diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzmfgw" "b/data_all_eng_slimpj/shuffled/split2/finalzzmfgw" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzmfgw" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\\label{sec:intro}\n\nThe mean-field (MF) theories provide us with good first approximation\nof ground states in various quantum many-body systems.\nFor many-fermionic systems,\nwe cannot exaggerate importance of the Hartree-Fock (HF) theory,\nwhich determines the one-body fields self-consistently\nwith respecting the variational principle~\\cite{ref:RS80}.\nThe HF theory is appropriate for defining\nsingle-particle (s.p.) orbitals of the constituent particles\nin the individual system.\nThe density-functional theory (DFT)~\\cite{ref:PY89,ref:ED11}\nhas mathematical and computational similarity to the HF theory,\nin the points that the ground state properties are obtained\nfrom the variational principle\nwith respect to the one-body fields,\nalthough there are no physical meaning in the s.p. orbitals.\nThe MF theory has been extended\nto the Hartree-Fock-Bogolyubov (HFB) theory~\\cite{ref:RS80},\nin which spontaneous breakdown of the particle-number conservation\n(\\textit{i.e.}, the global gauge symmetry) is taken into account\nwhile respecting the self-consistency and the variational principle.\nThe HFB theory is useful when two particles correlate as a pair\nand the system lies in the superconducting or the superfluid phase.\n\nThe random-phase approximation (RPA)~\\cite{ref:PB52}\ngives description of excited states,\nexcitation energies and transition strengths to be precise,\non top of the HF solution~\\cite{ref:Thou60}.\nIt is well linked to the linear response theory,\nand is obtained also as the small amplitude limit\nof the time-dependent HF theory~\\cite{ref:RS80}.\nAs in the similarity between the HF and the DFT,\nthe small amplitude limit of the time-dependent DFT~\\cite{ref:ED11}\nresembles the RPA.\nAnalogously to the RPA on the HF,\nthe quasiparticle-RPA is formulated on top of the HFB theory,\nwhich can describe excited states of systems in superfluidity.\n\nThe RPA and their extensions are basically built\nupon a solution in the corresponding MF theory.\nAs long as the MF state lies at a distinct minimum,\nthe RPA solutions can be regarded as excited states from the MF state.\nHowever, the RPA solutions could be unphysical\nif the MF state is actually at a saddle-point\nin the vector space defined in its vicinity.\nWhereas the RPA is extensively applied,\nstability of the RPA solutions is not always clear.\nIn localized self-bound systems like atomic nuclei,\nspontaneous symmetry breakdown (SSB) necessarily occurs\nin the MF regime~\\cite{ref:Thou60}.\nAn obvious example is the breaking of the translational symmetry,\nand another well-known example is the breaking of the rotational symmetry\nin deformed nuclei.\nThe SSB leads to a Nambu-Goldstone (NG) mode,\nwhich comes out as a zero-energy solution in the RPA.\nStructure of the vector space around the MF minimum\ncould be further complicated in the high-spin states~\\cite{ref:SM84}\nand along the path of the collective motion~\\cite{ref:HNMM}.\nIn numerical calculations within the MF theories,\nwe often assume certain symmetry and ignore some degrees of freedom (d.o.f.)\nto save computational resources.\nEven if a minimum is obtained in a MF calculation,\nit does not guarantee that it remains to be a minimum\nfor d.o.f. ignored in the calculation,\nas in a spherical HF calculation for a quadrupolarly deformed nucleus.\nWhereas stability of the RPA solutions in vicinity of the MF minimum\nwas investigated in Refs.~\\cite{ref:Thou61,ref:TV62}\nby Thouless \\textit{et al.},\nmore general arguments with rigorous mathematical treatment are desired\nfor a variety of extensions of the RPA developed to date.\n\nIn this article,\nI reanalyze properties of RPA solutions mathematically,\nin terms of the linear algebra.\nUnphysical solutions as well as physical solutions are examined\nin some detail.\nAlthough one may consider that unphysical solutions are just meaningless,\nthey are useful for understanding properties of the RPA equation\nmore profoundly.\nAs a result, they help us to comprehend\nin what manner physical solutions and NG modes come out.\nThe present analysis will be of practical significance as well\nin coding programs for numerical calculations in the RPA,\nbecause one often has to prepare for various situations in numerical studies,\nwithout knowing structure of the vector space around the MF state sufficiently.\n\n\\section{RPA equation}\\label{sec:RPAeq}\n\nThe RPA equation is written as\n\\begin{equation}\\begin{split}\n \\sum_\\beta \\big[A_{\\alpha\\beta}\\,X^{(\\nu)}_\\beta\n + B_{\\alpha\\beta}\\,Y^{(\\nu)}_\\beta\\big] &= \\omega_\\nu\\,X^{(\\nu)}_\\alpha\\,,\\\\\n \\sum_\\beta \\big[B_{\\alpha\\beta}^\\ast\\,X^{(\\nu)}_\\beta\n + A_{\\alpha\\beta}^\\ast\\,Y^{(\\nu)}_\\beta\\big] &= -\\omega_\\nu\\,Y^{(\\nu)}_\\beta\\,,\n\\end{split}\\label{eq:RPAeq-a}\\end{equation}\nwhere $\\alpha,\\beta$ represent particle-hole bases on the HF solution\nor two quasiparticle bases on the HFB solution.\nThe matrices $A$ and $B$ are obtained\nfrom the residual interaction and the HF s.p. (or the HFB q.p.) energies,\nand satisfy\n\\begin{equation} A=A^\\dagger\\,,\\quad B=B^T\\,. \\label{eq:sym-AB}\\end{equation}\nIn order to cope with a variety of physical situations,\nI start discussions only from this structure of the RPA equation,\nwithout any further assumptions.\nThe solution of Eq.~(\\ref{eq:RPAeq-a}) is\ncomprised of $\\omega_\\nu$ and $(X^{(\\nu)}, Y^{(\\nu)})$,\nwhich correspond to the energy and the wave function\nof the $\\nu$-th excited state.\nIt is imposed that $(X^{(\\nu)}, Y^{(\\nu)})$ obeys the \\textit{normalization},\n\\begin{equation}\\begin{split}\n\\sum_\\alpha \\big[X^{(\\nu)}_\\alpha X^{(\\nu')\\ast}_\\alpha\n - Y^{(\\nu)}_\\alpha Y^{(\\nu')\\ast}_\\alpha\\big] &= \\delta_{\\nu\\nu'}\\,,\\\\\n\\sum_\\alpha \\big[X^{(\\nu)}_\\alpha Y^{(\\nu')}_\\alpha\n - Y^{(\\nu)}_\\alpha X^{(\\nu')}_\\alpha\\big] &= 0\\,.\n\\end{split}\\label{eq:norm-a}\\end{equation}\nThe dimension of the $A$ and $B$ matrices is denoted by $D$.\n$D$ is finite in many practical calculations.\nAlthough $D$ can be infinite in principle,\nthe arguments here will cover infinite $D$ as a limiting case.\n\nProperties of the RPA solutions are better argued\nin terms of the $2D\\times 2D$ matrices,\n\\begin{equation}\n \\mathsf{S}:=\\begin{pmatrix} A&B\\\\ B^\\ast&A^\\ast \\end{pmatrix}\\,,~\n \\mathsf{N}:=\\begin{pmatrix} 1&0\\\\ 0&-1 \\end{pmatrix}\\,,~\n \\mathsf{\\Sigma}_x := \\begin{pmatrix} 0&1\\\\1&0 \\end{pmatrix}\\,.\n\\label{eq:matrices}\\end{equation}\nFor the matrices $\\mathsf{N}$ and $\\mathsf{\\Sigma}_x$,\n$\\mathsf{N}^2=\\mathsf{\\Sigma}_x^2=\\mathsf{1}$\nand $\\mathsf{\\Sigma}_x\\,\\mathsf{N}+\\mathsf{N}\\,\\mathsf{\\Sigma}_x=\\mathsf{0}$\nhold.\nThe matrix $\\mathsf{S}$ is nothing but\nthe curvature matrix at the HF (or HFB) solution,\nand is known as the stability matrix.\nWith the matrices defined in Eq.~(\\ref{eq:matrices}),\nthe RPA equation (\\ref{eq:RPAeq-a}) is expressed as\n\\begin{equation} \\mathsf{S}\\,\\vect{x}_\\nu=\\omega_\\nu\\mathsf{N}\\,\\vect{x}_\\nu\\,;\\quad\n \\vect{x}_\\nu=\\begin{pmatrix} X^{(\\nu)}\\\\ Y^{(\\nu)}\\end{pmatrix}\\,,\n\\label{eq:RPAeq-b}\\end{equation}\nand the normalization condition (\\ref{eq:norm-a}) as\n\\begin{equation} \\vect{x}_\\nu^\\dagger\\,\\mathsf{N}\\,\\vect{x}_{\\nu'}=\\delta_{\\nu\\nu'}\\,,~\n \\vect{x}_\\nu^T\\mathsf{\\Sigma}_x\\,\\mathsf{N}\\,\\vect{x}_{\\nu'}=0\\,.\n\\label{eq:norm-b}\\end{equation}\nThe number of the solutions satisfying (\\ref{eq:norm-b})\nmatches the dimension $D$ of the $A,B$ matrix in ideal cases.\nHowever, it is not obvious whether there exist $D$ solutions\nthat satisfy the normalization condition (\\ref{eq:norm-b}).\n\n\\begin{definition}\\label{def:solvable}\nIf there exist $D$ independent solutions of $(\\omega_\\nu, \\vect{x}_\\nu)$\nthat satisfy Eqs.~(\\ref{eq:RPAeq-b}), (\\ref{eq:norm-b}) and $\\omega_\\nu>0$,\nthe RPA equation is said \\textbf{fully solvable}.\n\\end{definition}\n\n\\section{General properties of RPA solutions}\\label{sec:general}\n\n\\subsection{Dualities and eigenvalues}\\label{subsec:duality}\n\nThe RPA equation (\\ref{eq:RPAeq-b}) is equivalent\nto the eigenvalue problem of the matrix $\\mathsf{N\\,S}$.\nAn eigensolution is defined by a set of an eigenvalue\nand an eigenvector $(\\omega_\\nu,\\vect{x}_\\nu)$.\nAt the same time, Eq.~(\\ref{eq:RPAeq-b}) reads\nthe eigenvalue problem of $\\mathsf{S\\,N}$\nfor an eigensolution $(\\omega_\\nu,\\mathsf{N}\\,\\vect{x}_\\nu)$.\nThis duality is important to derive basic properties of the RPA solutions.\nThe relevant duality will be established later,\nand is called \\textit{LR-duality} in this article,\nbecause it connects left and right basis vectors.\n\nAnother important ingredient is the symmetry of $\\mathsf{S}$.\nThe structure of $\\mathsf{S}$ given in Eq.~(\\ref{eq:matrices})\nwith (\\ref{eq:sym-AB}) is characterized by\n\\begin{equation} \\mathsf{S}=\\mathsf{S}^\\dagger\\,,\\quad\n \\mathsf{\\Sigma}_x\\,\\mathsf{S}^\\ast\\,\\mathsf{\\Sigma}_x = \\mathsf{S}\\,.\n\\label{eq:S-prop}\\end{equation}\nIt should be noted that, owing to the first equation of (\\ref{eq:S-prop}),\nan eigenvector of $\\mathsf{S\\,N}$ associated with an eigenvalue $\\omega_\\nu$\nimmediately gives a left eigenvector of $\\mathsf{N\\,S}$\ncorresponding to the eigenvalue $\\omega_\\nu^\\ast$,\nbecause $\\mathsf{S\\,N}\\,\\vect{y}_\\nu = \\omega_\\nu\\,\\vect{y}_\\nu$\nis equivalent to $\\vect{y}_\\nu^\\dagger\\,\\mathsf{N\\,S}\n= \\omega_\\nu^\\ast\\,\\vect{y}_\\nu^\\dagger$.\n\\begin{proposition}\\label{theor:sym-eigen}\nIf $\\omega_\\nu$ is an eigenvalue of $\\mathsf{N\\,S}$,\n$-\\omega_\\nu$ is also an eigenvalue with equal degeneracy.\nSo is $\\omega_\\nu^\\ast$.\n\\end{proposition}\n\\begin{proof}\nEquation~(\\ref{eq:RPAeq-b}) gives the secular equation, whose l.h.s. is\n\\begin{equation}\\begin{split}\n \\det(\\mathsf{S}-\\omega\\,\\mathsf{N})\n &= \\det\\big([\\mathsf{S}-\\omega\\,\\mathsf{N}]^T\\big)\n = \\big[\\det(\\mathsf{S}-\\omega^\\ast\\,\\mathsf{N})\\big]^\\ast\\\\\n &= \\det(\\mathsf{\\Sigma}_x\\,\\mathsf{S}^\\ast\\,\\mathsf{\\Sigma}_x\n -\\omega\\,\\mathsf{N})\n = \\det(\\mathsf{S}^\\ast\n -\\omega\\,\\mathsf{\\Sigma}_x\\,\\mathsf{N}\\,\\mathsf{\\Sigma}_x)\\\\\n &= \\det(\\mathsf{S}^\\ast+\\omega\\,\\mathsf{N})\n = \\big[\\det(\\mathsf{S}+\\omega^\\ast\\,\\mathsf{N})\\big]^\\ast\\,,\n\\end{split}\\label{eq:secular-lhs}\\end{equation}\nbecause of Eq.~(\\ref{eq:S-prop}).\nThus the equations $\\det(\\mathsf{S}-\\omega\\,\\mathsf{N})=0$,\n$\\det(\\mathsf{S}-\\omega^\\ast\\,\\mathsf{N})=0$\nand $\\det(\\mathsf{S}+\\omega^\\ast\\,\\mathsf{N})=0$ are all equivalent,\nensuring correspondence of the eigenvalues $\\omega$, $\\omega^\\ast$\nand $-\\omega^\\ast$.\nThis leads to the eigenvalue $-\\omega$ as well.\n\\end{proof}\n\\noindent\nIt is straightforward to show the following corollary\nfrom the second equation of (\\ref{eq:S-prop}).\n\\begin{corollary}\\label{lem:sym-sigma}\nIf $(\\omega_\\nu, \\vect{x}_\\nu)$ is an eigensolution of Eq.~(\\ref{eq:RPAeq-b}),\n$(-\\omega_\\nu^\\ast, \\mathsf{\\Sigma}_x\\vect{x}_\\nu^\\ast)$ is also a solution.\n\\end{corollary}\n\\noindent\nThis relation indicates another kind of duality,\nwhich is called \\textit{UL-duality} to distinguish from the LR-duality,\nsince it is related to interchange (with taking complex conjugate)\nof the upper and the lower components of the basis vectors.\nWith respect to the normalization of Eq.~(\\ref{eq:norm-b}),\nthe following relation is obtained,\n\\begin{equation} (\\mathsf{\\Sigma}_x\\vect{x}_\\nu^\\ast)^\\dagger\\,\\mathsf{N}\\,\n (\\mathsf{\\Sigma}_x\\vect{x}_{\\nu'}^\\ast)\n = -\\vect{x}_{\\nu'}^\\dagger\\,\\mathsf{N}\\,\\vect{x}_\\nu\\,. \\end{equation}\nThe second equation of (\\ref{eq:norm-b}) is interpreted as a relation\nbetween $\\mathsf{\\Sigma}_x\\vect{x}_\\nu^\\ast$ and $\\vect{x}_{\\nu'}$,\n$(\\mathsf{\\Sigma}_x\\vect{x}_\\nu^\\ast)^\\dagger\\,\\mathsf{N}\\,\\vect{x}_{\\nu'}=0$.\n\nFor an arbitrary vector\n${\\displaystyle\\vect{x}=\\begin{pmatrix} X\\\\ Y\\end{pmatrix}}$,\nwe have $\\vect{x}^\\dagger\\,\\mathsf{N}\\,\\vect{x}\n= X^\\dagger\\,X - Y^\\dagger\\,Y$.\nIn this respect the `norm' is not positive-definite.\n\\begin{definition}\\label{def:normalizable}\nA vector $\\vect{x}$ is said \\textbf{normalizable}\nwhen $\\vect{x}^\\dagger\\,\\mathsf{N}\\,\\vect{x}>0$.\n\\end{definition}\n\\noindent\nIndeed, if $\\vect{x}^\\dagger\\,\\mathsf{N}\\,\\vect{x}=r^2\\,(>0)$,\n$\\vect{x}\/r$ is normalized in the respect of Eq.~(\\ref{eq:norm-b}).\nAs noted above,\nit is not guaranteed that an eigenvector $\\vect{x}_\\nu$ of $\\mathsf{N\\,S}$\ncan be normalized.\nThe normalizability of an eigenvector $\\vect{x}_\\nu$\nis a key to whether the RPA equation is solvable.\n\nIn arguments with respect to the RPA~\\cite{ref:Thou61},\nthe `orthogonality' between two vectors $\\vect{x}$ and $\\vect{y}$\nis sometimes defined when $\\vect{x}^\\dagger\\,\\mathsf{N}\\,\\vect{y}=0$,\nby regarding $\\mathsf{N}$ as the metric.