\in \\widehat \\EuScript{X}-\\{*\\}} v(x,0)=\\max c_i$,\n  while\n  $$ v(x,t) = \\sum c_i \\mbox{\\bf P}_{x,t} (I_{A_i}) \\le \\max c_i,$$ \n  because $I_{A_i}$ are disjoint, $\\mbox{\\bf P}_{x,t}$-a.s.  for all $(x,t)\\in \\EuScript{X}^*\\times {\\mathbb Z}_+$.   \n\\end{proof} \n\\begin{proof}[Proof of Theorem \\ref{th:newmain}] \nWithout loss of generality, we can assume that $\\theta$ is such that  $\\sum_{x\\in \\EuScript{X}} \\theta (x) u(x)<\\infty$. Thus, extend $u$ to $\\EuScript{X}^*$ by letting $u(*) = \\sum_{x\\in \\EuScript{X}} \\theta (x) u(x)$, and let $\\widehat u$ be the restriction of $u$ to $\\widehat \\EuScript{X}$. By assumption, there exists $0\\le f\\in L^1(\\mu_1^{\\widehat {p^\\tau}})$ such that \n$$\n\\widehat u (x) = \\int_{\\partial_m( \\widehat{\\EuScript{X}} , \\widehat{p^\\tau})}  K^{\\widehat{p^\\tau}}(x,\\zeta) f (\\zeta) d \\mu_1^{\\widehat {p^\\tau}}(\\zeta).\n$$ \nThere exists  a nondecreasing sequence of nonnegative simple functions   $(f_n:n\\in{\\mathbb N})$ on $\\partial_m( \\widehat{\\EuScript{X}} , \\widehat{p^\\tau})$ such that $f_n \\nearrow f$. Letting \n$$\\widehat u_n (x) =  \\int_{\\partial_m( \\widehat{\\EuScript{X}} , \\widehat{p^\\tau})} K^{\\widehat{p^\\tau}}(x,\\zeta) f_n (\\zeta) d \\mu_1^{\\widehat{p^\\tau}}(\\zeta),$$\nit follows from Lemma \\ref{lem:inv_to_tail} that there exists a combination $\\sum c_i {\\bf 1}_{I_{A_i}}$, $I_{A_i}\\in {\\cal T} (\\EuScript{X}^*,p^*)$, such that\n$$ \\widehat u_n (x) = \\sum c_i \\mbox{\\bf P}_{x,0} (I_{A_i}),\\quad x \\in \\widehat \\EuScript{X}-\\{*\\}.$$\n\nLet $v_n(x,t) = \\sum c_i \\mbox{\\bf P}_{x,t} (I_{A_i}),~(x,t)\\in \\EuScript{X}^*\\times {\\mathbb Z}_+$. Then $v_n \\in  S^\\infty(\\EuScript{X}^*,p^*)$.  \nSince by Lemma \\ref{lem:supremum},  \n$\\|v_n\\|_\\infty\\le \\sup_{x\\in \\widehat \\EuScript{X}-\\{*\\}} \\|v_n(x,0)\\|_\\infty\\le \\|\\widehat u\\|_\\infty$, we can extract a subsequence $(v_{n_k})$ which converges pointwise. Clearly, $v_\\infty (x,0)=u(x)$ on $\\widehat \\EuScript{X}-\\{*\\}$. However, it also follows from dominated convergence that  $v_\\infty \\in S^\\infty(\\EuScript{X}^*,p^*)$. Furthermore, \n$\\|v_\\infty\\|_\\infty \\le \\|\\widehat u\\|_\\infty$.\n\\end{proof} \n\n\\begin{remark} Here is an outline of an alternative proof to Theorem  \\ref{th:newmain}, based entirely on the construction of the Poisson boundary presented in Section \\ref{sec:Poisson} and  avoiding the notion of Martin boundary. The proof was suggested  by the referee. \\\\ \n\nLet $u\\in H^\\infty(\\EuScript{X},p^{\\tau})$, and let $\\phi_u$ be the element in $L^\\infty({\\cal PB}(\\EuScript{X},p^\\tau))$ obtained through \\eqref{eq:martin-approach}:\n\\begin{equation}\n\\label{eq:y_embed}\n\\phi_u ({\\boldsymbol{y}}) = \\lim_{n\\to\\infty} u(y_n), \\quad \\mbox{\\bf P}_{\\theta}\\mbox{-a.s.},\n\\end{equation}\nNow $\\boldsymbol{y}$ is a deterministic function of $\\boldsymbol{x}$: $\\boldsymbol{y} = \\boldsymbol{y}(\\boldsymbol{x})$ through $y_n = x_{\\tau_n}$, and therefore one can rewrite the lefthand side as a function of $\\boldsymbol{x}$, ${\\widetilde{\\phi_u}}(\\boldsymbol{x}) = \\phi_u(\\boldsymbol{y}(\\boldsymbol{x}))$. \nThe  Borell-Cantelli argument in the heart of Lemma \\ref{lem:inv_to_tail} gives a sequence $(\\rho_{n_i}:i=1,2,\\dots)$ of ${\\cal F}_{n_i}^\\infty(\\EuScript{X})$-measurable random variables with $n_i \\nearrow \\infty$, and $\\rho_{n_i}  \\in \\langle \\tau \\rangle$, eventually $\\mbox{\\bf P}_{\\theta}$-a.s.  Therefore the righthand side of \\eqref{eq:y_embed} is  $\\lim_{i\\to\\infty} u(x_{\\rho_{n_i}})$,  $\\mbox{\\bf P}_{\\theta}$-a.s., and is therefore  $\\overline{{\\cal T}}(\\EuScript{X},p)$-measurable. Thus, we have obtained an embedding \n$$H^{\\infty}(\\EuScript{X},p^\\tau) \\ni u \\hookrightarrow \\widetilde{\\phi_u} \\in L^\\infty ({\\cal TB}(\\EuScript{X},p)).$$\n\\end{remark} \n\nWe now prove Theorem \\ref{th:martin}. \n\\begin{proof}[Proof of Theorem \\ref{th:martin}]\n\tBy the assumption on  the support of $\\mu_u^{\\widehat p^\\tau}$, there exists a sequence $f_n\\in L^\\infty (\\mu_{\\bf 1}^{\\widehat {p^\\tau}})$ such that $f_n d\\mu_1^{\\widehat{p^\\tau}}$ converges weakly to $\\mu^{\\widehat {p^{\\tau}}}_{u}$ (see Proposition \\ref{pr:approx}). Since for each $x\\in \\widehat \\EuScript{X}$, the  mapping $\\zeta \\to  K^{\\widehat{p^\\tau}}(x,\\zeta)$ is bounded and continuous on the compact metric space $\\partial ({\\widehat \\EuScript{X}},{\\widehat {p^\\tau}} )$, it follows that each of the functions \n\t$$u_n (x) = \\int K^{\\widehat{p^\\tau}}(x,\\zeta) f_n (\\zeta) d\\mu_{\\bf 1}^{\\widehat {p^\\tau}},~x\\in{\\widehat \\EuScript{X}}$$ \n\tis in $H^\\infty(\\widehat{\\EuScript{X}},\\widehat{p^\\tau})$ and  the sequence $(u_n :n\\in{\\mathbb N})$  converges pointwise to $u$. By Theorem \\ref{th:newmain}, there exists a function $0\\le \\overline{u}_n \\in S^\\infty(\\EuScript{X},p)$ such that  $\\overline{u}_n(x,0)=u_n (x),~x \\in \\EuScript{X}$.   \n\tSince $p \\overline{u}_n(x,t+1)=\\overline{u}_n (x,t),~x\\in \\EuScript{X}$,  it follows that $p(x,y) \\overline{u}_n(y,t+1)\\le \\overline{u}_n(x,t)$ whenever $p(x,y)>0$, or \n\t$$ \\overline{u}_n (y,t+1)\\le \\inf \\{ \\frac{\\overline{u}_n(x,t)}{p(x,y)}: x~, p(x,y)>0\\}.$$\n\tIterating, and using the fact that $p$ is irreducible, we have that for every $(y,t)\\in \\EuScript{X}\\times {\\mathbb Z}_+$ there exist $x=x(y,t)\\in \\EuScript{X}$ and a constant $c(y,t)>0$, both not depending on $n$, such that \n\t$\\overline{u}_n(y,t) \\le c(y,t)\\overline{u}_n(x(y,t),0)$. Since $\\overline{u}_n (x,0)$ converges to $u(x)$ as $n\\to\\infty$ for every $x\\in \\EuScript{X}$,  it follows that the sequence of nonnegative numbers $(\\overline{u}_n (y,t):n\\in{\\mathbb N})$ is bounded. As a result, there exists a subsequence $(n_j:j\\in{\\mathbb N})$ such that $\\overline{u}_{n_j}(x,t)$ converges to a finite limit at all $(x,t)\\in \\EuScript{X}\\times{\\mathbb Z}_+$. Denote this limit function by $\\overline{u}$. Clearly, $\\overline{u}(x,0)=u(x)$. Since $p$ is locally finite, \n\t$$ p \\overline{u} (x,t+1)= \\sum_{y} p(x,y)\\lim_{j\\to\\infty} \\overline{u}_{n_j} (y,t+1) = \\lim_{j\\to\\infty} p \\overline{u}_{n_j}(x,t+1)=\\lim_{j\\to\\infty}\\overline{u}_{n_j}(x,t)= \\overline{ u}(x,t),$$ \n\tcompleting the proof. \t \t  \n\\end{proof} \n\n\n\n\n\\section{Examples}\\label{sec:examples}\nIn this section, we provide some examples.\n\\subsection{Deterministic Stopping times} \nSuppose that $\\tau=c$. Let $\\rho = 1$, we have that $\\mbox{\\bf P}_x (\\rho_n \\in \\langle \\tau \\rangle) =1$ whenever $n+1$ is multiple of $c$.