\nHowever, in this article I use the usual definition\nthat $\\vect{x}$ and $\\vect{y}$ are orthogonal\nwhen $\\vect{x}^\\dagger\\,\\vect{y}=0$.\n\nThe hermiticity of $\\mathsf{S}$ leads to the relation\nbetween the solutions of Eq.~(\\ref{eq:RPAeq-b}),\n\\begin{equation} \\vect{x}_\\nu^\\dagger\\,\\mathsf{S}\\,\\vect{x}_{\\nu'}\n = \\omega_\\nu^\\ast\\,\\vect{x}_\\nu^\\dagger\\,\\mathsf{N}\\,\\vect{x}_{\\nu'}\n = \\omega_{\\nu'}\\,\\vect{x}_\\nu^\\dagger\\,\\mathsf{N}\\,\\vect{x}_{\\nu'}\\,.\n\\label{eq:norm-relation}\\end{equation}\nEquation~(\\ref{eq:norm-relation}) concludes:\n\\begin{lemma}\\label{lem:bi-orthogonality}\nFor eigensolutions $(\\omega_\\nu, \\vect{x}_\\nu)$\nand $(\\omega_{\\nu'}, \\vect{x}_{\\nu'})$ of the RPA equation (\\ref{eq:RPAeq-b}),\n$\\omega_\\nu^\\ast=\\omega_{\\nu'}$\nor $\\vect{x}_\\nu^\\dagger\\,\\mathsf{N}\\,\\vect{x}_{\\nu'}=0$ follows.\n\\end{lemma}\n\\noindent\nThe case of $\\nu=\\nu'$ was argued in Ref.~\\cite{ref:Thou61}.\n\nIt is now possible to classify solutions of Eq.~(\\ref{eq:RPAeq-b}).\n\\begin{proposition}\\label{prop:eigenvalues}\nEigenvalues (denoted by $\\omega_\\nu$) of $\\mathsf{N\\,S}$ come out\nin one of the following manners:\n\\begin{enumerate}\n\\item\\label{item:positive}\n $\\omega_\\nu>0$ with a normalizable eigenvector $\\vect{x}_\\nu$,\n in association with another eigensolution\n $(-\\omega_\\nu, \\mathsf{\\Sigma}_x\\vect{x}_\\nu^\\ast)$.\n\\item\\label{item:negative}\n $\\omega_\\nu>0$ with an unnormalizable eigenvector,\n in association with an eigenvalue $-\\omega_\\nu$ that could be normalizable.\n\\item\\label{item:pure-imag}\n a pair of pure imaginary eigenvalues, $\\pm\\omega_\\nu$\n with $\\mathrm{Re}(\\omega_\\nu)=0$, $\\mathrm{Im}(\\omega_\\nu)\\ne 0$.\n\\item\\label{item:quartet}\n a quartet of complex eigenvalues, $\\pm\\omega_\\nu, \\pm\\omega_\\nu^\\ast$\n with $\\mathrm{Re}(\\omega_\\nu)\\ne 0$, $\\mathrm{Im}(\\omega_\\nu)\\ne 0$.\n\\item\\label{item:null} a null eigenvalue.\n\\end{enumerate}\n\\end{proposition}\n\\noindent\nA solution belonging to Class~\\ref{item:positive}\nof Prop.~\\ref{prop:eigenvalues} may be called \\textit{physical solution},\nwhile a solution to one of \\ref{item:negative}\\,--\\,\\ref{item:quartet}\n\\textit{unphysical solution}.\nClass~\\ref{item:null} is closely connected to the NG mode,\nand a solution belonging to it will be called \\textit{NG-mode solution}.\nFocusing on solutions in vicinity of a MF minimum,\nThouless did not discuss\nsolutions of Classes~\\ref{item:negative} and \\ref{item:quartet}\nin Refs.~\\cite{ref:Thou61,ref:TV62}.\n\n\\subsection{Basis vectors in Jordan blocks}\\label{subsec:Jordan}\n\nSeveral eigenvalues of $\\mathsf{N\\,S}$ could be degenerate,\nand degenerate eigenvalues may give rise to Jordan blocks.\nThis possibility is examined in this subsection.\nSuppose that a Jordan block is generated from an eigenvector $\\vect{x}_\\nu$.\nA basis vector of the Jordan block is denoted by $\\vect{\\xi}_k^{(\\nu)}$,\nwhich is obtained by\n\\begin{equation} \\mathsf{S}\\,\\vect{\\xi}_{k+1}^{(\\nu)}\n = \\omega_\\nu\\,\\mathsf{N}\\,\\vect{\\xi}_{k+1}^{(\\nu)}\n + ic_k^{(\\nu)}\\,\\mathsf{N}\\,\\vect{\\xi}_k^{(\\nu)}\\,.\\quad\n (c_k^{(\\nu)}\\in\\mathbf{C})\n\\label{eq:Jordan}\\end{equation}\nThe constant $c_k^{(\\nu)}$ is subject to normalization and to relative phase\nof $\\vect{\\xi}_k^{(\\nu)}$ and $\\vect{\\xi}_{k+1}^{(\\nu)}$.\nAlthough it is taken to be $ic_k^{(\\nu)}=1$ in the Jordan normal form,\nanother normalization will be adopted in Sec.~\\ref{subsec:imag-sol}.\nStarting from $\\vect{\\xi}_1^{(\\nu)}=\\vect{x}_\\nu$,\n$\\vect{\\xi}_{k+1}^{(\\nu)}$ and $c_k^{(\\nu)}\\,(\\ne 0)$ can be fixed\nsuccessively for $k\\,(\\geq 1)$.\nThis chain of equations ends at a certain $k$,\nwhere no $\\vect{\\xi}_{k+1}^{(\\nu)}\\,(\\ne\\vect{0})$ exists.\nDimension of individual Jordan block is determined\nby how long the chain continues, and is denoted by $d_\\nu$.\nEquation~(\\ref{eq:Jordan}) defines a Jordan block of $\\mathsf{S\\,N}$ as well,\nwhose basis vectors are $\\mathsf{N}\\,\\vect{\\xi}_k^{(\\nu)}$.\n\nMost of the following arguments will cover the case of $d_\\nu=1$;\n\\textit{i.e.}, the case that an eigenvector $\\vect{x}_\\nu$\ndoes not generate a Jordan block.\n\\begin{lemma}\\label{lem:Jordan-sym}\nIt there is a Jordan block for an eigenvalue $\\omega_\\nu$ with dimension $d_\\nu$,\nso are for $-\\omega_\\nu$ and $\\omega_\\nu^\\ast$ with equal dimension $d_\\nu$.\n\\end{lemma}\n\\begin{proof}\nA Jordan block for the eigenvalue $\\omega_\\nu$ of $\\mathsf{S\\,N}$\ndirectly corresponds to a Jordan block for $\\omega_\\nu^\\ast$ of $\\mathsf{N\\,S}$,\nby regarding the right basis vectors of $\\mathsf{S\\,N}$\nas the left basis vectors of $\\mathsf{N\\,S}$.\nTherefore the dimensions corresponding to $\\omega_\\nu$ and $\\omega_\\nu^\\ast$\nmust be equal.\\\\\nIt follows from Eqs.~(\\ref{eq:S-prop}) and (\\ref{eq:Jordan}) that,\n\\begin{equation} \\mathsf{S}\\,\\mathsf{\\Sigma}_x\\vect{\\xi}_{k+1}^{(\\nu)\\ast}\n = -\\omega_\\nu^\\ast\\,\\mathsf{N}\\,\\mathsf{\\Sigma}_x\\vect{\\xi}_{k+1}^{(\\nu)\\ast}\n + ic_k^{(\\nu)\\ast}\\,\\mathsf{N}\\,\\mathsf{\\Sigma}_x\\vect{\\xi}_k^{(\\nu)\\ast}\\,,\n\\label{eq:Jordan-sym}\\end{equation}\nverifying that there exists a Jordan block for $-\\omega_\\nu^\\ast$\nwith equal dimension.\nThen, so is it for $-\\omega_\\nu$.\n\\end{proof}\n\\begin{definition}\\label{def:ULdual}\nThe basis vectors $\\vect{\\xi}_k^{(\\nu)}$\nand $\\mathsf{\\Sigma}_x\\vect{\\xi}_k^{(\\nu)\\ast}$\nare said to be \\textbf{UL-dual} of each other.\n\\end{definition}\n\\noindent\nIf $\\vect{\\xi}_k^{(\\nu)}=\\mathsf{\\Sigma}_x\\vect{\\xi}_k^{(\\nu)\\ast}$,\nit is said to be \\textit{self UL-dual}.\n\nIn general, a single eigenvalue $\\omega_\\nu$ may give plural Jordan blocks.\nI hereafter reserve the subscript $\\nu$\nfor representing individual Jordan block,\nwhich is connected to a single eigenvector $\\vect{x}_\\nu$ of $\\mathsf{N\\,S}$,\nrather than the eigenvalue.\nThe following proposition and lemmas are closely connected\nto the structure of the Jordan block,\n\\begin{equation} \\begin{pmatrix} \\omega_\\nu & ic_1^{(\\nu)} & 0 & \\cdots & 0 \\\\\n 0 & \\omega_\\nu & ic_2^{(\\nu)} & \\ddots & 0 \\\\\n \\vdots & \\vdots & \\ddots & \\ddots & \\vdots \\\\\n 0 & 0 & 0 & & ic_{d_\\nu-1}^{(\\nu)} \\\\\n 0 & 0 & 0 & \\cdots & \\omega_\\nu \\end{pmatrix}\\,,\n\\label{eq:Jordan-block}\\end{equation}\nwhich is represented with the left and right basis vectors\nthat form the inverse matrix of each other apart from normalization.\nExplicit proofs of Lemma~\\ref{lem:Jordan-orthogonality} to\nProp.~\\ref{theor:one-to-one_basis} are given\nin Appendix~\\ref{app:proof-Jordan},\nwhich are instructive and useful\nto confirm their compatibility with later propositions and lemmas.\n\\begin{lemma}\\label{lem:Jordan-orthogonality}\nUnless eigenvalues $\\omega_\\nu^\\ast$ and $\\omega_{\\nu'}$ are equal,\nany basis vector of $\\mathsf{N\\,S}$ associated with $\\omega_\\nu$\n(either an eigenvector or a basis vector belonging to a Jordan block)\nis orthogonal to any basis vector of $\\mathsf{S\\,N}$\nassociated with $\\omega_{\\nu'}$.\n\\end{lemma}\n\nThe above lemma suggests that\na Jordan block of $\\mathsf{N\\,S}$ for a specific $\\omega_\\nu$\nand a Jordan block of $\\mathsf{S\\,N}$ for $\\omega_\\nu^\\ast$ are paired.\n\\begin{lemma}\\label{lem:Jordan-normalizability}\nIt is possible to take so that\neach basis vector in a Jordan block of $\\mathsf{N\\,S}$\ncould overlap with no more than one basis vector\nthat contained in a Jordan block of $\\mathsf{S\\,N}$.\nIn order for the overlap not to vanish,\ndimensions of these Jordan blocks of $\\mathsf{N\\,S}$ and $\\mathsf{S\\,N}$\nmust be equal.\n\\end{lemma}\n\\noindent\nThe next corollary follows\nfrom the proof of Lemma~\\ref{lem:Jordan-normalizability}\ngiven in Appendix~\\ref{app:sub-A2}:\n\\begin{corollary}\\label{cor:non-Jordan}\nIf two eigenvectors of $\\vect{x}_\\nu$ and $\\vect{x}_{\\nu'}$ of $\\mathsf{N\\,S}$\nsatisfies $\\vect{x}_\\nu^\\dagger\\,\\mathsf{N}\\,\\vect{x}_{\\nu'}\\ne 0$,\nboth of them do not constitute Jordan blocks (\\textit{i.e.}, $d_\\nu=d_{\\nu'}=1$).\n\\end{corollary}\n\\begin{proposition}\\label{theor:one-to-one_basis}\nIt is possible to produce a complete set of basis vectors of $\\mathsf{N\\,S}$,\nby a proper transformation if necessary,\nso that each of them could overlap with only one basis vector of $\\mathsf{S\\,N}$.\nOne-to-one correspondence is established\nbetween Jordan blocks of $\\mathsf{N\\,S}$ and $\\mathsf{S\\,N}$\nthat contain basis vectors having non-vanishing overlaps.\nThe correspondence between basis vectors of $\\mathsf{N\\,S}$ and $\\mathsf{S\\,N}$\nis also one to one.\n\\end{proposition}\n\\noindent\nThis proposition is an expression that the left and the right basis vectors\ngiving the Jordan representation constitute the inverse matrix of each other.\n\nProposition~\\ref{theor:one-to-one_basis} enables us to define\na basis vector of $\\mathsf{S\\,N}$\nthat is dual of individual basis vector of $\\mathsf{N\\,S}$.\nLet us denote the basis vector dual to $\\vect{\\xi}_k^{(\\nu)}$\nby $\\bbar{\\vect{\\xi}}_k^{(\\nu)}$.\nThen $\\bbar{\\vect{\\xi}}_k^{(\\nu)\\dagger}\\,\\mathsf{N}\\,\\vect{\\xi}_{k'}^{(\\nu')}$\ndoes not vanish only for $\\nu=\\nu'$ and $k=k'$.\nThis is a realization of the Jordan block of (\\ref{eq:Jordan-block}).\n$\\bbar{\\vect{\\xi}}_k^{(\\nu)}$ is the $(d_\\nu+1-k)$-th basis vector\nof the Jordan block for an eigenvector\n$\\bbar{\\vect{x}}_\\nu:=\\bbar{\\vect{\\xi}}_{d_\\nu}^{(\\nu)}$,\nwhich is associated with the eigenvalue $\\omega_\\nu^\\ast$.\nInverting Eq.~(\\ref{eq:Jordan}), we obtain\n\\begin{equation} \\mathsf{S}\\,\\bbar{\\vect{\\xi}}_{k-1}^{(\\nu)}\n = \\omega_\\nu^\\ast\\,\\mathsf{N}\\,\\bbar{\\vect{\\xi}}_{k-1}^{(\\nu)}\n - ic_{k-1}^{(\\nu)\\ast}\\,\n \\frac{\\vect{\\xi}_{k-1}^{(\\nu)\\dagger}\\,\\mathsf{N}\\,\\bbar{\\vect{\\xi}}_{k-1}^{(\\nu)}}\n {\\vect{\\xi}_k^{(\\nu)\\dagger}\\,\\mathsf{N}\\,\\bbar{\\vect{\\xi}}_k^{(\\nu)}}\\,\n \\mathsf{N}\\,\\bbar{\\vect{\\xi}}_k^{(\\nu)}\\,,\n\\label{eq:Jordan-inv}\\end{equation}\nas confirmed in Appendix~\\ref{app:sub-A4}.\nThis establishes the LR-duality of basis vectors.\n\\begin{definition}\\label{def:LRdual}\nThe basis vectors $\\vect{\\xi}_k^{(\\nu)}$ and $\\bbar{\\vect{\\xi}}_k^{(\\nu)}$\nare said to be \\textbf{LR-dual} of each other.\n\\end{definition}\n\\noindent\nWhen $\\vect{\\xi}_k^{(\\nu)}=\\bbar{\\vect{\\xi}}_k^{(\\nu)}$,\nit is said \\textit{self LR-dual}.\nIt is reasonable to define $\\bbar{\\vect{\\xi}}_k^{(\\nu)}$\nso that it would fulfill\n$\\big|\\bbar{\\vect{\\xi}}_k^{(\\nu)\\dagger}\\,\\mathsf{N}\\,\\vect{\\xi}_{k'}^{(\\nu')}\\big|\n=\\delta_{\\nu\\nu'}\\,\\delta_{kk'}$.\nIt is not necessarily convenient to assume\n$\\bbar{\\vect{\\xi}}_k^{(\\nu)\\dagger}\\,\\mathsf{N}\\,\\vect{\\xi}_{k'}^{(\\nu')}\n=\\delta_{\\nu\\nu'}\\,\\delta_{kk'}$, though possible.\nThe normalizability in Def.~\\ref{def:normalizable}\nis now perceived as a part of the self LR-duality.\n\n\\subsection{Redefining eigenvalue problem}\\label{subsec:redefine}\n\nBecause $\\mathsf{S}$ is hermitian,\nit is diagonalizable with an appropriate unitary matrix $\\mathsf{U}$,\n\\begin{equation} \\mathsf{S}=\\mathsf{U}^{-1}\\,\\mathrm{diag.}(\\lambda_i)\\,\\mathsf{U}\\,.\\quad\n (\\lambda_i\\in\\mathbf{R})\n\\label{eq:S-diag}\\end{equation}\nBy using this expression, $\\mathsf{S}^{1\/2}$ can be taken as\n\\begin{equation} \\mathsf{S}^{1\/2}=\\mathsf{U}^{-1}\\,\\mathrm{diag.}(\\lambda_i^{1\/2})\\,\\mathsf{U}\\,.\n\\label{eq:S-half}\\end{equation}\nThough $\\lambda_i^{1\/2}$ is two-valued, it is not important\nwhich value is adopted.\nA new matrix is now defined,\n\\begin{equation} \\tilde{\\mathsf{S}}:=\\mathsf{S}^{1\/2}\\,\\mathsf{N\\,S}^{1\/2}\\,.\n\\label{eq:S-tilde}\\end{equation}\nObviously, $\\tilde{\\mathsf{S}}$ is hermitian\nonly when $\\lambda_i\\geq 0$ for any $i\\,(=1,\\cdots,2D)$.\n\n\\begin{lemma}\\label{lem:tilde-S_eigen}\nAll the eigenvalues and eigenvectors of $\\mathsf{N\\,S}$\ncorrespond to those of $\\tilde{\\mathsf{S}}$, and vice versa.\n\\end{lemma}\n\\begin{proof}\nIf $\\mathsf{S}\\,\\vect{x}_\\nu=\\omega_\\nu\\mathsf{N}\\,\\vect{x}_\\nu$,\n$\\tilde{\\mathsf{S}}\\,(\\mathsf{S}^{1\/2}\\vect{x}_\\nu)\n =\\omega_\\nu(\\mathsf{S}^{1\/2}\\vect{x}_\\nu)$ follows.\nTherefore $(\\omega_\\nu, \\mathsf{S}^{1\/2}\\vect{x}_\\nu)$\ngives an eigensolution of $\\tilde{\\mathsf{S}}$\nunless $\\mathsf{S}^{1\/2}\\vect{x}_\\nu=\\vect{0}$.\nConversely, if $\\tilde{\\mathsf{S}}\\,\\vect{y}_\\nu=\\omega_\\nu\\vect{y}_\\nu$,\n$\\mathsf{S}\\,(\\mathsf{N\\,S}^{1\/2}\\vect{y}_\\nu)\n =\\omega_\\nu\\,\\mathsf{N}\\,(\\mathsf{N\\,S}^{1\/2}\\vect{y}_\\nu)$ follows.\\\\\nIf $\\mathsf{S}^{1\/2}\\vect{x}_\\nu=\\mathbf{0}$,\nwe have $\\mathsf{S}\\,\\vect{x}_\\nu=\\tilde{\\mathsf{S}}\\,\\vect{x}_\\nu=0$,\nindicating $\\vect{x}_\\nu$ is an eigenvector corresponding to the null eigenvalue\nboth of $\\mathsf{N\\,S}$ and $\\tilde{\\mathsf{S}}$.