\n\n\\subsection{First Passage times}\\label{sec:hitting}\nLet $A$ be a recurrent set for $p$ and $\\tau = \\inf\\{n \\ge1: x_n \\in A\\}$. Setting $\\rho = \\tau$ satisfies the condition of Assumtption \\ref{as:asymp}. This is the generalization of Furstenberg's result for random walks on groups,  when $A$ is a recurrent subgroup  \\cite{Furstenberg1973}.\nWe will now show how this can be used to study equivalence of bounded harmonic functions on product spaces. Suppose that $p$ is an irreducible and  transient  transition function on $X \\times Y$, where $X$ and $Y$ are nonempty countable sets. $X$ may be finite. We will  assume that there exists $o\\in X$ such that the set  $A=\\{o\\}\\times Y$ is $p$--recurrent. Write $\\boldsymbol{x}=(\\boldsymbol{x}(1),\\boldsymbol{x}(2))$ for the corresponding Markov chain,  where $\\boldsymbol{x}(1)$ is an $X$-valued process, and $\\boldsymbol{x}(2)$ is a $Y$-valued process. Note that in general neither $\\boldsymbol{x}(1)$ nor $\\boldsymbol{x}(2)$ are Markov chains. Let $\\tau$ denote the first passage time to $A$: \n$$ \\tau = \\inf\\{t\\ge 1: {x}_t(1) = o\\}.$$ \nThen $\\rho=\\tau$ satisfies Assumption \\ref{as:asymp}.\nObserve next that $p^\\tau$ induces a transition function $\\gamma$  on $Y$ through the relation $$\\gamma(y,y') = p^{\\tau}((o,y),(o,y')),~x,y\\in Y.$$\n Given $x\\in X$ and $y \\in Y$, let $v((x,y)) = \\mbox{\\bf E}_{(x,y)} [ u({y}_{\\tau})]$. Clearly,  $v((o,y)) = \\gamma u (y) = u(y)$. Therefore, $p^{\\tau}v((o,y))=\\gamma u(y) = u(y) = v ((o,y))$. Next, if $x\\ne o$, we have \n$$p^\\tau v ((x,y)) = \\sum_{y'}p^\\tau((x,y),(o,y')) \\mbox{\\bf E}_{(o,y')} [ u({ y}_{\\tau})]=\\sum_{y'}p^{\\tau}((x,y),(o,y')) u(y')=v((x,y)).$$  \n      \nOne easy example is $X={\\mathbb Z}$,  $Y={\\mathbb Z}^{d-1}$, $d\\ge 3$,  and   $p$ being the simple symmetric random walk on $X \\times Y = {\\mathbb Z}^d$ and $o=0$.  It is known that the Markov chain is steady, and that all bounded harmonic functions are constants, so in particular, $S^\\infty(\\EuScript{X},p)$ consists only of constant functions. Furthermore, the first (or any component) is recurrent. Thus  $\\gamma$ is a transition function on  ${\\mathbb Z}^{d-1}$ which is symmetric, but not nearest neighbor (in fact, it is easy to see that $\\sum_{y\\in {\\mathbb Z}^{d-1}} \\gamma(0,y)|y|=\\infty$). From Theorem \\ref{th:newmain} we therefore obtain that all bounded harmonic functions for $\\gamma$ are constants. \n\n\\subsection{Additive functionals} \nRecall that an additive functional for a Markov chain is a real-valued process $I= (I_n:n \\ge 0)$, such that $I_{n+k} = I_n + I_k \\circ S^n$ and $I_k$ is measurable with respect to the sigma-algebra generated by the first $k+1$ coordinate functions for all $k$ (enough for $k=0,1$). An example for an additive functional is $I_n = \\sum_{k\\le n} f(x_k)$ where $f:{\\EuScript{X}}\\to {\\mathbb R}$ is any function. \\\\\nLet $(I_n:n \\ge 0)$ be an additive functional for $\\boldsymbol{x}$ that satisfies $\\displaystyle\\lim_{n\\to\\infty} I_n = \\infty$ $\\mbox{\\bf P}_x$-a.s. Let $\\tau = \\inf \\{n\\ge 1: I_n > I_{n-1}\\}$. Setting $\\rho = \\tau$ clearly satisfies the condition. Note that the above example is a special case, with the additive functional counting the number of visits to $A$. On the other hand, letting $\\tau = \\inf \\{n: I_n \\ge a\\}$ for some fixed $a$, in general does not satisfy the condition.\n\n\\subsection{A generic choice for $\\rho$} \nLet $\\EuScript{X}$ be a  free semigroup generated by the finite nonempty set ${\\cal G}$. That is, the elements of ${\\EuScript{X}}$ are finite sequences of elements in ${\\cal G}$, the empty sequence included, denoted by $\\emptyset$. Given $x \\in {\\EuScript{X}}$, we write $ x g$ for the sequence in ${\\EuScript{X}}$ obtained by concatenating  $g$ to $x$ from the right.  If $y=xg$, we write $g=x^{-1}y$  and refer to $g$ as the increment. Assume that for any $x \\in \\EuScript{X}$ and $g\\in {\\cal G}$ the transition function $p$ is invariant under the action of semigroups that is  $p(x,xg) = p_g$, where $g\\to p_{g}$ is any probability measure on ${\\cal G}$. \nWe will also assume that $\\tau$ is a stopping time invariant under the action of semigroup in the following sense\n\\begin{equation} \n\\label{eq:invariant_tau}\\tau (\\omega_0,\\omega_1, \\dots) =\\tau (x\\omega_0,x\\omega_1, \\dots) = \\tau(\\emptyset, \\omega_0^{-1}\\omega_1,\\omega_0^{-1} \\omega_2,\\dots)\n\\end{equation}\nfor all $x$ in the semigroup $\\EuScript{X}$.\nThat is, $\\tau$ is  a function of the consecutive increments rather than the actual path.\nLet $\\tau$ satisfy the following two additional conditions: \n\n\\begin{itemize}\n\\item[i)] $\\tau$ is bounded by $M \\in {\\mathbb Z}_+$; \n\\item[ii)] $\\mbox{\\bf P}_\\emptyset (\\tau =1)> 0$. \n\\end{itemize}\nLet $A=\\{g\\in {\\cal G}:\\tau(\\emptyset,g)=1\\}$. \nObserve that from assumption (ii), $\\sum_{g\\in A} p_g>0$, and therefore $\\mbox{\\bf P}_x$-a.s., given  a path $(\\omega_0,\\omega_1,\\dots)$, its associated sequence of increments $(\\omega_0^{-1}\\omega_1,\\omega_1^{-1}\\omega_2,\\dots)$ contains infinitely many  runs (of consecutive increments) in $A$ longer than any fixed $k$. By (i), any ``time'' interval of length longer than $M$ contains at least one element in $\\langle \\tau\\rangle$ before its last element. If, in addition, all increments corresponding to this time interval are in $A$, it necessarily follows that the last element is in $\\langle \\tau \\rangle$. This simple idea translates to the following definition of $\\rho$:\n$$\\rho(\\boldsymbol{x}) = \\inf\\left\\{n>3M: x_{i-1}^{-1}x_i \\in A,~\\mbox{ for all } i =n-2M,\\dots,n  \\right\\}.$$\nFrom the definition of the random walk, $\\rho$ is finite $\\mbox{\\bf P}_x$-a.s. for any $x\\in \\EuScript{X}$. Also, because $\\tau \\le M$, between time   $\\rho_n - 2M$ and $\\rho_n$ there exists at least one element of $\\langle \\tau \\rangle$, call the first such element $\\tau_m$. Since all consecutive increments until time $\\rho_n$ are in $A$,  all elements of the sequence $(\\tau_m, \\tau_m+1, \\dots, \\rho_n)$ are in $\\langle \\tau \\rangle$.  Thus  $\\mbox{\\bf P}_x (\\rho_n \\in \\langle \\tau \\rangle )=1$ and Assumption \\ref{as:asymp} is satisfied. \n\nThe second author and Kaimanovich show that the Poisson boundary of random walks on countable groups preserved under any stopping times. Moreover, they show that the results hold for randomized stopping times with finite logarithmic moment. They first lift random walks on a countable group to a random walk on a free finitely or infinitely countable generated) semigroup whose first step of the random walk is distributed on the generators of the free semigroup, and show their results in this setup and then apply them to prove the results for any countable groups, see \\cite{Behrang} and \\cite{Forghani-Kaimanovich2016} for more details. However, our result in the above example provides a different proof for special stopping times on finitely generated free semigroups.