\n\\end{proof}\n\nThus the RPA equation is equivalent\nto the eigenvalue problem of $\\tilde{\\mathsf{S}}$;\n$\\tilde{\\mathsf{S}}\\,\\vect{x}_\\nu=\\omega_\\nu\\vect{x}_\\nu$.\nThis redefinition is applied to investigate solvability of the RPA equation\nin Sec.~\\ref{subsec:real-sol}.\n\n\\section{Decomposition of vector space}\\label{sec:decomp}\n\nThe whole vector space $\\mathcal{V}$,\nin which the stability matrix $\\mathsf{S}$ is defined,\ncan be decomposed via the basis vectors produced by $\\mathsf{N\\,S}$\nor those by $\\mathsf{S\\,N}$.\nThis furnishes further discussion on properties of the RPA solutions.\n\nRecalling the LR-duality explored in Sec.~\\ref{sec:general},\nwe obtain the projector\nwhich separates out the direction along a certain basis vector\nof $\\mathsf{N\\,S}$, as in Ref.~\\cite{ref:TV62},\n\\begin{equation} \\mathsf{\\Lambda}_{\\nu,k} :=\n \\frac{\\vect{\\xi}_k^{(\\nu)}\\,\\bbar{\\vect{\\xi}}_k^{(\\nu)\\dagger}\\,\\mathsf{N}}\n {\\bbar{\\vect{\\xi}}_k^{(\\nu)\\dagger}\\,\\mathsf{N}\\,\\vect{\\xi}_k^{(\\nu)}}\\,.\n\\label{eq:proj-nuk}\\end{equation}\nObviously $\\mathsf{\\Lambda}_{\\nu,k}\\,\\mathsf{\\Lambda}_{\\nu',k'}\n=\\delta_{\\nu\\nu'}\\,\\delta_{kk'}\\,\\mathsf{\\Lambda}_{\\nu,k}$.\nThe projector separating out the subspace\ncorresponding to the Jordan block (including the $d_\\nu=1$ case)\ngenerated from the eigenvector $\\vect{x}_\\nu$ is obtained by\n\\begin{equation} \\mathsf{\\Lambda}_\\nu := \\sum_{k=1}^{d_\\nu} \\mathsf{\\Lambda}_{\\nu,k}\\,. \\end{equation}\nThe projector $\\mathsf{\\Lambda}_\\nu$ defines a subspace $\\mathcal{W}_\\nu$,\n\\begin{equation} \\mathcal{W}_\\nu := \\bigg\\{ \\sum_{k=1}^{d_\\nu} a_k\\,\\vect{\\xi}_k^{(\\nu)};\n a_k\\in\\mathbf{C}\\bigg\\}\\,, \\label{eq:subspace-nu}\\end{equation}\nfor which $\\mathsf{\\Lambda}_\\nu\\mathcal{W}_\\nu=\\mathcal{W}_\\nu$\nand $(\\mathsf{1}-\\mathsf{\\Lambda}_\\nu)\\mathcal{W}_\\nu=\\emptyset$ (empty set)\nare satisfied.\nThe completeness is expressed as\n\\begin{equation} \\sum_\\nu \\mathsf{\\Lambda}_\\nu = \\mathsf{1}\\,,\\quad\n \\mathcal{V} = \\bigoplus_\\nu \\mathcal{W}_\\nu\\,.\\label{eq:proj-nu}\\end{equation}\n\nIn association with the LR-duality, one may consider\n\\begin{equation} \\bbar{\\mathcal{W}}_\\nu := \\bigg\\{ \\sum_{k=1}^{d_\\nu}\n a_k\\,\\bbar{\\vect{\\xi}}_k^{(\\nu)}; a_k\\in\\mathbf{C}\\bigg\\}\\,.\n\\label{eq:subspace-nubar}\\end{equation}\nThe projector corresponding to $\\bbar{\\mathcal{W}}_\\nu$\nis given by $\\mathsf{N\\,\\Lambda}_\\nu^\\dagger\\,\\mathsf{N}$.\nIn order for the arguments in Sec.~\\ref{sec:general} to be applicable\neven after a certain projection,\nit is desired to respect the UL-duality, as well as the LR-duality.\nTherefore the UL-dual subspace is also considered,\n\\begin{equation} \\mathsf{\\Sigma}_x\\mathcal{W}_\\nu^\\ast := \\bigg\\{ \\sum_{k=1}^{d_\\nu}\n a_k\\,\\mathsf{\\Sigma}_x\\vect{\\xi}_k^{(\\nu)\\ast}; a_k\\in\\mathbf{C}\\bigg\\}\\,.\n\\label{eq:subspace-signu}\\end{equation}\nThe projector relevant to $\\mathsf{\\Sigma}_x\\mathcal{W}_\\nu^\\ast$ is obtained by\n\\begin{equation} \\sum_{k=1}^{d_\\nu} \\frac{\\mathsf{\\Sigma}_x\\vect{\\xi}_k^{(\\nu)\\ast}\\,\n \\bbar{\\vect{\\xi}}_k^{(\\nu)T}\\mathsf{\\Sigma}_x\\,\\mathsf{N}}\n {\\bbar{\\vect{\\xi}}_k^{(\\nu)T}\\mathsf{\\Sigma}_x\\,\\mathsf{N}\\,\n \\mathsf{\\Sigma}_x\\vect{\\xi}_k^{(\\nu)\\ast}}\n= \\sum_{k=1}^{d_\\nu} \\frac{\\mathsf{\\Sigma}_x\\vect{\\xi}_k^{(\\nu)\\ast}\\,\n \\bbar{\\vect{\\xi}}_k^{(\\nu)T}\\,\\mathsf{N}\\,\\mathsf{\\Sigma}_x}\n {\\bbar{\\vect{\\xi}}_k^{(\\nu)T}\\,\\mathsf{N}\\,\\vect{\\xi}_k^{(\\nu)\\ast}}\n= \\mathsf{\\Sigma}_x\\,\\mathsf{\\Lambda}_\\nu^\\ast\\,\\mathsf{\\Sigma}_x\\,. \\end{equation}\nThe subspace $\\mathsf{\\Sigma}_x\\bbar{\\mathcal{W}}_\\nu^\\ast$\nand its relevant projector are defined as well.\nDepending on $\\omega_\\nu$,\nsome of $\\mathcal{W}_\\nu$, $\\bbar{\\mathcal{W}}_\\nu$,\n$\\mathsf{\\Sigma}_x\\mathcal{W}_\\nu^\\ast$\nand $\\mathsf{\\Sigma}_x\\bbar{\\mathcal{W}}_\\nu^\\ast$ could be identical.\nWe shall use a collective index $[\\nu]$\nto stand for their direct sum,\n\\begin{equation} \\mathcal{W}_{[\\nu]} := \\mathcal{W}_\\nu\\oplus\\bbar{\\mathcal{W}}_\\nu\n \\oplus\\mathsf{\\Sigma}_x\\mathcal{W}_\\nu^\\ast\n \\oplus\\mathsf{\\Sigma}_x\\bbar{\\mathcal{W}}_\\nu^\\ast\\,,\\end{equation}\napart from their overlap.\nThe projector $\\mathsf{\\Lambda}_{[\\nu]}$ on $\\mathcal{W}_{[\\nu]}$\nis defined by sum of the projectors.\nThough in restricted cases,\na similar projection is considered in Ref.~\\cite{ref:Don99}.\nIt is straightforward to show the following properties\nof $\\mathsf{\\Lambda}_{[\\nu]}$,\n\\begin{equation} \\mathsf{\\Lambda}_{[\\nu]}\n =\\mathsf{N\\,\\Lambda}_{[\\nu]}^\\dagger\\,\\mathsf{N}\\,,\\quad\n \\mathsf{\\Lambda}_{[\\nu]}\\,\\mathsf{\\Lambda}_{[\\nu']}\n =\\delta_{[\\nu],[\\nu']}\\,\\mathsf{\\Lambda}_{[\\nu]}\\,,\\quad\n \\mathsf{\\Sigma}_x\\,\\mathsf{\\Lambda}_{[\\nu]}^\\ast\\,\\mathsf{\\Sigma}_x\n = \\mathsf{\\Lambda}_{[\\nu]}\\,.\n\\label{eq:proj-prop}\\end{equation}\n\\begin{lemma}\\label{lem:dim-Lambda}\n$d_{[\\nu]}:=\\dim \\mathcal{W}_{[\\nu]}$ must be even.\n\\end{lemma}\n\\begin{proof}\nUnless $\\mathcal{W}_\\nu=\\bbar{\\mathcal{W}}_\\nu\n=\\mathsf{\\Sigma}_x\\mathcal{W}_\\nu^\\ast\n=\\mathsf{\\Sigma}_x\\bbar{\\mathcal{W}}_\\nu^\\ast$,\nthis is obvious from Lemma~\\ref{lem:Jordan-sym}.\nIf $\\mathcal{W}_\\nu=\\bbar{\\mathcal{W}}_\\nu\n=\\mathsf{\\Sigma}_x\\mathcal{W}_\\nu^\\ast\n=\\mathsf{\\Sigma}_x\\bbar{\\mathcal{W}}_\\nu^\\ast$,\n$\\omega_\\nu=0$ and $\\bbar{\\vect{\\xi}}_k^{(\\nu)}=\\vect{\\xi}_{d_\\nu+1-k}^{(\\nu)}$,\nas is clear from the argument\nwith respect to Prop.~\\ref{theor:one-to-one_basis}\nin Appendix~\\ref{app:sub-A3}.\nThen, if $d_\\nu$ is odd, we have\n$\\bbar{\\vect{\\xi}}_{(d_\\nu+1)\/2}^{(\\nu)}=\\vect{\\xi}_{(d_\\nu+1)\/2}^{(\\nu)}$,\ncontradictory to Lemma~\\ref{lem:pure-imag-Jordan}\n(Eq.~(\\ref{eq:imag-norm}), to be more precise),\nwhich leads to $\\bbar{\\vect{\\xi}}_{(d_\\nu+1)\/2}^{(\\nu)\\dagger}\\,\\mathsf{N}\\,\n\\vect{\\xi}_{(d_\\nu+1)\/2}^{(\\nu)}=0$.\nSee also the arguments in Appendix~\\ref{app:sub-B1}.\n\\end{proof}\n\nFor an arbitrary $2D\\times 2D$ matrix $\\mathsf{M}$,\nits projection onto the subspace $\\mathcal{W}_{[\\nu]}$\nis obtained by\n\\begin{equation} \\mathsf{M}_{[\\nu]} := \\mathsf{\\Lambda}_{[\\nu]}^\\dagger\\,\\mathsf{M}\\,\n \\mathsf{\\Lambda}_{[\\nu]}\\,. \\end{equation}\nOf next interest is how the projection affects\nthe RPA equation and the dualities.\n\\begin{proposition}\\label{theor:SN-proj}\nThe projections of $\\mathsf{S}$ and $\\mathsf{N}$ satisfy\n\\begin{equation} \\mathsf{\\Lambda}_{[\\nu]}^\\dagger\\,\\mathsf{S}\\,\\mathsf{\\Lambda}_{[\\nu']}\n = \\delta_{[\\nu],[\\nu']}\\,\\mathsf{S}_{[\\nu]}\\,,\\quad\n \\mathsf{\\Lambda}_{[\\nu]}^\\dagger\\,\\mathsf{N}\\,\\mathsf{\\Lambda}_{[\\nu']}\n = \\delta_{[\\nu],[\\nu']}\\,\\mathsf{N}_{[\\nu]}\\,. \\nonumber\\end{equation}\n\\end{proposition}\n\\begin{proof}\nFrom Eq.~(\\ref{eq:proj-prop}),\n\\begin{equation}\n \\mathsf{\\Lambda}_{[\\nu]}^\\dagger\\,\\mathsf{N}\\,\\mathsf{\\Lambda}_{[\\nu']}\n = \\mathsf{N}\\,\\mathsf{\\Lambda}_{[\\nu]}\\,\\mathsf{\\Lambda}_{[\\nu']}\n = \\delta_{[\\nu],[\\nu']}\\,\\mathsf{N}\\,\\mathsf{\\Lambda}_{[\\nu]}\n = \\delta_{[\\nu],[\\nu']}\\,\\mathsf{\\Lambda}_{[\\nu]}^\\dagger\\,\\mathsf{N}\\,\n \\mathsf{\\Lambda}_{[\\nu]}\\,.\\end{equation}\nConcerning $\\mathsf{S}_{[\\nu]}$, Eq.~(\\ref{eq:Jordan}) yields\n\\begin{equation} \\bbar{\\vect{\\xi}}_k^{(\\nu)\\dagger}\\,\\mathsf{S}\\,\\vect{\\xi}_{k'}^{(\\nu')}\n = \\delta_{\\nu\\nu'}\\,\\big[\\omega_\\nu\\,\\delta_{kk'}\n + ic_k^{(\\nu)}\\,\\delta_{k+1,k'}\\big]\\,\n \\bbar{\\vect{\\xi}}_k^{(\\nu)\\dagger}\\,\\mathsf{N}\\,\\vect{\\xi}_k^{(\\nu)}\\,,\n\\label{eq:Jordan-me}\\end{equation}\nand therefore\n\\begin{equation}\\begin{split}\n \\mathsf{\\Lambda}_{[\\nu]}^\\dagger\\,\\mathsf{S}\\,\\mathsf{\\Lambda}_{[\\nu']}\n &= \\mathsf{N}\\,\\mathsf{\\Lambda}_{[\\nu]}\\,\\mathsf{N\\,S}\\,\\mathsf{\\Lambda}_{[\\nu']}\n = \\mathsf{N}\\,\\bigg[\\sum_{\\nu\\in[\\nu]}\\sum_{k=1}^{d_\\nu}\n \\frac{\\vect{\\xi}_k^{(\\nu)}\\,\\bbar{\\vect{\\xi}}_k^{(\\nu)\\dagger}\\,\\mathsf{N}}\n {\\bbar{\\vect{\\xi}}_k^{(\\nu)\\dagger}\\,\\mathsf{N}\\,\\vect{\\xi}_k^{(\\nu)}}\\bigg]\\,\n \\mathsf{N\\,S}\\,\\bigg[\\sum_{\\nu'\\in[\\nu']}\\sum_{k'=1}^{d_{\\nu'}}\n \\frac{\\vect{\\xi}_{k'}^{(\\nu')}\\,\\bbar{\\vect{\\xi}}_{k'}^{(\\nu')\\dagger}\\,\\mathsf{N}}\n {\\bbar{\\vect{\\xi}}_{k'}^{(\\nu')\\dagger}\\,\\mathsf{N}\\,\\vect{\\xi}_{k'}^{(\\nu')}}\\bigg]\\\\\n &= \\delta_{[\\nu],[\\nu']}\\,\\mathsf{N}\\,\\sum_{\\nu\\in[\\nu]}\\bigg[\n \\omega_\\nu\\sum_{k=1}^{d_\\nu}\\mathsf{\\Lambda}_{\\nu,k}\n +i\\sum_{k=1}^{d_\\nu}c_k^{(\\nu)}\\,\n \\frac{\\vect{\\xi}_k^{(\\nu)}\\,\\bbar{\\vect{\\xi}}_{k+1}^{(\\nu)\\dagger}\\,\\mathsf{N}}\n {\\bbar{\\vect{\\xi}}_{k+1}^{(\\nu)\\dagger}\\,\\mathsf{N}\\,\\vect{\\xi}_{k+1}^{(\\nu)}}\\bigg]\n = \\delta_{[\\nu],[\\nu']}\\,\\mathsf{S}_{[\\nu]}\\,.\n\\end{split}\\label{eq:Snu-separation}\\end{equation}\nThe last equality follows\nbecause the quantity summed over $\\nu$ does not depend on $[\\nu']$.\n\\end{proof}\n\\begin{lemma}\\label{lem:proj-sym}\n$\\mathsf{S}_{[\\nu]}$ inherits the symmetry properties of Eq.~(\\ref{eq:S-prop}),\n\\begin{equation} \\mathsf{S}_{[\\nu]}=\\mathsf{S}_{[\\nu]}^\\dagger\\,,\\quad\n\\mathsf{\\Sigma}_x\\,\\mathsf{S}_{[\\nu]}^\\ast\\,\\mathsf{\\Sigma}_x=\\mathsf{S}_{[\\nu]}\\,.\n\\nonumber\\end{equation}\n\\end{lemma}\n\\begin{proof}\nThe hermiticity of $\\mathsf{S}_{[\\nu]}$ is obvious from its definition.\nThe second equation is proven as\n\\begin{equation} \\mathsf{\\Sigma}_x\\,\\mathsf{S}_{[\\nu]}^\\ast\\,\\mathsf{\\Sigma}_x\n = \\mathsf{\\Sigma}_x\\,\\mathsf{\\Lambda}_{[\\nu]}^T\\,\\mathsf{S}^\\ast\\,\n \\mathsf{\\Lambda}_{[\\nu]}^\\ast\\,\\mathsf{\\Sigma}_x\n = \\mathsf{\\Lambda}_{[\\nu]}^\\dagger\\,\\mathsf{\\Sigma}_x\\,\\mathsf{S}^\\ast\\,\n \\mathsf{\\Sigma}_x\\,\\mathsf{\\Lambda}_{[\\nu]}\n = \\mathsf{\\Lambda}_{[\\nu]}^\\dagger\\,\\mathsf{S}\\,\\mathsf{\\Lambda}_{[\\nu]}\n = \\mathsf{S}_{[\\nu]}\\,,\n\\end{equation}\nfrom Eq.~(\\ref{eq:proj-prop}).\n\\end{proof}\n\\noindent\nOwing to Prop.~\\ref{theor:SN-proj} and Lemma~\\ref{lem:proj-sym},\n$\\mathsf{S}_{[\\nu]}$ defines the RPA equation within $\\mathcal{W}_{[\\nu]}$\nwith maintaining both the LR- and UL-dualities\\footnote{\nA $d_{[\\nu]}$-dimensional representation of $\\mathsf{M}_{[\\nu]}$\nis provided by the matrix element\n$\\boldsymbol{\\xi}_k^{(\\nu)\\dagger}\\,\\mathsf{N}\\,\\boldsymbol{\\xi}_{k'}^{(\\nu')}\n\/ \\sqrt{|(\\bar{\\boldsymbol{\\xi}}_k^{(\\nu)\\dagger}\\,\\mathsf{N}\\,\n\\boldsymbol{\\xi}_k^{(\\nu)})\\,(\\bar{\\boldsymbol{\\xi}}_{k'}^{(\\nu')\\dagger}\\,\n\\mathsf{N}\\,\\boldsymbol{\\xi}_{k'}^{(\\nu')})|}$.\n}.\nThus the RPA equation is decomposed to the equation in each subspace $[\\nu]$.\nFurthermore, Prop.~\\ref{theor:SN-proj} ensures\nthat the same holds for a direct sum of subspaces\n$\\mathcal{W}_{[\\nu]}\\oplus\\mathcal{W}_{[\\nu']}$,\nvia $(\\mathsf{\\Lambda}_{[\\nu]}+\\mathsf{\\Lambda}_{[\\nu']})^\\dagger\\,\n\\mathsf{S}\\,(\\mathsf{\\Lambda}_{[\\nu]}+\\mathsf{\\Lambda}_{[\\nu']})\n=\\mathsf{S}_{[\\nu]+[\\nu']}$, and so forth.\n\nThe whole space $\\mathcal{V}$ is thus decomposed\ninto the direct sum of $\\mathcal{W}_{[\\nu]}$,\n\\begin{equation} \\mathcal{V} = \\bigoplus_{[\\nu]} \\mathcal{W}_{[\\nu]}\\,.\\end{equation}\nLet us denote the complementary space of $\\mathcal{W}_{[\\nu]}$\nby $\\mathcal{W}_{[\\nu]^{-1}}$\nand the relevant projector by $\\mathsf{\\Lambda}_{[\\nu]^{-1}}$;\n\\begin{equation} \\mathcal{W}_{[\\nu]^{-1}} := \\bigoplus_{[\\nu']\\,(\\ne[\\nu])} \\mathcal{W}_{[\\nu']}\\,,\\quad\n \\mathsf{\\Lambda}_{[\\nu]^{-1}} := \\mathsf{1}-\\mathsf{\\Lambda}_{[\\nu]}\n = \\sum_{[\\nu']\\,(\\ne[\\nu])} \\mathsf{\\Lambda}_{[\\nu']}\\,.\n\\label{eq:proj-comple}\\end{equation}\nAnalogously to $\\mathsf{S}_{[\\nu]}$,\n$\\mathsf{S}_{[\\nu]^{-1}}:=\\mathsf{\\Lambda}_{[\\nu]^{-1}}^\\dagger\\,\\mathsf{S}\\,\n\\mathsf{\\Lambda}_{[\\nu]^{-1}}$ defines the RPA equation in $\\mathcal{W}_{[\\nu]^{-1}}$,\nkeeping the LR- and UL-dualities\nand eliminating the solutions within $\\mathcal{W}_{[\\nu]}$,\nyet without influencing the solutions in $\\mathcal{W}_{[\\nu]^{-1}}$.\nTherefore, all the disclosed properties of the RPA solutions in $\\mathcal{V}$\nare transferred to the solutions of the RPA equation\nwithin $\\mathcal{W}_{[\\nu]^{-1}}$.\nIt is noted that the determinant of $\\mathsf{S}$ is decomposed as well,\n\\begin{equation} \\det\\,\\mathsf{S} = \\prod_{[\\nu]} (\\det\\,\\mathsf{S}_{[\\nu]})\\,,\n\\label{eq:detS-decomp}\\end{equation}\nwhere $(\\det\\,\\mathsf{S}_{[\\nu]})$ on the r.h.s.\nis the $d_{[\\nu]}$-dimensional determinant.\nThe l.h.s of the secular equation, $\\det(\\mathsf{S}-\\omega\\,\\mathsf{N})$,\ncan be expressed in an analogous manner.\n\n\\section{Properties of each class of solutions}\n\\label{sec:stable&unstable}\n\nProperties of the RPA solutions are further analyzed\nfor individual class of Prop.~\\ref{prop:eigenvalues}.\nIt deserves commenting here on degeneracy.