\n\n\\subsection{ $\\boldsymbol\\rho$ is not $\\boldsymbol\\tau$}\nWe want to show that also in general choosing $\\rho=\\tau$ will not satisfy Assumption~\\ref{as:asymp}.  Let $\\EuScript{X}$ be the free  semigroup generated by ${\\cal G}=\\{a,b\\}$. Then for every $x$ we define  transition function   on ${\\EuScript{X}}$ by $$ p(x,xa)=p(x,xb)=\\frac12.$$ \nFor any sample path $\\boldsymbol{x}=(x_0,x_1,x_2,\\cdots)$ and $x$ in $\\EuScript{X}$, define the stopping time  \n$$\\tau(\\boldsymbol{x})=\\begin{cases}\n1 & x_0=x,\\ x_1=xa\\\\\n2 & x_0=x,\\ x_1=xb.\n\\end{cases}$$\n\n\nWe claim that $\\tau$ does not satisfy Assumption~\\ref{as:asymp}.  By contradiction,  suppose that  $\\rho=\\tau$. \n Then \n\\begin{align*} \n\t \\mbox{\\bf P}_x (\\rho_n \\in \\langle \\tau \\rangle )&=\\mbox{\\bf P}_x(\\rho_n \\in  \\langle \\tau \\rangle | n \\in \\langle \\tau \\rangle  )\\mbox{\\bf P}_x( n \\in \\langle \\tau \\rangle)+\\mbox{\\bf P}_x(\\rho_n \\in  \\langle \\tau \\rangle | n \\not \\in \\langle \\tau \\rangle  )\\mbox{\\bf P}_x( n \\not\\in \\langle \\tau \\rangle)\\\\\n\t &= 1 \\times \\mbox{\\bf P}_x( n \\in \\langle \\tau \\rangle) + \\frac 34 (1- \\mbox{\\bf P}_x( n \\in \\langle \\tau \\rangle) )\\\\\n\t & = \\frac 14(3+\\mbox{\\bf P}_x( n \\in \\langle \\tau \\rangle)),\n\t\\end{align*}\n\twhere the $\\frac 34$ factor on the second line is because if $n \\not\\in \\langle \\tau \\rangle$, then $n+1 \\in \\langle \\tau \\rangle$ and so  $\\rho_n\\in  \\langle \\tau \\rangle$ if and only if either: i) $\\rho_n=n+1$, that is $x_n^{-1}x_{n+1}=a$; or ii)  $\\rho_n=n+2$ and $n+2\\in \\langle \\tau \\rangle$, that is  $x_{n}^{-1}x_{n+1} = b$ and $x_{n+1}^{-1}x_{n+2}=a$. Finally, letting $\\alpha_n = \\mbox{\\bf P}_x (n \\in \\langle \\tau \\rangle)\n$, we have $\\alpha_1=\\frac 12$, $\\alpha_2= \\frac 34$ and by conditioning on the first increment, for $n\\ge 3$, $\\alpha_n=\\frac 12 \\alpha_{n-1}+ \\frac 12 \\alpha_{n-2}$. As a result, $\\lim_{n\\to\\infty}\\alpha_n = \\frac 23$. \tIt follows that \n$$ \\displaystyle\\lim_{n\\to\\infty}\\mbox{\\bf P}_x (\\rho_n \\in \\langle \\tau \\rangle )= \\frac{11}{12}<1.$$ \n\n\\subsection{Delayed stopping} \nWe construct a Markov chain and a corresponding stopping time with the following property. There exist states such that for every choice of  $\\rho$, $\\mbox{\\bf P}_x (\\rho_n \\in \\langle \\tau \\rangle)<1$, for all $n$ large. Nevertheless,  we can define   $\\rho$ so that   $\\displaystyle\\lim_{n\\to\\infty} \\mbox{\\bf P}_x (\\rho_n \\in \\langle \\tau\\rangle) =1$ for all $x\\in \\EuScript{X}$. \\\\\n Let ${\\EuScript{X}}={\\mathbb N}\\times \\{0,1\\}$. Let $r$ be any irreducible and transient transition function on ${\\mathbb N}$. We will also assume that $r$ is lazy, that is $r(x,x) =\\frac 12$ for all $x$.  Let $s$ be a transition function on $\\{0,1\\}$ such that  $0$ is an absorbing state, $s(0,0)=1$, and $s(1,0)=s(0,1)=\\frac12$.\n For $n_1,n_2 \\in {\\mathbb N}$ and $d_1,d_2 \\in \\{0,1\\}$, let \n$$\np((n_1,d_1),(n_2,d_2)) = r(n_1,n_2)s(d_1,d_2).\n$$\nThe space of sample paths of the Markov chain  with transition function  $p$ can be expressed in terms of the component processes. That is, if $\\boldsymbol{x}$ denotes that chain, then  $\\boldsymbol{x}=(\\boldsymbol{x}(1),\\boldsymbol{x}(2))$, with $\\boldsymbol{x}(1)$ and  $\\boldsymbol{x}(2)$ are two independent sample paths with respect to $r$ and $s$, receptively.  Assume that $A$ and $B$ are disjoint recurrent sets for $r$. Then clearly, both $A$ and $B$ are infinite.  Let $\\sigma = \\inf\\{n\\ge 1: x_n(1) \\in A\\}$, and continue inductively $\\sigma_1=\\sigma,~ \\sigma_{n+1} = \\sigma\\circ {S^{\\sigma_n}}$.  Define  \n$$\n\\tau = \\begin{cases} \n\\sigma & x_0(2) =0\\\\ \\sigma_{\\sigma} & x_0(2)=1 \n\\end{cases}\n$$ \nIn other words, if $x_0(2)=0$, we stop when $\\boldsymbol{x}(1)$ hits $A$. Otherwise, we wait until $\\sigma$, and then stop after $\\boldsymbol{x}(1)$ hits  $A$ an additional $\\sigma-1$ more times. Let $m$ be in $B$. Suppose that $\\rho:{\\EuScript{X}}^{{\\mathbb Z}_+}\\to {\\mathbb Z}_+$. \nWe begin by observing that \n$$\\mbox{\\bf P}_{(m,1)}(\\rho_n \\in \\langle \\tau\\rangle) \\le \\mbox{\\bf P}_{(m,1)}(\\sigma \\le n) + \\mbox{\\bf P}_{(m,1)} (\\rho_n \\in \\langle \\tau\\rangle,\\sigma>n).$$\nNow since $x_0(2)=1$, it follows from the definition of $\\tau$, that under $\\mbox{\\bf P}_{(m,1)}$, we have $\\tau=\\sigma_{{\\sigma}}$. On the event $\\sigma>n$, this automatically implies $\\tau_1>n$, and in particular, if $\\rho_n \\in \\langle \\tau\\rangle $, we must have $\\rho_n > \\sigma+ (\\sigma-1) > 2n$. As a result, the event $\\{\\rho_n \\in \\langle \\tau \\rangle\\}\\cap  \\{\\sigma>n \\} $ is contained in the event $\\{\\rho \\circ S^n > n\\}\\cap \\{\\sigma>n \\}$. This, with the Markov property imply:  \n\\begin{align} \\nonumber \\mbox{\\bf P}_{(m,1)}(\\rho_n \\in \\langle \\tau\\rangle) &\\le  \\mbox{\\bf P}_{(m,1)}(\\sigma \\le n)  + \\mbox{\\bf P}_{(m,1)}(\\rho\\circ S^n>n,\\sigma>n) \\\\\n\\label{eq:as} \n& =  \\mbox{\\bf P}_{(m,1)}(\\sigma \\le n) +  \\mbox{\\bf E}_{(m,1)} [\\mbox{\\bf P}_{\\boldsymbol{x}_n}(\\rho>n),\\sigma>n].\n\\end{align}\nSince we assume that $\\rho$ is finite $\\mbox{\\bf P}_{(m,0)}$ a.s., there exists $n_0\\in{\\mathbb N}$ such that  for all $n\\ge n_0$,  $\\mbox{\\bf P}_{(m,0)}(\\rho>n) <\\frac 12$. Under $\\mbox{\\bf P}_{m,1}$, the event \n$$\nC_n=\\{x_0(1)=x_1(1)=\\dots=x_n(1)\\}\\cap \\{x_n(2) = 0\\}\n$$ has probability $2^{-n} \\times (1-2^{-n})$, and is  contained in  $\\{\\sigma>n\\}$. Thus for $n\\ge n_0$, \n\\begin{align*} \n \\mbox{\\bf E}_{(m,1)} [\\mbox{\\bf P}_{\\boldsymbol{x}_n}(\\rho>n), \\sigma>n] &\\le  \\mbox{\\bf P}_{(m,1)} ( C_n)\\times  \\mbox{\\bf P}_{(m,0)}(\\rho>n)+ \\mbox{\\bf P}_{(m,1)} (C_n^c,\\sigma>n) \\\\\n & <\\frac 12 \\mbox{\\bf P}_{(m,1)}(C_n) +  \\mbox{\\bf P}_{(m,1)} (C_n^c ,\\sigma>n)\\\\\n & =  \\frac 12 \\mbox{\\bf P}_{(m,1)}(C_n) + \\mbox{\\bf P}_{(m,1)} (\\sigma>n) - \\mbox{\\bf P}_{(m,1)}(C_n,\\sigma>n)\\\\\n & =\\mbox{\\bf P}_{(m,1)} (\\sigma>n)  - \\frac 12 \\mbox{\\bf P}_{(m,1)}(C_n),\n \\end{align*}\n where the last equality is due to the fact that under $\\mbox{\\bf P}_{(m,1)}$, $C_n \\subseteq \\{\\sigma>n\\}$. Plug this inequality into  \\eqref{eq:as} to obtain \n $$  \\mbox{\\bf P}_{(m,1)}(\\rho_n \\in \\langle \\tau\\rangle) \\le 1 - \\frac 12 2^{ -n} (1-2^{-n}),~n\\ge n_0.