\nTo the author's best knowledge,\npossibility of degeneracy, particularly of Jordan blocks,\nhas not been examined well, except several specific NG modes.\nAlthough degeneracy occurs even in physical solutions\nunder presence of certain symmetry\n(\\textit{e.g.}, degeneracy with respect to magnetic quantum numbers\nunder the rotational symmetry),\nit does not give rise to Jordan blocks.\nThis is obvious when the conservation law allows us\nto separate the RPA equation into the equations\naccording to the quantum numbers.\nHowever, it is not trivial whether the same holds\nfor a variety of extensive applications of the RPA.\nFor instance, energy levels are highly degenerate in continuum,\nas in the continuum RPA~\\cite{ref:SB75}.\nConsideration of the degeneracy could be relevant to\nhow we can take the continuous limit from arguments on discrete levels.\nFor the NG mode, Thouless restricted himself\nto the case of two-dimensional Jordan blocks.\nWhile higher-dimensional blocks are not very likely to emerge\nin physical situations,\nit will be meaningful to distinguish physical situations\nfrom facts with rigorous mathematical proof.\n\n\\subsection{Solutions for real eigenvalues}\\label{subsec:real-sol}\n\nLet us first consider Classes~\\ref{item:positive} and \\ref{item:negative}\nof Prop.~\\ref{prop:eigenvalues}.\n\n\\begin{proposition}\\label{theor:stability}\nIf the stability matrix $\\mathsf{S}$ is positive-definite,\nthe RPA equation is fully solvable.\nIf the RPA equation is fully solvable, $\\mathsf{S}$ is positive-definite.\n\\end{proposition}\n\\noindent\nAlthough the first part of this proposition was already proven\nin Ref.~\\cite{ref:Thou61},\nI prove it again in combination with the second part.\n\\begin{proof}\nSuppose that $\\mathsf{S}$ is positive-definite.\nThen $\\tilde{\\mathsf{S}}$ in Sec.~\\ref{subsec:redefine}\nis hermitian and therefore diagonalizable\nby a certain matrix $\\tilde{\\mathsf{X}}$,\n$\\tilde{\\mathsf{S}}\\,\\tilde{\\mathsf{X}}\n=\\tilde{\\mathsf{X}}\\,\\tilde{\\mathsf{\\Omega}}$,\nwhere $\\tilde{\\mathsf{\\Omega}}$ is a diagonal matrix.\n$\\tilde{\\mathsf{X}}$ can be unitary, but we shall take another normalization.\nThe eigenvalues in $\\tilde{\\mathsf{\\Omega}}$ are all real and non-zero,\nsince $\\det\\tilde{\\mathsf{S}}=\\det(\\mathsf{N\\,S})\\ne 0$.\nProposition~\\ref{theor:sym-eigen} tells us\nthat they are pairwise, $\\pm\\omega_\\nu$ ($\\nu=1,\\cdots,D$).\nWe here take ${\\displaystyle\\tilde{\\mathsf{\\Omega}}:=\\begin{pmatrix}\n \\mathrm{diag.}(\\omega_\\nu)&0\\\\ 0&-\\mathrm{diag.}(\\omega_\\nu) \\end{pmatrix}}$\nso that $\\omega_\\nu>0$,\nand define $\\mathsf{\\Omega}:=\\mathsf{N}\\,\\tilde{\\mathsf{\\Omega}}$,\nwhich is diagonal and positive-definite.\nLet us adopt the normalization of $\\tilde{\\mathsf{X}}$\nas $\\tilde{\\mathsf{X}}^\\dagger\\,\\tilde{\\mathsf{X}}=\\mathsf{\\Omega}$\n(\\textit{i.e.}, $\\tilde{\\mathsf{X}}\\,\\mathsf{\\Omega}^{-1\/2}$ is unitary)\nand define $\\mathsf{X}:=\\mathsf{S}^{-1\/2}\\,\\tilde{\\mathsf{X}}$.\nThis derives\n\\begin{equation} \\mathsf{S\\,X}=\\mathsf{N\\,X\\,N\\,\\Omega}\\,,\\quad\n \\mathsf{X}^\\dagger\\,\\mathsf{N\\,X}=\\mathsf{N}\\,, \\label{eq:RPAeq-c}\\end{equation}\nproving that the RPA equation is fully solvable because,\nif we write $\\mathsf{X}=(\\vect{x}_1,\\cdots,\\vect{x}_D,\n\\mathsf{\\Sigma}_x\\vect{x}_1^\\ast,\\cdots,\\mathsf{\\Sigma}_x\\vect{x}_D^\\ast)$,\nEq.~(\\ref{eq:RPAeq-c}) yields Eqs.~(\\ref{eq:RPAeq-b}) and (\\ref{eq:norm-b})\nfor $\\nu=1,\\cdots,D$.\\\\\nThis part of the proposition is proven also from Eq.~(\\ref{eq:norm-relation})\nwith $\\nu=\\nu'$ and Corollary~\\ref{cor:non-Jordan}.\nIf $\\mathsf{S}$ is positive-definite,\n$\\vect{x}_\\nu^\\dagger\\,\\mathsf{S}\\,\\vect{x}_\\nu>0$,\nderiving $\\omega_\\nu>0$ and $\\vect{x}_\\nu^\\dagger\\,\\mathsf{N}\\,\\vect{x}_\\nu>0$\nor both negative, for any solution of the RPA equation (\\ref{eq:RPAeq-b}).\nThen Corollary~\\ref{cor:non-Jordan} ensures\nthat no eigenvector constitutes Jordan blocks.\\\\\nConversely, if the RPA equation is fully solvable as in Eq.~(\\ref{eq:RPAeq-c}),\n$\\det\\mathsf{X}\\ne 0$ and $\\omega_\\nu>0$ for $^\\forall\\nu\\,(=1,\\cdots,D)$.\nThen, since\n\\begin{equation} \\mathsf{S}=\\mathsf{U}^{-1}\\,\\mathrm{diag.}(\\lambda_i)\\,\\mathsf{U}\n =(\\mathsf{N\\,X\\,N})\\,\\mathsf{\\Omega}\\,(\\mathsf{N\\,X\\,N})^\\dagger\\,,\\end{equation}\nit follows that\n\\begin{equation} \\mathrm{diag.}(\\lambda_i)\n =(\\mathsf{U\\,N\\,X\\,N})\\,\\mathsf{\\Omega}\\,(\\mathsf{U\\,N\\,X\\,N})^\\dagger\n\\end{equation}\nand therefore, by expressing $\\mathsf{U\\,N\\,X\\,N}=(\\chi_{i\\nu})$,\n\\begin{equation} \\lambda_i =\\sum_\\nu \\omega_\\nu\\,|\\chi_{i\\nu}|^2 >0\\,;\n\\label{eq:pos-def-reverse}\\end{equation}\nnamely $\\mathsf{S}$ is positive-definite.\nNotice $\\big|\\det(\\mathsf{U\\,N\\,X\\,N})\\big|=\\big|\\det\\,\\mathsf{X}\\big|\\ne 0$,\nwhich excludes the possibility of $\\lambda_i=0$\nin Eq.~(\\ref{eq:pos-def-reverse}).\n\\end{proof}\n\n\\noindent\nTherefore the arguments on physical solutions by Thouless are applicable\neven under the presence of degeneracy as in the continuum.\n\nFor solutions of Class~\\ref{item:negative},\na positive eigenvalue $\\omega_\\nu$ is accompanied by\nan eigenvector $\\vect{x}_\\nu$ with $\\vect{x}_\\nu^\\dagger\\mathsf{N}\\,\\vect{x}_\\nu<0$\nor $\\vect{x}_\\nu^\\dagger\\mathsf{N}\\,\\vect{x}_\\nu=0$.\nIn the former case, its UL-dual partner is normalizable\nbut corresponds to the eigenvalue $-\\omega_\\nu\\,(<0)$.\nThe submatrix $\\mathsf{S}_{[\\nu]}$ of this solution is negative-definite,\nas exemplified in Appendix~\\ref{app:2*2}.\nTherefore the stability matrix $\\mathsf{S}$\nhas two negative eigenvalues at least.\nIn the $\\vect{x}_\\nu^\\dagger\\mathsf{N}\\,\\vect{x}_\\nu=0$ case,\n$\\vect{x}_\\nu$ forms a Jordan block~\\cite{ref:Neergard},\nwhose UL-dual partner associated with $-\\omega_\\nu\\,(<0)$\nbelongs to another Jordan block.\nProbably for this reason, Thouless ignored this class of solutions,\nhaving focused on his arguments near the stability.\n\n\\subsection{Solutions for complex eigenvalues}\\label{subsec:imag-sol}\n\nComplex eigenvalues belong to Classes~\\ref{item:pure-imag}\nand \\ref{item:quartet} of Prop.~\\ref{prop:eigenvalues}.\nI next discuss properties of solutions of Class~\\ref{item:pure-imag}.\n\n\\begin{lemma}\\label{lem:pure-imag-eigen}\nFor an eigenvalue $\\omega_\\nu$ of $\\mathsf{N\\,S}$\nwith $\\mathrm{Re}(\\omega_\\nu)=0$,\nany corresponding eigenvector can be taken so as to satisfy\n$\\mathsf{\\Sigma}_x\\vect{x}_\\nu^\\ast=e^{-i\\phi}\\,\\vect{x}_\\nu$ ($\\phi\\in\\mathbf{R}$).\nConversely, if an eigenvector $\\vect{x}_\\nu$ satisfies\n$\\mathsf{\\Sigma}_x\\vect{x}_\\nu^\\ast=e^{-i\\phi}\\,\\vect{x}_\\nu$,\n$\\mathrm{Re}(\\omega_\\nu)=0$ holds for its corresponding eigenvalue.\n\\end{lemma}\n\\begin{proof}\n$\\mathrm{Re}(\\omega_\\nu)=0$ is equivalent to $\\omega_\\nu=-\\omega_\\nu^\\ast$.\nTherefore, from Corollary~\\ref{lem:sym-sigma},\nboth $\\vect{x}_\\nu$ and $\\mathsf{\\Sigma}_x\\vect{x}_\\nu^\\ast$\nbelong to the equal eigenvalue $\\omega_\\nu$,\nwhether they are linearly dependent or independent.\nThen a linear combination of them,\n$\\vect{y}_\\nu:=\\alpha\\,\\vect{x}_\\nu + \\beta\\,\\mathsf{\\Sigma}_x\\vect{x}_\\nu^\\ast$\n($\\alpha,\\beta\\in\\mathbf{C}$),\nis also an eigenvector associated with the eigenvalue $\\omega_\\nu$.\nAssuming $\\beta=e^{i\\phi}\\,\\alpha^\\ast$,\nwe verify $\\mathsf{\\Sigma}_x\\vect{y}_\\nu^\\ast=e^{-i\\phi}\\,\\vect{y}_\\nu$.\nWhen $\\vect{x}_\\nu$ and $\\mathsf{\\Sigma}_x\\vect{x}_\\nu^\\ast$\nare linearly independent,\nwe obtain two independent vectors\nby adopting , \\textit{e.g.}, $\\alpha=-\\beta=1$ and $\\alpha=\\beta=i$.\\\\\nIf $\\omega_\\nu\\ne-\\omega_\\nu^\\ast$,\nthe associating eigenvectors\n$\\vect{x}_\\nu$ and $\\mathsf{\\Sigma}_x\\vect{x}_\\nu^\\ast$\nmust be linearly independent;\nnamely $\\mathsf{\\Sigma}_x\\vect{x}_\\nu^\\ast=e^{-i\\phi}\\,\\vect{x}_\\nu$ is impossible.\n\\end{proof}\n\\noindent\nIt is noted here that $\\mathrm{Re}(\\omega_\\nu)=0$ covers\nthe solutions of Class~\\ref{item:null} as well as \\ref{item:pure-imag}\nin Prop.~\\ref{prop:eigenvalues}.\n\nIf $\\mathsf{\\Sigma}_x\\vect{x}_\\nu^\\ast=e^{-i\\phi}\\,\\vect{x}_\\nu$ is assumed\nfor the $\\mathrm{Re}(\\omega_\\nu)=0$ case,\nthe lower $D$-dimensional components of the RPA equation (\\ref{eq:RPAeq-b})\nis only a repetition of the upper $D$-dimensional components.\nWe can choose the phase $e^{i\\phi}$ arbitrarily,\nbecause it is controllable via a transformation\n$\\vect{y}_\\nu=e^{i\\theta}\\vect{x}_\\nu$.\nA convenient choice is $e^{i\\phi}=-1$, so that ${\\displaystyle\\vect{x}_\\nu\n= -\\mathsf{\\Sigma}_x\\vect{x}_\\nu^\\ast\n= \\begin{pmatrix} X^{(\\nu)}\\\\ -X^{(\\nu)\\ast}\\end{pmatrix}}$.\n\nLet us now take $c_k^{(\\nu)}\\in\\mathbf{R}$ in Eq.~(\\ref{eq:Jordan}).\n\\begin{lemma}\\label{lem:pure-imag-Jordan}\nFor an eigenvalue $\\omega_\\nu$ of $\\mathsf{N\\,S}$\nwith $\\mathrm{Re}(\\omega_\\nu)=0$,\nall corresponding basis vectors can satisfy\n$\\mathsf{\\Sigma}_x\\vect{\\xi}_k^{(\\nu)\\ast}=-\\vect{\\xi}_k^{(\\nu)}$;\n\\textit{i.e.}, ${\\displaystyle\\vect{\\xi}_k^{(\\nu)}\n = \\begin{pmatrix} \\Xi^{(\\nu,k)}\\\\ -\\Xi^{(\\nu,k)\\ast}\\end{pmatrix}}$.\n\\end{lemma}\n\\noindent\nThis lemma states that solutions of Classes~\\ref{item:pure-imag}\nand \\ref{item:null} can be self UL-dual,\ntogether with basis vectors generated from them.\n\\begin{proof}\nIn the case that $\\mathrm{Re}(\\omega_\\nu)=0$,\nEqs.~(\\ref{eq:Jordan}) and (\\ref{eq:Jordan-sym}) come\n\\begin{equation}\\begin{split}\n (\\mathsf{S}-\\omega_\\nu\\,\\mathsf{N})\\,\\vect{\\xi}_{k+1}^{(\\nu)}\n &= ic_k^{(\\nu)}\\,\\mathsf{N}\\,\\vect{\\xi}_k^{(\\nu)}\\,,\\\\\n (\\mathsf{S}-\\omega_\\nu\\,\\mathsf{N})\\,\\mathsf{\\Sigma}_x\\vect{\\xi}_{k+1}^{(\\nu)\\ast}\n &= ic_k^{(\\nu)}\\,\\mathsf{N}\\,\\mathsf{\\Sigma}_x\\vect{\\xi}_k^{(\\nu)\\ast}\\,.\n\\end{split}\\label{eq:Jordan-sym2}\\end{equation}\nFrom Lemma~\\ref{lem:pure-imag-eigen} and the argument above,\nwe can assume $\\mathsf{\\Sigma}_x\\vect{\\xi}_1^{(\\nu)\\ast}=-\\vect{\\xi}_1^{(\\nu)}$.\nIf $\\mathsf{\\Sigma}_x\\vect{\\xi}_k^{(\\nu)\\ast}=-\\vect{\\xi}_k^{(\\nu)}$,\nthe first equation of (\\ref{eq:Jordan-sym2}) indicates\nthat the second equation has a solution fulfilling\n$\\mathsf{\\Sigma}_x\\vect{\\xi}_{k+1}^{(\\nu)\\ast}=-\\vect{\\xi}_{k+1}^{(\\nu)}$.\nThe lemma is then proven inductively.\n\\end{proof}\n\\noindent\nCompatibility of this lemma with Prop.~\\ref{theor:one-to-one_basis}\nis confirmed in Appendix~\\ref{app:sub-B1}.\nUnder the above convention for the $\\mathrm{Re}(\\omega_\\nu)=0$ case,\nthe normalization condition of $\\vect{\\xi}_{k}^{(\\nu)}$ can be\n\\begin{equation} \\bbar{\\vect{\\xi}}_k^{(\\nu)\\dagger}\\,\\mathsf{N}\\,\\vect{\\xi}_{k'}^{(\\nu')}\n = \\bar{\\Xi}^{(\\nu,k)\\dagger}\\,\\Xi^{(\\nu',k')}\n - \\big[\\bar{\\Xi}^{(\\nu,k)\\dagger}\\,\\Xi^{(\\nu',k')}\\big]^\\ast\n = \\pm i\\delta_{\\nu\\nu'}\\,\\delta_{kk'}\\,\n\\label{eq:imag-norm}\\end{equation}\nalthough $\\vect{\\xi}_{k}^{(\\nu)}$ is not normalizable\nin the respect of Def.~\\ref{def:normalizable}.\n\nConsider solutions in vicinity of the stability,\nin which the stability matrix $\\mathsf{S}$ has a single negative eigenvalue.\nNear the stability,\nthe subspace providing negative $\\det\\,\\mathsf{S}_{[\\nu]}$ can be separated out\nby using the projector in Sec.~\\ref{sec:decomp},\nwhich should be two dimensional and therefore provides\n${\\displaystyle\\mathsf{S}_{[\\nu]}=\\begin{pmatrix} a& b\\\\\nb^\\ast& a\\end{pmatrix}}$ ($a\\in\\mathbf{R}$, $b\\in\\mathbf{C}$),\nas in Appendix~\\ref{app:2*2}.\nA pair of pure-imaginary eigenvalues is obtained,\nillustrating that the first unphysical solution\nemerges as Class~\\ref{item:pure-imag} of Prop.~\\ref{prop:eigenvalues}.\n\nUnlike the self LR-duality for real eigenvalues,\nthe self UL-duality does not forbid Jordan blocks,\nalthough most pure-imaginary eigenvalues are expected\nnot to form Jordan blocks.\nAn example of Jordan blocks is presented in Appendix~\\ref{app:4*4-pure-imag}.\n\nLet us turn to solutions of Class~\\ref{item:quartet}.\nQuartet solutions are a manifestation of the two types of dualities.\nThe possibility of quartet solutions\nwas first pointed out in Ref.~\\cite{ref:UR71}\nfor $\\mathsf{S}=\\mathsf{S}^\\ast$ cases,\nand mentioned in Ref.~\\cite{ref:SM84} in more general context.\nA minimal model for quartet solutions is constructed by taking $D=2$,\nand is analyzed in Appendix~\\ref{app:4*4-quartet}.\nFor quartet solutions $\\nu$, $d_{[\\nu]}$ is a multiple of four.\nHence, by denoting the solutions $\\pm\\alpha\\pm i\\beta$\n($\\alpha,\\beta\\in\\mathbf{R}$),\n$\\det\\,\\mathsf{S}_{[\\nu]}=\\det(\\mathsf{N}_{[\\nu]}\\,\\mathsf{S}_{[\\nu]})\n=(\\alpha^2+\\beta^2)^{d_{[\\nu]}\/2}>0$.\nAs $\\mathsf{S}$ cannot be positive-definite\non account of the latter part of Prop.~\\ref{theor:stability},\nEq.~(\\ref{eq:detS-decomp}) indicates\nthat $\\mathsf{S}$ has at least two negative eigenvalues\nfor quartet solutions to come out.