$$ \n \nNext we show that choosing $\\rho=\\sigma$ satisfies $\\displaystyle\\lim_{n\\to\\infty} \\mbox{\\bf P}_{x} (\\rho_n \\in \\langle \\tau \\rangle) =1$, for all $x$ (note:  we can also choose $\\rho=\\tau$). Let $N_{i,j}$ count the number of times $\\boldsymbol{x}(1)$ visits $A$ between times $i$ and $j$. That is, \n$$\nN_{i,j} = \\sum_{i \\le n \\le j} {\\bf 1}_A (x_n(1)).\n$$\nObserve that from the definition of $\\tau$, if $x^2_k =0$, then the times of the $k$-th, $k+1$-th, etc. hits of $A$ by $\\boldsymbol{x}(1)$ will all be in $\\langle \\tau \\rangle$. Letting $T = \\inf \\{n: \\boldsymbol{x}_n(2) =0\\}$, we have \n$$ \\mbox{\\bf P}_x (\\rho_n \\in \\langle \\tau \\rangle) \\ge \\sum_{k} \\mbox{\\bf P}_x ( T = k,N_{k,n}\\ge k).$$ \nSince $A$ is recurrent, $ N_{k,n}\\underset{n\\to\\infty}{ \\nearrow}  \\infty$, and so the result follows from monotone convergence. \n\n\\subsection{Splitting of Poisson Boundary}\\label{splitting}\nThis final example is of a transition function $p$ and a stopping time  where $S^\\infty (\\EuScript{X},p)$ consists only of constant functions, yet $H^\\infty(\\EuScript{X},p^{\\tau})$ contains at least two linearly independent elements. In particular, there does not exist $\\rho$ satisfying Assumption \\ref{as:asymp}.\\\\\n Let $p$ be the transition function of the nearest neighbor symmetric random walk on $\\EuScript{X}={\\mathbb Z}^d$, $d\\ge 3$ (the assumption on the dimension is to ensure transience of $p$). By Ney and Spitzer \\cite{Ney-Spitzer}, both the  Poisson and Martin boundary with respect to $p$ are  trivial: all harmonic functions are constants. Write ${ x}=({ x}(1),\\dots, { x}(d))$. Define stopping times $T_{+},T_-,T_0$ as follows: \n$$T_+ = \\inf\\{n \\ge 1:{ x}_n(1)={x}_{0}(1)+ 1\\} \\hspace{1cm} T_- = \\inf\\{n \\ge 1:{ x}_n(1)={x}_{0}(1)- 1\\}$$\nand \n$$ T_0=\\inf\\{n\\ge 1:{x}_n(1) ={ x}_0(1)\\}.$$ \nFinally, let $T_{+,2} = T_{0}\\circ S^{T_{+}}$ and $ T_{-,2} = T_{0}\\circ S^{T_{-}}$.  In words, $T_\\pm$ is the first time ${x}(1)$ is one unit to the right (for $+$) or to the left (for $-$) of its starting point, and $T_{\\pm,2}$ is the second time ${x}(1)$ is one unit to the right ($+$) or one unit to the left ($-$) from its starting location. \n\nSet \n$$\n\\tau=\\begin{cases} T_+ \\wedge T_{-,2}\\mbox{ if } x_0(1)\\ge 0 \\\\ T_-\\wedge T_{+,2}\\mbox{ if } x_0(1) < 0. \\end{cases}\n$$\n\nObserve that by symmetry, $\\mbox{\\bf P}(T_+<T_-)=\\frac 12$, and $\\mbox{\\bf P}(T_+< T_{-,2})= \\frac 12+ \\frac 12 \\times \\frac 12 \\times \\frac 12=\\frac 58$. It immediately follows that $x_{\\tau}(1)$ is a nearest neighbor Markov chain on ${\\mathbb Z}$ with the following transition probabilities: \n$$ \nr(x,y)= \\begin{cases} \\frac 58 & x \\ge 0,~ y= x+1 \\mbox{ or } x<0, ~y=x-1 \\\\ \\frac 38 & x\\ge 0,~ y=x-1\\mbox{ or } x<0,~ y=x+1.\\end{cases}\n$$ \n\nIn particular, ${ x}^{\\tau}(1)$ is transient with positive drift on the positive half line and negative drift on the negative half line. Clearly,  $\\{\\lim  { x}_{\\tau_n}(1)=+\\infty\\}$ is invariant with respect to $\\boldsymbol{x}^{\\tau}$, therefore the function $\\mbox{\\bf P}_x (\\lim {x}_{\\tau_n}(1) = +\\infty)$ is a $p^{\\tau}$-harmonic function. It is easy to see that this function is not constant. Therefore the conclusion of Theorem \\ref{th:newmain} does not hold, which in turn implies that there does not exist a $\\rho$ satisfying Assumption \\ref{as:asymp}. Note that with this construction, $p^\\tau$ is irreducible.   