\n\n\\subsection{NG-mode solutions}\\label{subsec:NG-sol}\n\nThe simplest example of the NG-mode solution\nis given in Appendix~\\ref{app:2*2}, by the $2\\times 2$ stability matrix.\nIt illustrates that the null eigenvalue\nis often associated with a two-dimensional Jordan block,\nas indicated by Thouless~\\cite{ref:Thou61}.\nThe NG modes that generate two-dimensional Jordan blocks\nhave well been investigated~\\cite{ref:RS80,ref:Thou61,ref:TV62}.\nHowever, in the example of Appendix~\\ref{app:2*2},\nthere is a trivial case of $\\mathsf{S}=\\mathsf{0}$\nin which two $d_\\nu=1$ eigenvectors are present for the null eigenvalues.\nMoreover, an example of 4-dimensional Jordan block\nis seen in Appendix~\\ref{app:4*4-null}.\nLikely or not, it is difficult to exclude the possibilities\nother than the two-dimensional Jordan block for the null eigenvalue\nonly from mathematical viewpoints.\n\n\\begin{corollary}\\label{cor:deg-null}\nIf there exists a null eigenvalue for $\\mathsf{N\\,S}$,\nit must have even number of degeneracy.\n\\end{corollary}\n\\begin{proof}\nBecause of Prop.~\\ref{theor:sym-eigen},\nthe number of non-zero eigenvalues must be even, up to their degeneracies.\nMoreover, Lemma~\\ref{lem:Jordan-sym} ensures\nthat sum of dimensions of Jordan blocks for non-zero eigenvalues is even.\nThe total dimension of $\\mathsf{N\\,S}$ is $2D$,\nwhich concludes the degeneracy of the null eigenvalue must be even.\nAlso proven from Lemma~\\ref{lem:dim-Lambda}.\n\\end{proof}\n\nWhen SSB occurs, there must be NG-mode solutions\ncorresponding to the broken symmetry;\n\\textit{e.g.}, the linear momentum in the SSB with respect to the translation\nand the angular momentum in the SSB with respect to the rotation\nin deformed nuclei.\nFor specific NG-mode solutions with such physical interpretations,\ntheir properties can further be explored,\nthough I do not pursue this direction in this article.\n\nThe null eigenvalues may lie at the intersection\nof the self LR- and the self UL-dualities.\nAlthough there is no single eigenvector having both of the self dualities\nas indicated by Lemma~\\ref{lem:dim-Lambda},\nthere could be an even-dimensional Jordan block\nin which the LR-duality closes by its basis vectors,\nwhile keeping the self UL-duality of Lemma~\\ref{lem:pure-imag-Jordan}.\nIn such cases the Jordan block, instead of the basis vectors,\nmay be said self LR-dual.\n\\begin{proposition}\\label{theor:null-self-dual}\nFor even-dimensional Jordan blocks associated with a null eigenvalue,\nit is possible to produce basis vectors $\\big\\{\\vect{\\eta}_k;\nk=1,\\cdots,d_\\nu\\big\\}$ having double self duality,\n$\\mathsf{\\Sigma}_x\\vect{\\eta}_k^\\ast=-\\vect{\\eta}_k$ and\n$\\bbar{\\vect{\\eta}}_k=\\vect{\\eta}_{d_\\nu+1-k}$.\n\\end{proposition}\n\\noindent\nThis proposition is proven in Appendix~\\ref{app:sub-B2}.\nAn example of the transformation shown in Appendix~\\ref{app:sub-B2}\nis given by the NG mode of the angular momentum,\nunder SSB with respect to the rotation.\nEven though the Jordan blocks corresponding to $J_\\pm$\nare the LR-dual of each other,\ntheir linear combinations provide those corresponding to $J_x$ and $J_y$,\neach of which could be self LR-dual.\n\nLet us focus on the case that the basis vectors simultaneously fulfill\n$\\mathsf{\\Sigma}_x\\vect{\\xi}_k^{(\\nu)\\ast}=-\\vect{\\xi}_k^{(\\nu)}$\nand $\\bbar{\\vect{\\xi}}_k^{(\\nu)}=\\vect{\\xi}_{d_\\nu+1-k}^{(\\nu)}$, with even $d_\\nu$.\nThen the projector of Eq.~(\\ref{eq:proj-nuk})\nhas the relation $\\mathsf{N\\,\\Lambda}_{\\nu,k}^\\dagger\\,\\mathsf{N}\n=\\mathsf{\\Lambda}_{\\nu,d_\\nu+1-k}$,\nand therefore obeys\n\\begin{equation}\\begin{split} &\\mathsf{\\Lambda}_{\\nu,k}+\\mathsf{\\Lambda}_{\\nu,d_\\nu+1-k}\n =\\mathsf{N}\\,\\big[\\mathsf{\\Lambda}_{\\nu,k}\n +\\mathsf{\\Lambda}_{\\nu,d_\\nu+1-k}\\big]^\\dagger\\,\\mathsf{N}\\,,\\\\\n&\\mathsf{\\Sigma}_x\\,\\big[\\mathsf{\\Lambda}_{\\nu,k}\n +\\mathsf{\\Lambda}_{\\nu,d_\\nu+1-k}\\big]^\\ast\\,\\mathsf{\\Sigma}_x\n = \\mathsf{\\Lambda}_{\\nu,k}+\\mathsf{\\Lambda}_{\\nu,d_\\nu+1-k}\\,\n\\label{eq:null-proj2}\\end{split}\\end{equation}\nalthough the relation analogous to Prop.~\\ref{theor:SN-proj}\n(Eq.~(\\ref{eq:Snu-separation}), in particular)\ndoes not necessarily hold.\nLike the arguments using $\\mathsf{\\Lambda}_{[\\nu]}$ in Sec.~\\ref{sec:decomp},\n$\\mathsf{\\Lambda}_{\\nu,k}+\\mathsf{\\Lambda}_{\\nu,d_\\nu+1-k}$\nproduces a subspace keeping both the LR- and UL-dualities.\nFor doubly self-dual Jordan blocks,\nthe basis vectors for the NG-mode may be removed two by two via the projectors,\nwith minimal d.o.f. coupled to them.\nWithin the two-dimensional subspace\ngiven by $\\mathsf{\\Lambda}_{\\nu,d_\\nu\/2}+\\mathsf{\\Lambda}_{\\nu,d_\\nu\/2+1}$,\nthe prescription proposed in Refs.~\\cite{ref:Thou61,ref:TV62}\n(and well summarized in Refs.~\\cite{ref:RS80,ref:Row70})\nis applicable.\n\nIf $d_\\nu=\\mathrm{odd}$, there should be two Jordan blocks,\nwhich are the partner of the LR- (or UL-) duality of each other.\nTo separate them, one may apply the projector $\\Lambda_{[\\nu]}$\nintroduced in Sec.~\\ref{sec:decomp}.\n\n\\section{Summary}\\label{sec:summary}\n\nProperties of solutions of the RPA equation is reanalyzed\nin terms of the linear algebra.\nAs well as eigensolutions, cases in which the matrix $\\mathsf{N\\,S}$\n(and $\\mathsf{S\\,N}$) forms Jordan blocks are examined.\nTwo types of dualities of eigenvectors and basis vectors,\nwhich are called LR- and UL-dualities in this article,\nare pointed out and explored.\nThese dualities are useful to clarify properties of the RPA solutions.\nProjection respecting the dualities is developed.\n\nEigenvalues given by the RPA equation are classified\ninto five classes, in Prop.~\\ref{prop:eigenvalues}.\nAs pointed out by Thouless, all solutions are physical ones\nif the stability matrix is positive-definite.\nIts opposite is also true (in absence of NG modes),\nbeing useful to judge stability of a MF solution\nfrom numerical calculations in the RPA.\nThese solutions are singled out, not constituting Jordan blocks,\nand have the self LR-duality while are paired by the UL-duality.\nEigenvectors and basis vectors for pure-imaginary eigenvalues\ncan be made self UL-dual, and paired by the LR-duality.\nWith no self dualities, quartet solutions manifest two types of the dualities.\nNG-mode solutions, which are associated with the null eigenvalue\nand often related to the spontaneous symmetry breaking,\nlie at intersection of the two self dualities.\nHowever, a single vector cannot be both self LR-dual and self UL-dual.\nOnly even-dimensional Jordan blocks can have double self dualities.\nThe well-known prescription of separating out the NG modes\ncould be applicable to such cases.\n\n\\section*{Acknowledgment}\n\nThe author is grateful to K.~Matsuyanagi, K.~Neerg\\aa rd, H.~Kurasawa, Y.R.~Shimizu,\nJ.~Terasaki and T.~Inakura for discussions.\nThis work is financially supported in part\nby JSPS KAKENHI Grant Number~24105008 and Grant Number~16K05342.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{ Introduction and Terminology} \nIn 1996, Matheson and Tarjan [MT] proved that if $G$ is a plane triangulated disc then\n$\\gamma (G)\\le |V(G)|\/3$.\n(In particular, then, the same bound on $\\gamma$ applies to any triangulation of the plane.)\nPlummer and Zha [PZ] proved that if $G$ is a triangulation of the projective plane, then $\\gamma (G)\\le |V(G)|\/3$\nand if $G$ is a triangulation of either the torus or Klein bottle, then $\\gamma (G)\\le \\lceil |V(G)|\/3\\rceil$.\nThe latter result was sharpened by Honjo et al. [HKN] who showed that $\\gamma (G)\\le |V(G)|\/3$\nfor graphs embedded in these two surfaces.\nThey also showed that for any surface $\\Sigma$, there is a positive integer $\\rho (\\Sigma )$ such that\nif $G$ is embedded as a triangulation of $\\Sigma$ and the embedding has face-width at least $\\rho (\\Sigma )$,\nthen $\\gamma (G)\\le |V(G)|\/3$.\nEven more recently, Furuya and Matsumoto [FM] generalized this result by showing that $\\gamma (G)\\le |V(G)|\/3$,\nfor every triangulation $G$ of any closed surface.\\par\n\nIn [MT], Matheson and Tarjan also conjectured that their bound of $n\/3$ could be improved; namely, if $n$ is sufficiently large, then any triangulation of the plane with $n$ vertices would have $\\gamma\\le n\/4$.\nThe \ntriangle has $\\gamma (K_3)=1 = n\/3$, the octahedron shown in Figure 1.1(i) has $\\gamma=2=n\/3$ and the 7-vertex graph shown in Figure 1.1(ii) has $\\gamma = 2\/7>1\/4$, so this shows that\none must assume $n\\ge 8$ in order for the Matheson-Tarjan conjecture to be true.\n\n\n\n\\begin{figure}[!hbtp] \n\\begin{center}\n\\includegraphics[scale=.75]{2Counterexamples.eps} \n\\centerline{\\bf Figure 1.1}\n\\end{center}\n\\end{figure}\n \n \nMore generally, in [PZ] the first and third authors of the present paper conjectured that if $G$ is a\ntriangulation of {\\it any}\nnon-spherical surface, then\n$\\gamma (G)\\le n\/4$.\nBoth these conjectures involving the $n\/4$ bound remain unsettled.\\par \n \n \n\nIn 2010, King and Pelsmajer [KP] proved the Matheson-Tarjan bound of n\/4 holds in the plane \ncase when the maximum degree of the triangulation is 6.\\par\n\nAn outerplanar graph is a graph embedded in the plane in such a way that all vertices of the graph lie on the boundary of the external face.\nAn outerplanar graph is {\\it maximal} (outerplanar) if it is not possible to add any new edge to $G$ without destroying outerplanarity.\nIn 2013, Campos and Wakabayashi [CW] proved that if $G$ is a maximal outerplanar graph with at least $n\\ge 4$ vertices, then $\\gamma (G)\\le (n+t)\/4$, where $t$ is the number of vertices of degree 2 in $G$.\n\n\nIt was proved independently by N\\\"unning [N] and by Sohn and Yuan [SY] that for any graph $G$ with\n$n$ vertices and minimum degree $\\delta (G)\\ge 4$, $\\gamma (G)\\le 4n\/11$.\nIn the present paper we will show that if $G$ is a plane triangulation on $n\n$ vertices which has a Hamilton cycle and $\\delta (G)\\ge 4$, then $\\gamma (G)\\le \\max\\{\\lceil 2n\/7\\rceil, \\lfloor 5n\/16\\rfloor\\}$.\nIt follows immediately that if $G$ is a 4-connected plane triangulation on $n\n$ vertices, then\n $\\gamma(G)\\le \\max\\{\\lceil 2n\/7\\rceil, \\lfloor 5n\/16\\rfloor\\}$.\nNote that if $n\\notin \\{6,8,9,11,12,15,19,22,25\\}$, $\\lceil 2n\/7\\rceil \\le \\lfloor 5n\/16\\rfloor$, so this result can be restated to\nsay that if $G$ is a 4-connected plane triangulation on $n$ vertices and $n\\ge 26$, then $\\gamma (G)\\le \\lfloor 5n\/16\\rfloor$. \n\n\n\\section{Preferred Hamilton cycles in plane triangulations}\n\nLet $G$ be a plane triangulation with $\\delta (G)\\ge 4$ and suppose $G$ contains a Hamilton cycle $H$.\nWe can think of $H$ bounding a triangulated inner subgraph $G_{int}$ and a triangulated outer subgraph\n$G_{ext}$ such that $G_{int}\\cap G_{ext}= H$.\nSuppose $v\\in V(G)$.\nWe denote by $i\\deg v$ (respectively, $o\\deg v$) the degree of vertex $v$ in $G_{int}$ (resp. in $G_{ext}$).\n\\par\n\nWe will need to pay particular attention to those vertices which have $i\\deg v = 2$ or $o\\deg v=2$.\nWe will call these vertices {\\it 2-vertices}.\nExamples are shown in Figure 2.1 where $i\\deg v_1=2$ and $o\\deg v_2=2$ respectively and hence each is an example of\na 2-vertex.\\par\n\n\\begin{figure}[!hbtp]\\refstepcounter{figure}\n\\begin{center}\n\\includegraphics[scale=.45]{DomHam1.eps} \n\\centerline{\\bf Figure 2.1}\n\\end{center}\n\\end{figure}\n\nLet $G$ be a Hamiltonian plane triangulation. Such a graph may have many Hamilton cycles.\nWe now show how to select such a cycle with certain properties we shall need later on.\\par\n\nTo this end, let begin with any Hamilton cycle $H$ in $G$.\nA triangle $T$ of $G_{int}$ will be called {\\it internal} (with respect to $H$) if $E(T)\\cap E(H)=\\emptyset$.\n\\par\n\\medskip\n\\noindent\n{\\bf Lemma 2.1:} Let $G$ be a plane triangulation with $\\delta (G)\\ge 4$ which contains a Hamilton cycle.\nSuppose $\\gamma (G)\\ge 2$.\nThen there exists a Hamilton cycle in $G$ containing no three consecutive 2-vertices.\\par\n\\medskip\\noindent\n{\\bf Proof:} First, suppose that $G$ contains four consecutive 2-vertices.\nSince $G$ is a triangulation, these four 2-vertices must alternate between $ideg=2$ and $o\\deg=2$.\nSo let us suppose that we have four consecutive 2-vertices forming a subpath of $H$, which we will denote by\n$wxyz$, such that \n$i\\deg w = i\\deg y = 2$ and $o\\deg x = o\\deg z=2$.\nLet the predecessor of $w$ be $a$ and the successor of $z$ be $b$.\nNote that $a\\ne b$, for if $a=b$, then $G=K_5$ contradicting the hypothesis that $G$ is planar.\n\n\n\\begin{figure}[!hbtp] \n\\begin{center}\n\\includegraphics[scale=.4]{DomHam2.eps} \n\\centerline{\\bf Figure 2.2}\n\\end{center}\n\\end{figure}\n\n \nForm a new Hamilton cycle $H'$ substituting for the path $awxyzb$ the path $awyxzb$ as indicated by the dashed path shown in Figure 2.2.\nWith respect to the original Hamilton cycle $H$, $w,x,y$ and $z$ are 2-vertices.\nHowever, with respect to $H'$, only $x,y$ and (possibly) $b$ are 2-vertices.\n(Note that the vertices $w$ and $z$ are {\\it not} 2-vertices with respect to $H'$ because the degrees of $w$ and$z$ are both at least 4.)\\par\n\nNext suppose that $H$ does not contain four consecutive 2-vertices, but suppose it does contain three consecutive 2-vertices.\nLet three such consecutive 2-vertices be, in order, $x,y$ and $z$.\nLet the predecessor of $x$ be $a$ and the successor of $z$ be $b$.\nAgain, $a\\ne b$ as before.\nLet the neighbor of $b$ on $H$ different from $z$ be denoted as $b'$.\\par\n\n\\begin{figure}[!hbtp] \n\\begin{center}\n\\includegraphics[scale=.35]{DomHam3.eps} \n\\centerline{\\bf Figure 2.3}\n\\end{center}\n\\end{figure}\n\nNow suppose $b$ has another neighbor via an edge in $G_{int}$ other than $y,z$ and $b'$.\nThen replace the path $axyzb$ by $axzyb$ in $H$ to obtain a new Hamilton cycle $H'$.\nThen whereas $H$ has $x,y$ and $z$ as 2-vertices, $H'$ has vertices $y$ and $z$\nas 2-vertices.\nBut neither $x$ and $b$ is a 2-vertex with respect to $H'$ since the degree of $x$ is at least 4 and because of our assumption on the fourth neighbor of $b$.\n\n\nThus $H'$ has one less triple of successive 2-vertices than does $H$.\n(See Figure 2.3.)\n\n\n \n\nSuppose, on the other hand, that the only neighbors of vertex $b$ via edges in $G_{int}$ are $y,z$ and $b'$.\nHence $y$ is adjacent to $b'$.\nIn this case, vertex $b$ will be a new 2-vertex with respect to $H'$, where $H'$ is obtained from $H$ be replacing the path $axyzb$\nby the path $axzyb$ and hence the total number of 2-vertices in $H'$ is the same as in $H$.\n\nMore generally, suppose that $y$ is adjacent to $b'=b_0, b_1,\\ldots,b_k,b_{k+1}=b'$, but not adjacent to $b_{k+2}$,\nwhere $b_0b_1\\cdots b_kb_{k+1}b'$ is a subpath of $H$.\nIn this case we replace the path $axyzb_0b_1,\\cdots b_kb'$ with\n$axzb_0b_1\\cdots b_kyb'$ to obtain a new Hamilton cycle $H'$.\nNote that none of $b=B_1,b_2,\\ldots,b_k$ is a 2-vertex relative to $H'$ since none of these vertices is a\n2-vertex with respect to $H$.\nMoreover, whereas $H$ contained consecutive 2-vertices $x,y,z$, these have been replaced with $y$ and $b_k$ which are 2-vertices with respect to cycle $H'$.\\par\n\nNote that it is possible that $y$ is adjacent via an edge in $G_{int}$ to every vertex in $G$\n(cf. Figure 2.4), but in that case\n$\\gamma (G)=1$.\n\n\n\\begin{figure}[!hbtp]\n\\begin{center}\n\\includegraphics[scale=.43]{DomHam4.eps} \n\\centerline{\\bf Figure 2.4}\n\\end{center}\n\\end{figure} \n\n\\section{ $(H,A,B,O)$-graphs}\\par\n\n\\medskip\n\nIn this section we introduce the concept of an $(H,A,B,O)$-graph and study some of its properties.\\par\n\nLet $G$ be a\n(not necessarily planar)\ngraph on $n$ vertices with a Hamilton cycle $H = x_1x_2\\cdots x_nx_1$.\\par\n\nA subgraph of $G$ consisting of a 3-cycle $x_ix_{i+1}x_{i+2}x_i$, where $\\deg_G (x_{i+1})=2$, will be\ndenoted by\n{\\it $A$}.\nA subgraph on four vertices consisting of the path $x_ix_{i+1}x_{i+2}x_{i+3}$, together with the edges\n$x_ix_{i+2}$ and $x_{i+1}x_{i+3}$\nsuch that $\\deg_G(x_{i+1}) = \\deg_{G}x_{i+2}=3$\nwill be denoted by {\\it $B$}.\nA third type of configuration, denoted by $O$, consists of a single edge with endvertices included\nand such that if $x_i, i=1,2,$ is an endvertex of an $O$, then there is no edge of the form $x_{i-1}x_{i+1}$.\nFurther, if\nan $A$ is immediately preceded and immediately succeeded by $O$s on the Hamilton cycle $H$, we will say that this $A$ is {\\it isolated}.\\par\n\\medskip\n\\noindent\n{\\bf Def.:} An {\\it $(A,B)$-string} is a connected maximal induced subgraph of $G$ consisting of $A$s and $B$s.\nA {\\it mixed} $(A,B)$-string is a string containing at least one $A$ and at least one $B$.\\par\n\\medskip\n\n\n\nWe now suppose $G$, together with a Hamilton cycle $H$ contained in $G$, satisfies the following three assumptions.\\par\n\\medskip\n\\noindent\n(1) \nSuppose all edges in $E(G)-E(H)$ are of the form $x_ix_{i+2}$, ($\\mod n$).\nSuch edges will be called {\\it 2-chords}.\\par\n\\medskip\n\n\n\\noindent\n(2) Suppose further that $E(G)$ consists only of edges lying in the Hamilton cycle $H$ together with edges lying in either an\n$A$ or a $B$.\n\n\\par\n\\medskip\n\\noindent\n(3) Finally, assume that $G$ consists only of $A$s, $B$s and $O$s, where\nany pair intersect in at most one vertex.\\par\n\\medskip\nSuch a graph $G$, together with a Hamilton cycle $H$ in $G$, is called an\n{\\it (H,A,B,O)-graph}.\n(The reader is referred to Figure 3.1 for an example of such a graph.)\n\n\n\\begin{figure}[!hbtp]\\refstepcounter{figure}\n\\begin{center}\n\\includegraphics[scale=.45]{AsectionBsectionGraph.eps} \n\\centerline{\\bf Figure 3.1}\n\\end{center}\n\\end{figure}\n\n\nNote that an $(H,A,B,O)$-graph is planar.\nNote also that it follows from Lemma 2.1 that every outerplanar triangulation with $\\delta\\ge 4$ which contains a Hamilton cycle must contain a spanning $(H,A,B,O)$-subgraph.\\par\n\n\n\nLet ${\\cal K}$ denote the collection of all $(H,A,B,O)$-graphs with at least seven vertices and let\n${\\cal K}_{\\lceil (n+1)\/2\\rceil}$ denote the subclass of ${\\cal K}$ consisting of those members of ${\\cal K}$ with\n$n = |V(K)|$ vertices and in which the\nHamilton cycle $H$ possesses at least $(n+1)\/2$ chords.\nLet $K$ be a member of ${\\cal K}_{\\lceil (n+1)\/2\\rceil}$.\nWe now introduce seven operations,\ncalled {\\it reductions}.\nIn each of these reductions we replace a certain subgraph of $K\\in {\\cal K}_{\\lceil (n+1)\/2\\rceil}$ with a certain smaller\nsubgraph so as\nto produce a new graph $K'\\in {\\cal K}_{\\lceil (n'+1)\/2\\rceil}$ for some $n'(n-6)\/2=(n'+1)\/2$ and $K'\\in {\\cal K}_{\\lceil(n'+1)\/2\\rceil}$.\\par\n\\medskip\n\\noindent\n{\\bf Claim:}\n$\\gamma (K)\\le \\gamma (K')+2$ and if $\\gamma (K')\\le (2\/7)|V(K')|$, then\n$\\gamma (K)\\le (2\/7)|V(K)|$.\n\\par\n\\medskip\n\\noindent\n{\\it Proof of Claim:}\nLet $D'$ be a minimum dominating set in $K'$.\nIf 1 is in $D'$, or 1 is not in $D'$, but is dominated from above, then add 4 and 8 to dominate $K$.\nIf 1 is not in $D'$, but is dominated from below in $K'$, then add 2 and 6 to dominate $K$.\nThis proves the Claim.\\par\n\\par\n\\medskip\n\\noindent\n{\\bf Reduction 5.} Suppose $n\\ge 15$.\nReplace the 13-vertex configuration by a 5-vertex configuration consisting of an $O$ and a $B$ to obtain $K'$ as shown in Figure 3.7.\\par\n\\medskip\n\n\n\\begin{figure}[!hbtp] \n\\begin{center}\n\\includegraphics[scale=.6]{figure37.eps} \n\\centerline{\\bf Figure 3.7}\n\\end{center}\n\\end{figure}\n\n\nIn this case, $n'=n-8\\ge 7$ and $t'=t-4\\ge (n+1)\/2-4=(n'+1)\/2$.\nHence $K'\\in {\\cal K}_{\\lceil (n'+1)\/2\\rceil}$.\\par\n\\medskip\n\\noindent\n{\\bf Claim:}\n$\\gamma (K)\\le \\gamma (K')+2$ and\n$\\gamma (K)\\le 2|V(K)|\/7$, if $\\gamma(K')\\le 2|V(K')|\/7$.\n\\par\n\\medskip\n\\noindent{\\it Proof of Claim:}\nLet $D'$ be a dominating set in $K'$.\nIf 1 and $13\\in D'$, then add 6 and 8 to dominate $K$.\nIf $1\\in D'$, but $13\\notin D'$,\nthen add 6 and 10 to dominate $K$.\n\\medskip\n\\noindent\nIf $1\\notin D'$, but $13\\in D'$, then one of 10, 11 or 12 belongs to $D'$ in order to dominate the vertex $10\\in K'$.\nDenote this vertex by $v$.\nThen add 2, 4, and 8 to $D'-\\{v\\}$\nto dominate $K$.\nFinally suppose $1\\notin D'$ and $13\\notin D'$.\nIf both 1 and 13 are dominated by vertices from $\\{10, 11, 12\\}$, then $D'$ contains two vertices from $\\{10,11,12\\}$\nwhich we will call $v$ and $w$.\nThen add 3, 6, 8 and 11 to $D'-\\{v,w\\}$ to dominate $K$.\n\\par\nSo suppose at most one of 1 and 13 is dominated by vertices from $\\{10,11,12\\}$.\nThen $D'$ contains at least one of 10, 11 and 12 which we will call $v$.\nThen add 3, 6 and 10 to $D'-\\{v\\}$ to dominate $K$ if 13 is dominated by a vertex different from 10,11 and 12.\nOtherwise, add 4, 8 and 11 to $D'-\\{v\\}$ to dominate $K$.\nThis completes the proof of the Claim.\\par\n\n\\medskip\n\\noindent\n{\\bf Reduction 6.} Suppose $n\\ge 21$.\nReplace the 15-vertex configuration by a single vertex to obtain $K'$ as shown in Figure 3.8.\nThen again $K'\\in {\\cal K}_{\\lceil(n'+1)\/2\\rceil}$.\\par\n\nIn this case, $n'=n-14$, $t'=t-7\\ge (n+1)\/2-7=(n-13)\/2=(n'+1)\/2$.\n\n\\medskip\n\\noindent\n{\\bf Claim:}\n$\\gamma (K)\\le \\gamma (K')+4$ and\nif $\\gamma (K')\\le (2\/7)|V(K')|$, then $\\gamma (K)\\le (2\/7)|V(K)|$.\\par\n\\medskip\n\n\n\\begin{figure}[!hbtp] \n\\begin{center}\n\\includegraphics[scale=.6]{figure38.eps} \n\\centerline{\\bf Figure 3.8}\n\\end{center}\n\\end{figure}\n\n\n\\noindent\n{\\it Proof of Claim:}\nLet $D'$ be a minimum dominating set for $K'$.\nIf $15\\notin D'$,\nadd 3, 6, 9 and 13 to $D'$.\nThis completes the proof of the Claim.\nOur final reduction will be applied only in the final stages of the proof of Theorem 3.1.\\par\n\\medskip\n\\noindent\n{\\bf Reduction 7.} Suppose $n\\ge 21.$\nReplace the 12-vertex configuration by a single vertex to obtain $K'$ as shown in Figure 3.9.\n\\par\n\\bigskip\n\n\\begin{figure}[!hbtp] \n\\begin{center}\n\\includegraphics[scale=.6]{figure39.eps} \n\\centerline{\\bf Figure 3.9}\n\\end{center}\n\\end{figure}\n \nIn this case, $|V(K')|=|V(K)|-11$.\n\\par\n\\medskip\n\\noindent\n{\\bf Claim:}\n$\\gamma (K)\\le \\gamma (K')+3$ and if $\\gamma (K')\\le\n 2|V(K')|\/7\n\n$, then $\\gamma (K)\\le \n2|V(K)|\/7\n$.\\par\n\\medskip\n\\noindent\n{\\it Proof of Claim:}\nLet $D'$ be a minimum dominating set for $K'$.\nIf $12\\in D'$, then add $1, 5, 7$ and $12$ to dominate $K$.\nIf $12\\notin D'$, then add $2, 5$ and $10$ to dominate $K$.\nThus $\\gamma (K)\\le \\gamma (K')+3\\le\n2|V(K')|\/7\n+3 =\n2(|V(K)|-11)\/7\n+3\\le\n2|V(K)|\/7\n$\nand the Claim is proved.\n\\par\n\\medskip\nWe are now prepared for our main result about $(H,A,B,O)$-graphs.\\par\n\\medskip\n\n\n\\noindent\n{\\bf Theorem 3.1.} If $K$ is an $(H,A,B,O)$-graph on $n$ vertices and $K$ has at least $(n+1)\/2$ 2-chords,\nthen $\\gamma (K)\\le \\lceil 2n\/7\\rceil$.\\par\n\\medskip\n\n\\noindent\n{\\bf Proof:} Let $K$ be an $(H,A,B,O)$-graph with \n$n$ vertices and at least $(n+1)\/2$ 2-chords. Note that $K$ is simple.\nIt follows that $n\\ge 5$ since \n$K$ has at least $(n+1)\/2$ 2-chords. Suppose on the contrary that $K$ is a minimum counterexample. \n\n\n\\medskip\n\n{\\bf Claim 1}: {\\sl $K$ has $n\\ge 21$ vertices.} \\medskip\n\n\\noindent {\\it Proof of Claim 1}: If not, suppose $5\\le n\\le 20$. Since $K$ has at least $ (n+1)\/ 2 $ 2-chords, by the Pigenhole Principle, there are two\n2-chords sharing a common endvertex $v$.\nThen $v$ dominates five consecutive vertices on the \nHamilton cycle.\nBut each vertex on the Hamilton cycle domintates at least three consecutive \nvertices.\nHence $\\gamma(K)\\le 1+\\lceil (n-5)\/3\\rceil$. It can be easily checked that $1+\\lceil (n-5)\/3\n\\rceil\\le \\lceil 2n\/7\\rceil$ for $5\\le n\\le 20$. So $\\gamma(K)\\le\n\\lceil 2n\/7\\rceil$, contradicting the assumption that $K$ is a counterexample. \nThis completes the proof of Claim 1.\n\\medskip\n \n\nIn the following, we may therefore assume that $K$ has at least $n\\ge 21$ vertices.\nSo Switch operations and Reductions 1-7 \ncan be applied to $K$.\\medskip\n \n\n\n{\\bf Claim 2}: {\\sl $K$ is composed of mixed $(A,B)$-strings only and each such mixed $(A,B)$-string $S$ satisfies (i), (ii) and (iii) below:\n\n(i) $S$ starts and ends with an $A$; \n\n(ii) any two $A$s in $S$ are separated by at least two consecutive $B$s in $S$;\n\n(iii) every such $S$\nis separated from the next mixed $(A,B)$-string by exactly one $O$.}\\medskip\n\n\\noindent{\\it Proof of Claim 2}: We prove Claim 2 via five subclaims.\n\\medskip\n\n(2.1) Every $(A,B)$-string in $K$ starts and ends with an $A$. \\medskip\n\nSuppose to the contrary that $S$ is an $(A,B)$-string which starts or ends with a $B$.\nApplying Reduction 3, let \n$K'$ be the resulting $(H,A,B,O)$-graph which has four fewer vertices than $K$.\nSince $K$ is a minimum counterexample, $\\gamma(K')\\le \\lceil 2|V(K')|\/7\\rceil$.\nBy the claim following Reduction 3, we have $\\gamma(K)\\le \\lceil 2n\/7\\rceil$,\nwhich contradicts the assumption that $K$ is a minimum counterexample.\nSubclaim (2.1) follows.\\medskip\n\n\n(2.2) No $(A,B)$-string of $K$ contains two consecutive $A$s. \\medskip\n\nSuppose to the contrary that $K$ has an $(A,B)$-string with two consecutive $A$s. \nApplying Reduction 2, let $K'$ be the new \n$(H,A,B,O)$-graph with $|V(K')|=n-4$ vertices. \nThen $\\gamma(K')\\le \\lceil 2|V(K')|\/7\\rceil$ since $K$\nis a minimum counterexample.\nBy the claim following Reduction 2, it follows that $\\gamma(K)\\le \\lceil 2n\/7\\rceil$, a contradiction\nand subclaim (2.2) is proved. \\medskip\n\n(2.3) $K$ contains at least one mixed $(A,B)$-string and no mixed\n$(A,B)$-string contains a segment of the form $ABA$.\\medskip\n\nAgain by way of contradiction, suppose $K$ does not contain a mixed $(A,B)$-string.\nThen by (2.1) and (2.2), all $(A,B)$-strings of\n$K$ are isolated $A$s.\nBut then the number of 2-chords of $K$ is the number of \n$A$s in $K$, which is at most $n\/3$, contradicting the assumption that $K$ has at least $(n+1)\/2$\n2-chords. Hence $K$ contains at least one mixed $(A,B)$-string. \n\nIf $K$ has a mixed $(A,B)$-string with a segment of the form $ABA$, apply Reduction 1 to the segment\n$ABA$.