As simpler, yet not irreducible example can be obtained by  defining $\\tau$ as the first time $x(1)$ is one unit away from $0$ than where it started at. In this case, $x_{\\tau}(1)$ is the Markov chain on ${\\mathbb Z}$ which jumps from $0$ to $\\pm 1$ with probability $1\/2$ each, and jumps from $x$ to $x+\\mbox{sgn}(x)$ with probability $1$ for all $x\\ne 0$.\n\\section{Appendix}\n\\begin{lemma}\n\t\\label{lem:partition}\n\tLet $\\mu$ be a Borel measure on a compact metric space $(M,d)$. Then for  every $\\delta$, there exists a number $n\\in{\\mathbb N}$ and Borel subsets  $N_0,N_1,\\dots,N_n$ of $M$ such that \n\t\\begin{enumerate} \n\t\t\\item $\\cup_{j=0}^n N_j = M$. \n\t\t\\item $N_j \\cap N_i =\\emptyset$ for $i\\ne j$. \n\t\t\\item $N_0$ is contained in the complement of $\\mbox{Supp}(\\mu)$. \n\t\t\\item For $j=1,\\dots,n$, the diameter of $N_j$ is $< \\delta$.  \n\t\t\\item $N_j,~j=1,\\dots,n$ contains a neighborhood of an element in the support of $\\mu$. \n\t\\end{enumerate} \n\\end{lemma}\n\\begin{proof}\n\tLet $B(x)$ denote the open ball of center $x$ with radius $\\delta\/2$. \n\tThe collection of balls $(B(x):x\\in \\mbox{Supp}(\\mu))$ is an open cover of the compact set $\\mbox{Supp}(\\mu)$. Therefore it possesses a finite subcover indexed by centers $x_1,\\dots,x_n$.  \n\tLet $N_1 =\\{y\\in B(x_1):d(y,x_1)\\le \\min_{j\\ne 1} d(y,x_j)\\}$. Then $x_1\\in N_1$, $N_1$ has nonempty interior, and $x_j\\not \\in N_1$ for all $j\\ne 1$. Continue inductively, letting \n\t$$N_{i+1}=\\{y\\in B(x_{i+1}):d(y,x_{i+1})\\le \\min_{j\\ne i+1} d(y,x_j)\\}-\\cup_{j\\le i}N_i.$$\n\tThus, $x_{i+1}\\in N_{i+1}$, $N_{i+1}$ has nonempty interior, and $x_j\\not \\in N_{i+1}$ for all $j\\le i+1$. Finally, let $N_0=M- \\cup_{j=1}^n N_j$. \n\\end{proof}  \n\\begin{prop}\n\t\\label{pr:approx}\n\tLet $\\mu$ and $\\nu$ be two probability measures on the Borel sets of  a compact metric space $(M,d)$, satisfying $\\mbox{Supp}(\\nu)\\subseteq \\mbox{Supp}(\\mu)$. Then there exists a sequence $(f_k:k\\in{\\mathbb N})$ of nonnegative simple functions such that $f_n d\\mu$ converges weakly to $\\nu$. That is, \n\t$$ \\lim_{n\\to\\infty} \\int g f_n d\\mu = \\int g d\\nu$$ \n\tfor every continuous $g$ on $M$. \n\\end{prop}\n\\begin{proof} \n\tBy lemma \\ref{lem:partition} for every $k\\in{\\mathbb N}$ we can find $n=n(k)$ and a partition  $N_0,N_1,\\dots,N_n$ of $M$ such that the diameter of $N_1,\\dots,N_n$ is $<\\frac{1}{k}$, $\\nu(N_0)=\\mu(N_0)=0$ and $\\mu(N_j)>0$ for $j=1,\\dots,n$. \n\tLet $f_k$ be the simple function equal to $0$ on $N_0$ and to $\\nu(N_j)\/\\mu(N_j)$ on $N_j$ for  $j=1,\\dots,n$. Then $\\int f_k d \\mu =1$. Next fix a continuous function $g$ on $M$. Then by compactness, $g$ is uniformly continuous. Fix $\\epsilon>0$, and choose $k$ to such that \n\t$|g(x) - g(y)|<\\epsilon$ whenever $d(x,y)<\\frac{1}{k}$.  Let $N_0,\\dots,N_n$ as above, and let $x_1,\\dots,x_n$ be arbitrary elements in $N_1,\\dots,N_n$, respectively.  Then \n\t$$| \\int g d\\nu - \\sum_{i=1}^n g(x_i) \\nu(N_i)| <\\epsilon$$ \n\tand  \n\t$$ |\\int g f_k d\\mu - \\sum_{i=1}^n g(x_i) \\nu(N_i) | <\\epsilon,$$ \n\tand therefore \n\t$$ |\\int gf_k d \\mu - \\int g d\\nu|<2\\epsilon,$$ \n\tcompleting the proof. \n\\end{proof} \n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}