\nLet $K'$ be the new $(H,A,B,O)$-graph.\nThen $|V(K')|<|V(K)|$. Since $K$ is a minimum\ncounterexample, it follows that $\\gamma(K')\\le \\lceil 2|V(K')|\/7\\rceil$.\nBy the claim following\nReduction 1, we have $\\gamma(K)\\le \\lceil 2n\/7\\rceil$, contradicting the assumption that $K$ is a\ncounterexample.\nSubclaim (2.3) thus follows. \\medskip\n\n\n(2.4) For every vertex of degree 2 there is a 2-chord joining its neighbors; i.e., these three vertices form an $A$.\n\\medskip\n\nBy way of contradiction, suppose that $K$ has a vertex $v$ of degree 2 which is not contained in any $A$.\nIn other words, $v$ is the intersection of two consecutive $O$s. \nLet $w$ be a neighbor of $v$ in the Hamilton cycle.\\par\n\nFirst, assume that $w$ is also a vertex of degree 2.\nThen $K$ contains three consective $O$s.\nUsing Switch operations, we may assume \nthat one of $v$ and $w$, say $v$, has a neighbor in a mixed $(A,B)$-string $S$.\n(We know such a mixed $(A,B)$-string exists by (2.3).)\nIf $v$ had no such neighbor, we could\napply Switch operations to bring the degree 2 vertices closer to the mixed $(A,B)$-string.\nHence, the three consecutive $O$s are followed by a mixed $(A,B)$-string $S$.\nBy (2.1) and (2.2), \nwe know that $S=AB\\cdots$; that is $S$ starts with an $A$ followed by a $B$. \nApply a Switch operation to $OOOAB\\cdots$ to obtain the segment $AOOOB\\cdots$.\nThen we have a new $(H,A,B,O)$-graph $K'$ such that $|V(K')|=|V(K)|=n$ and $K'$ has a string\nstarting with a $B$.\nBy (2.1), $K'$ is not a counterexample and hence $\\gamma(K')\\le \\lceil 2n\/7\\rceil$.\nBy the claim following the definition of a Switch operation, it follows that $\\gamma(K)\\le \\gamma(K')\\le \n\\lceil 2n\/7\\rceil$ contradicting the assumption that $K$ is a minimum counterexample. \n\nSo we may assume that both neighbors of $v$ have degree at least 3; i.e., $K$ contains two consecutive \n$O$s.\nBy (2.1), the two segments sharing vertices with the two consecutive $O$s\nare $A$s.\nIf one of the $A$s is isolated, \nthen apply Reduction 4 to obtain a new $(H,A,B,O)$-graph $K'$ such \nthat $|V(K')|<|V(K)|$.\nSince $K$ is a minimum counterexample, $\\gamma(K')\\le \\lceil 2|V(K')|\/7\\rceil$.\nBy the Claim following Reduction 4, we have $\\gamma(K)\\le \\lceil 2n\/7\\rceil$,\ncontradicting the assumption that $K$ being a minimum counterexample.\\par\nSo assume that neither of the $A$s is isolated.\nNow apply Reduction 5.\nAs before, we then have $\\gamma(K)\\le \\lceil 2n\/7\\rceil$, a contradiction, and (2.4) follows.\\medskip\n \n(2.5) Every $(A,B)$-string of $K$ is a mixed $(A,B)$-string.\\medskip\n\nSuppose to the contrary that $K$ contains an $(A,B)$-string which is not mixed.\nThen by (2.1) and (2.2) this string must consist of an isolated $A$.\nAmong all such isolated $A$s, choose one that is closest to some mixed $(A,B)$-string.\nIt then follows by (2.3) and (2.4) that $K$ contains a segment of the form $AOAOABB$.\nApply Reduction 6 to $AOAOABB$ and let $K'$ be the new $(H,A,B,O)$-graph.\nThen $|V(K')|<|V(K)|$.\nHence $\\gamma(K')\\le \\lceil 2|V(K')|\/7\\rceil$ since $K$ is a minimum counterexample.\nBy the claim following Reduction 6, we have $\\gamma(K)\\le \\lceil 2n\/7\\rceil$, a contradiction. \\medskip\n\nCombining (2.1)-(2.5), we can conclude that Claim 2 is proved.\n \\medskip\n\n{\\bf Claim 3}: {\\sl If $K$ contains at least six $B$s, then no three of these $B$s are consecutive.} \\medskip\n\n\\noindent{\\it Proof of Claim 3}: By way of contradiction, suppose that $K$ does contain three consecutive $B$s. \nLet $x_1$ be the number of $A$s and $x_2$ the number of $B$s.\nThen $x_2\\ge 6$ and since $K$ contains at least one mixed $(A,B)$-string, $x_1\\ge 2$.\\par\n\nLet $y$ be the number of mixed $(A,B)$-strings. By Claim 2, it follows that $y \\le (x_2-1)\/2$ since at least one mixed $(A,B)$-string contains at least three consecutive $B$s. \n\n\nSince $K$ contains three consecutive $B$s, $K$ contains a segment $ABBB$. \nApply Reduction 7 to contract the segment $ABBB$ to a single vertex (deleting loops) and let $K'$ be the\nresulting new $(H,A,B,O)$-graph. \nThen the number of 2-chords of $K'$ is $t'=(x_1-1)+2(x_2-3),$\nand the number of vertices of $K'$ is $|V(K')| = 2(x_1-1) +3(x_2-3) +y,$\nwhere $x_1-1$ and $x_2-3$ are the number of $A$s and $B$s in $K'$, respectively. \nBut $x_2\\ge 6$ and so it then follows that $x_2-3> (x_2-1)\/2\\ge y$.\nHence \n\n$${{t'}\\over{|V(K')|}}={{(x_1-1)+2(x_2-3)}\\over{2(x_1-1)+3(x_2-3)+y}} = \n {{[(x_1-1)+(x_2-3)] +(x_2-3)}\\over{2[(x_1-1)+(x_2-3)] + ((x_2-3)+y)}}\n > {{1}\\over{ 2}}\\ .$$\nSo the number of 2-chords in $K'$ is at least $(|V(K')|+1)\/2$. Since $|V(K')|<|V(K)|$ and $K$ is a minimum counterexample,\n$\\gamma(K')\\le \\lceil 2|V(K')|\/7 \\rceil $.\nBy the claim following Reduction 7, we have $\\gamma(K)\n\\le \\lceil 2n\/7\\rceil$, contradicting the fact that $K$ is a counterexample.\nThis completes the proof of Claim 3.\n\\medskip \n\nBy Claims 1-3, $S=ABBABBA\\cdots ABBA$ for every mixed $(A,B)$-string $S$ in $K$ or else\n$K$ has exactly five $B$s. \n\n\nFirst assume that $S=ABBABBA\\cdots ABBA$ for every mixed $(A,B)$-string in $K$.\nLet $x$ be the total number of segments $ABB$ in $K$, and $y$ be the number of mixed $(A,B)$-strings. \nThen $n=8x+3y$.\nNote that all vertices of the segment $ABB$\ncan be dominated by two vertices.\nSince $x\\ge y$, we have \n$$\\gamma(K)\\le 2x+y\\le 2(7x+4y)\/7 \\le 2(8x+3y)\/7=2n\/7,$$\nwhich contradicts the assumption that $K$ is a minimum counterexample.\n\nOn the other hand, if $K$ has exactly five $B$s,\nthen\n$K=ABBAOABBBAO$ or else \\hfill\\break $K=ABBABBBAO$\nby Claims 1 and 2.\nNote that the domination number of $ABBA$ is 3 and\nthe domination number of $ABBBA$ is 3. Hence $\\gamma(K)\\le 6\\le 2n\/7$, a contradiction. \nThis completes the\nproof of the theorem. \n\n\\section{Main Result}\\par\n\\medskip\nWe shall have need of the following theorem due to Campos and Wakabayashi [CW].\\par\n\\medskip\n\\noindent\n{\\bf Theorem 4.1:} If $G$ is an maximal outerplanar graph with $n\\ge 4$ vertices and with $t$ vertices of degree 2, then $\\gamma (G)\\le (n+t)\/4$.\\par\n\\medskip\nWe are now prepared for our main theorem.\\par\n\\medskip\n\\noindent\n{\\bf Theorem 4.2:} Let $G$ be a plane triangulation with $n$ vertices and $\\delta (G)\\ge 4$, and\nsuppose $G$ contains a Hamilton cycle $H$.\nThen\n$\\gamma (G)\\le \\max\\{\\lceil 2n\/7\\rceil, \\lfloor 5n\/16\\rfloor\\}$.\\par\n\\medskip\n\\noindent\n{\\bf Proof:} If $\\gamma (G)=1$, the result is trivial.\nSo suppose that $\\gamma (G)\\ge 2$. Since $\\delta(G)\\ge 4$, it follows that $n\\ge 6$. \nFirst we use Lemma 2.1 to replace the given Hamilton cycle $H$ with one which has no three consecutive 2-vertices.\\par\n\nSuppose now that at least one of $G_{int}$ and $G_{ext}$ has no more than $n\/4$ 2-chords,\nsay, without loss of generality, that $G_{int}$ has no more than $n\/4$ 2-chords.\nBut then $G_{int}$ is a maximal outerplanar graph with $t\\le n\/4$ vertices of degree 2 and so by Theorem 4.1,\n$\\gamma (G)\\le (n+t)\/4\\le (n+ n\/4)\/4= 5n\/16$.\\par\n\nSo suppose both $G_{int}$ and $G_{ext}$ have more than $n\/4$ 2-chords.\nThen the number of 2-chords of $G$ is larger than $n\/4+n\/4=n\/2$.\nIn this case, we may apply Theorem 3.1 to conclude that $\\gamma (G)\\le \\lceil (2n)\/7\\rceil$.\nThe proof of the theorem is complete. \n\\par\n\\medskip\n\nBy a well-known theorem of Tutte [T], every 4-connected planar graph contains a Hamilton cycle, \nso the following result is an immediate corollary of Theorem 4.2.\\medskip\n\n\\noindent\n{\\bf Corollary 4.3:} Let $G$ be a 4-connected plane triangulation with $n$ vertices.\nThen $\\gamma (G)\\le \\max\\{\\lceil 2n\/7\\rceil, \\lfloor 5n\/16\\rfloor\\}$.\\par\n\n\n\\section*{Concluding Remarks} \nDetermination of the domination number seems to be difficult even for plane triangulations.\nMatheson and Tarjan's conjecture has now been open for some eighteen years and there seems to be little significant progress toward\nverifying their conjectured upper bound of $1\/4|V(G)|$ for the general class of all planar triangulations.\nThe main result in the present paper makes some progress for those planar triangulations having minimum degree at least 4 and a Hamilton cycle.\n(Note that this class is larger than the class of 4-connected planar triangulations.)\nIn particular, the reductions used in our main proof would seem to constitute a new approach at least for the graphs in this subclass.\nHopefully, they will prove to shed some light on the general case.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction and main results}\n\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction and main results}\\label{s:introduction}\nLet $M$ be a complete, noncompact Riemannian manifold without boundary. It is called stochastically complete if\n\\begin{equation}\\label{d:stochastic}\\int_{M}p_t(x,y)d\\mathrm{vol}(y)=1,\\quad \\quad \\forall\\ t>0, x\\in M,\\end{equation} where $p_t(\\cdot,\\cdot)$ is the (minimal) heat kernel on $M.$\nYau \\cite {Yau78} first proved that any complete Riemannian manifold with a uniform lower bound of Ricci curvature is stochastically complete. Karp and Li \\cite{KarpLi} showed the stochastic completeness in terms of the following volume growth property: \\begin{equation}\\label{e:volume criterion}\\mathrm{vol}(B_r(x))\\leq Ce^{cr^2},\\quad\\quad\\mathrm{some}\\ x\\in M, \\forall\\ r> 0,\\end{equation} where $\\mathrm{vol}(B_r(x))$ is the volume of the geodesic ball of radius $r$ and centered at $x$. Varopoulos \\cite{Varopoulos}, Li \\cite{Li84} and Hsu \\cite{Hsu89} extended Yau's result to Riemannian manifolds with general conditions on Ricci curvature. So far, the optimal volume growth condition for stochastic completeness was given by Grigor'yan \\cite{Grigoryan86}. We refer to \\cite{Grigoryan99} for the literature on stochastic completeness of\nRiemannian manifolds. These results have been generalized to a quite general setting, namely, regular strongly local Dirichlet forms by Sturm \\cite{Sturm94}.\n\nCompared to local operators, graphs (discrete metric measure spaces) are nonlocal in nature and can be regarded as regular Dirichlet forms associated to jump processes. A general Markov semigroup is called a diffusion semigroup if chain rules hold for the associated infinitesimal generator, see Bakry, Gentil and Ledoux \\cite[Definition~1.11.1]{BakryGentilLedoux}, which is a property related to the locality of the generator. As a common point of view to many graph analysts, the absence of chain rules for discrete Laplacians is the main difficulty for the analysis on graphs. This causes many problems and various interesting phenomena emerge on graphs. A graph is called \\emph{stochastically complete} (or conservative) if an equation similar to \\eqref{d:stochastic} holds for the continuous time heat kernel, see Definition \\ref{d:stochastic complete graph}. The stochastic completeness of graphs has been thoroughly studied by many authors \\cite{DodziukMathai06,Dodziuk06,Wojciechowski08,Weber10,Wojciechowski1,Huang10,KellerLenz10,Wojciechowski2,KellerLenz12,GrigoryanHuangMasamune12,MasamuneUemuraWang12,KellerLenzWojciechowski}. In particular, the volume criterion \\eqref{e:volume criterion} with respect to the graph distance is no longer true for unbounded Laplacians on graphs, see \\cite{Wojciechowski2}. This can be circumvented by using intrinsic metrics introduced by Frank, Lenz and Wingert \\cite{FrankLenzWingert12}, see e.g. \\cite{GrigoryanHuangMasamune12,Folz14,Huang14}.\n\nGradient bounds of heat semigroups can be used to prove stochastic completeness. Nowadays, the so-called $\\Gamma$-calculus has been well developed in the framework of general Markov semigroups where $\\Gamma$ is called the ``carr\\'e du champ\" operator, see \\cite[Definition~1.4.2]{BakryGentilLedoux}. Given a smooth function $f$ on a Riemannian manifold, $\\Gamma(f)$ stands for $|\\nabla f|^2,$ see Section~\\ref{s:graphs} for the definition on graphs. Heuristically, on a Riemannian manifold $M$ if one can show the gradient bound for the heat semigroup\n\\begin{equation}\\label{e:gradient bounds}\\Gamma(P_t f)\\leq P_t(\\Gamma (f)),\\quad\\quad \\forall f\\in C_0^{\\infty}(M),\\end{equation} where $P_t=e^{t\\Delta_M}$ is the heat semigroup induced by the Laplace-Beltrami operator $\\Delta_M$ and $C_0^\\infty(M)$ is the space of compactly supported smooth functions on $M$, then the stochastic completeness follows from approximating the constant function $\\mathds{1}$ by compactly supported smooth functions. This approach has been systematically generalized to Markov diffusion semigroups, i.e. local operators, see \\cite{BakryGentilLedoux}. In this paper, we closely follow this strategy and prove the stochastic completeness for the non-diffusion case, i.e. graphs. This shows that the gradient-bound approach works even in nonlocal setting.\n\nWe introduce the setting of graphs and refer to Section~\\ref{s:graphs} for details.\nLet $(V,E)$ be a connected, undirected, (combinatorial) infinite graph with the set of vertices $V$ and the set of edges $E.$ We say $x,y\\in V$ are neighbors, denoted by $x\\sim y,$ if $(x,y)\\in E.$ The graph is called \\emph{locally finite} if each vertex has finitely many neighbors. In this paper, we only consider locally finite graphs. We assign a weight $m$ to each vertex, $m: V\\to (0,\\infty),$ and a weight $\\mu$ to each edge, $$\\mu:E\\to (0,\\infty), E\\ni (x,y)\\mapsto \\mu_{xy},$$ and refer to the quadruple $G=(V,E,m,\\mu)$ as a \\emph{weighted graph}. We denote by $$C_0(V):=\\{f:V\\to\\R|\\ \\{x\\in V| f(x)\\neq 0\\}\\ \\mathrm{is\\ of\\ finite\\ cardinality}\\}$$ the set of finitely supported functions on $V$ and by $\\ell^p(V,m),$ $p\\in [1,\\infty],$ the $\\ell^p$ spaces of functions on $V$ with respect to the measure $m.$\n\n\n\n\nFor any weighted graph $G=(V,E,m,\\mu)$, it associates with a Dirichlet form with respect to the Hilbert space $\\ell^2(V,m)$ corresponding to the Dirichlet boundary condition,\n\\begin{eqnarray}\\label{e:dirichlet form}Q^{(D)}:&&D(Q^{(D)})\\times D(Q^{(D)})\\to \\R \\nonumber\\\\ && (f,g)\\mapsto \\frac12\\sum_{x\\sim y}\\mu_{xy}(f(y)-f(x))(g(y)-g(x)),\\end{eqnarray} where the form domain $D(Q^{(D)})$ is defined as the completion of $C_0(V)$ under the norm $\\|\\cdot\\|_Q$ given by $$\\|f\\|^2_{Q}=\\|f\\|_{\\ell^2(V,m)}^2+\\frac12\\sum_{x\\sim y}\\mu_{xy}(f(y)-f(x))^2, \\ \\ \\forall f\\in C_0(V),$$ see Keller and Lenz \\cite{KellerLenz12}. For the Dirichlet form $Q^{(D)},$ its (infinitesimal) generator, denoted by $L,$ is called the (discrete) Laplacian. Here we adopt the sign convention such that $-L$ is a nonnegative operator. The associated $C_0$-semigroup is denoted by $P_t=e^{tL}:\\ell^2(V,m)\\to \\ell^2(V,m).$\nFor locally finite graphs, the generator $L$ acts as\n\\begin{align*}\n L f(x)=\\frac{1}{m(x)}\\sum_{y\\sim x}\\mu_{xy}(f(y)-f(x)), \\ \\ \\forall f\\in C_0(V),\n\\end{align*} see \\cite[Theorem~6~and~9]{KellerLenz12}.\nObviously, the measure $m$ plays an essential role in the definition of the Laplacian. Given the weight $\\mu$ on $E,$ typical choices of $m$ of particular interest are: \\begin{itemize}\\item $m(x)=\\sum_{y\\sim x}\\mu_{xy}$ for any $x\\in V$ and the associated Laplacian is called the normalized Laplacian.\n\\item $m(x)=1$ for any $x\\in V$ and the Laplacian is called combinatorial (or physical) Laplacian.\n\\end{itemize} Note that normalized Laplacians are bounded operators, so that these graphs are always stochastically complete, see Dodziuk \\cite{Dodziuk06} or Keller and Lenz \\cite{KellerLenz10}. Thus, the only interesting cases are combinatorial Laplacians, or more general unbounded Laplacians.\n\nFollowing the strategy in \\cite{BakryGentilLedoux}, to show stochastic completeness for the semigroups associated to unbounded Laplacians on graphs, it suffices to prove the gradient bounds as in\n\\eqref{e:gradient bounds}. For that purpose, we first introduce a completeness condition for infinite graphs: A graph $G=(V,E,m,\\mu)$ is called \\emph{complete} if there exists a nondecreasing sequence of finitely supported functions $\\{\\eta_k\\}_{k=1}^\\infty$ such that \\begin{equation}\\label{d:complete}\\lim_{k\\to\\infty}\\eta_k=\\mathds{1}\\ \\mathrm{and}\\ \\ \\Gamma(\\eta_k)\\leq \\frac{1}{k},\\end{equation} where $\\mathds{1}$ is the constant function $1$ on $V.$ Note that the measure $m$ plays a role in the definition of $\\Gamma,$ see Definition~\\ref{d:carre du}, so that it is essential to the completeness of a weighted graph. This condition was defined for Markov diffusion semigroups in \\cite[Definition~3.3.9]{BakryGentilLedoux}; here we adapt it to graphs. As is well-known, this condition is equivalent to the geodesic completeness for Riemannian manifolds, see \\cite{Strichartz83}. For the discrete setting, this condition is satisfied for a large class of graphs which possess intrinsic metrics, see Theorem \\ref{thm:complete intrinsic}.\n\nFor gradient bounds \\eqref{e:gradient bounds}, besides completeness we need curvature dimension conditions. For Markov diffusion semigroups, the curvature dimension conditions are defined via the $\\Gamma$ operator and the iterated operator denoted by $\\Gamma_2,$ see \\cite[eq.~1.16.1]{BakryGentilLedoux}. This approach, using curvature dimension conditions to obtain gradient bounds, was initiated in Bakry and \\'Emery \\cite{BakryEmery85}. The curvature dimension condition on graphs, the non-diffusion case, was first introduced by Lin and Yau \\cite{LinYau10} which serves as a combination of a lower bound of Ricci curvature and an upper bound of the dimension, see Definition \\ref{d:curvature dimension} for an infinite dimensional version $\\CCD (K,\\infty).$ For bounded Laplacians on graphs, Bauer et al. \\cite{Bauer13} introduced an involved curvature dimension condition, the so-called $\\CDE(K,n)$ condition, to prove the Li-Yau gradient estimate for heat semigroups. Also restricted to bounded Laplacians, Lin and Liu \\cite{LinLiu} proved the equivalence between the $\\CCD(K,\\infty)$ condition and the gradient bounds \\eqref{e:gradient bounds} for heat semigroups, see Liu and Peyerimhoff \\cite{LiuP14} for finite graphs. In this paper, under some mild assumptions, we prove the gradient bounds for unbounded Laplacians on graphs.\n\n\n\\begin{thm}[see Theorem~\\ref{thm:main}]\\label{thm:main theorem1}\n Let $G=(V,E,m,\\mu)$ be a complete graph and $m$ be non-degenerate, i.e. $\\inf_{x\\in V}m(x)>0.$ Then the following are equivalent:\n \\begin{enumerate}[(a)]\n \\item $G$ satisfies $\\CCD(K,\\infty).$\n \\item For any $f\\in C_0(V),$\n $$\\Gamma(P_t f)\\leq e^{-2Kt}P_t(\\Gamma (f)).$$\n \\end{enumerate}\n\\end{thm}\nSince it is not clear what volume growth is possible under the $\\CCD(K,\\infty)$ condition, our result cannot be derived from the criteria involving volume growth conditions. For unbounded Laplacians, the standard differential techniques for bounded Laplacians as in \\cite{LiuP14,LinLiu} fail due to essential difficulties in the summability of solutions to heat equations. For instance, we don't know whether $\\Gamma(P_t f)$ lies in the form domain (or, more strongly, in the domain of the generator), see Remark~\\ref{r:difficulty}. In order to overcome these difficulties, we add a mild assumption on the measure $m,$ i.e. the non-degenerancy of the measure, and critically utilize techniques from partial differential equations, see Lemma~\\ref{l:Caccippoli} for the Caccioppoli inequality and Theorem~\\ref{thm:w12}. The assumption of the non-degenerancy of the measure $m$ is mild since it is automatically satisfied for any combinatorial Laplacian.\n\nA direct consequence of the gradient bounds is the stochastic completeness for graphs satisfying the $\\CCD(K,\\infty)$ condition.\n\\begin{thm}\\label{thm:main theorem2} Let $G=(V,E,\\mu,m)$ be a complete graph satisfying the $\\CCD(K,\\infty)$ condition for some $K\\in \\R.$ Suppose that the measure $m$ is non-degenerate, then $G$ is stochastically complete.\n \\end{thm}\n\n\nThe paper is organized as follows: In next section, we set up basic notations of weighted graphs. The $\\Gamma$-calculus is introduced to define curvature dimension conditions. We define a new concept on the completeness of a graph and prove the completeness under the assumptions involving intrinsic metrics on graphs. In Section~\\ref{s:Caccioppoli}, we adopt some PDE techniques to prove a (discrete) Caccioppoli inequality for Poisson's equations. In section~\\ref{s:stochastic completeness}, we prove our main results: the\nequivalence of curvature dimension conditions and the gradient bounds for heat semigroups on complete graphs, Theorem~\\ref{thm:main theorem1}, and the stochastic completeness for graphs satisfying the curvature dimension condition, Theorem~\\ref{thm:main theorem2}.\n\n\n\n\\section{Graphs}\\label{s:graphs}\n\\subsection{Weighted graphs}\nLet $(V,E)$ be a (finite or infinite) undirected graph with the set of vertices $V$ and the set of edges $E$ where $E$ is a symmetric subset of $V\\times V.$ Two vertices $x,y$ are called neighbors if $(x,y)\\in E,$ in this case denoted by $x\\sim y.$ At a vertex $x,$ if $(x,x)\\in E,$ we say there is a self-loop at $x.$ In this paper, we do allow self-loops for graphs. A graph $(V,E)$ is called connected if for any $x,y\\in V$ there is a finite sequence of vertices, $\\{x_i\\}_{i=0}^n,$ such that $$x=x_0\\sim x_1\\sim \\cdots\\sim x_n=y.$$ In this paper, we only consider locally finite connected graphs.\n\nWe assign weights, $m$ and $\\mu,$ on the set of vertices $V$ and edges $E$ respectively and refer to the quadruple $G=(V,E,m,\\mu)$ as a \\emph{weighted graph}: Here $\\mu:E\\to (0,\\infty), E\\ni (x,y)\\mapsto \\mu_{xy}$ is symmetric, i.e. $\\mu_{xy}=\\mu_{yx}$ for any $(x,y)\\in E,$ and $m: V\\to (0,\\infty)$ is a measure on $V$ of full support. For convenience, we extend the function $\\mu$ on $E$ to the total set $V\\times V,$ $\\mu:V\\times V\\to [0,\\infty),$ such that $\\mu_{xy}=0$ for any $x\\not\\sim y.$\n\nFor functions defined on $V,$ we denote by $\\ell^p(V,m)$ or simply $\\ell^p_m,$ the space of $\\ell^p$ summable functions w.r.t. the measure $m$ and by $\\|\\cdot\\|_{\\ell^p_m}$ the $\\ell^p$ norm of a function.\nGiven a weighted graph $(V,E,m,\\mu)$, there is an associated Dirichlet form w.r.t. $\\ell^2_m$ corresponding to the Neumann boundary condition, see \\cite{HaeselerKellerLenzWojciechowski12},\n\\begin{eqnarray*}&&Q^{(N)}:D(Q^{(N)})\\times D(Q^{(N)})\\to \\R \\nonumber\\\\&&\\ \\ \\ (f,g)\\mapsto Q^{(N)}(f,g):=\\frac12\\sum_{x, y\\in V}\\mu_{xy}(f(y)-f(x))(g(y)-g(x)),\\end{eqnarray*} where $D(Q^{(N)}):=\\{f\\in \\ell^2_m|\\ \\sum_{x, y}\\mu_{xy}(f(y)-f(x))^2<\\infty\\}.$ For simplicity, we write $Q^{(N)}(f):=\\frac12\\sum_{x, y}\\mu_{xy}(f(y)-f(x))^2$ for any $f: V\\to\\R.$ Let $D(Q^{(D)})$ denote the completion of $C_0(V)$ under the norm $\\|\\cdot\\|_Q$ defined by $$\\|f\\|_{Q}=\\sqrt{\\|f\\|_{\\ell^2_m}^2+Q^{(N)}(f)}, \\ \\ \\forall f\\in C_0(V).$$ Another Dirichlet form $Q^{(D)},$ defined as the restriction of $Q^{(N)}$ to $D(Q^{(D)}),$ corresponds to the Dirichlet boundary condition, see \\eqref{e:dirichlet form} in Section~\\ref{s:introduction}.\n\nFor the Dirichlet form $Q^{(N)},$ there is a unique self-adjoint\noperator $L^{(N)}$ on $\\ell^2_m$ with\n$$D(Q^{(N)}) = \\mathrm{Domain\\ of\\ definition\\ of\\ } (-L^{(N)})^{\\frac12}$$\nand\n$$Q^{(N)}(f, g) = \\left\\langle(-L^{(N)})^{\\frac12}f,(-L^{(N)})^{\\frac12}g\\right\\rangle,\\quad f,g\\in D(Q^{(N)})$$ where $\\langle\\cdot,\\cdot\\rangle$ denotes the inner product in $\\ell^2_m.$\nThe operator $L^{(N)}$ is the infinitesimal generator associated to the Dirichlet form $Q^{(N)},$ also called the (Neumann) Laplacian. The associated $C_0$-semigroup on $\\ell^2_m$ is denoted by $P_t^{(N)}=e^{tL^{(N)}}.$ For the Dirichlet form $Q^{(D)}$, $L^{(D)}$ and $P_t^{(D)}$ are defined in the same way. In case that the Dirichlet forms corresponding to Neumann and Dirichlet boundary conditions coincide, i.e. $$Q^{(N)}=Q^{(D)},$$\nwe omit the superscripts and simply write $$Q=Q^{(N)}=Q^{(D)}, \\quad L=L^{(N)}=L^{(D)}\\quad \\mathrm{etc}.$$\n\n\nThe following integration by parts formula is useful in further applications, see \\cite[Corollary~1.3.1]{Fukushima}.\n\\begin{lemma}[Green's formula]\\label{l:green formula}\n Let $(V,E,m,\\mu)$ be a weighted graph. Then for any $f\\in D(Q^{(N)})$ and $g\\in D(L^{(N)})$,\n \\begin{equation}\\label{e:green formula}\\sum_{x\\in V} f(x)L^{(N)} g(x)m(x)=-Q^{(N)}(f,g).\\end{equation} A similar consequence holds for the case of Dirichlet boundary condition.\n\\end{lemma}\n\nFor locally finite graphs, we define the \\emph{formal Laplacian}, denoted by $\\Delta,$ as\n$$\\Delta f(x)=\\frac{1}{m(x)}\\sum_{y\\in X}\\mu_{xy}(f(y)-f(x))\\quad \\forall\\ f:V\\to\\R.$$ This formal Laplacian can be used to identify the generators defined before. A result of Keller and Lenz, \\cite[Theorem~9]{KellerLenz12}, states that\n\\begin{equation}\\label{e:generator action}L^{(D)}f=\\Delta f,\\quad \\quad \\forall\\ f\\in D(L^{(D)}),\\end{equation} and a similar result holds for Neumann condition, see \\cite{HaeselerKellerLenzWojciechowski12}. Note that$$\\Delta f\\in C_0(V), \\ \\ \\forall\\ f\\in C_0(V).$$ Different choices for the measure $m$ induce different Laplacians. The typical choices are normalized Laplacians and combinatorial Laplacians, see Section~\\ref{s:introduction}.\n\n\n\n\n\n\n\n\n\nThe measure $m$ on $V$ is called non-degenerate if \\begin{equation}\\label{d:nondegenerate}\\delta:=\\inf_{x\\in V}m(x)>0.\\end{equation} The nondegerancy of the measure $m$ yields a very useful fact for $\\ell^p(V,m)$ spaces.\n\\begin{prop}\\label{p:nondegenerate} Let $m$ be a non-degenerate measure on $V$ as in \\eqref{d:nondegenerate}. Then for any $f\\in \\ell^p(V,m),$ $p\\in [1,\\infty),$\n$$|f(x)|\\leq \\delta^{-\\frac1p}\\|f\\|_{\\ell^p_m}\\ \\ \\ \\forall x\\in V.$$ Moreover, for any $1\\leq p0$, is finite, then\n $G$ is a complete graph.\n\\end{thm}\n\\begin{proof}\n For any $00,$$ where $\\mathds{1}$ is the constant function $1$ on $V.$\n\\end{defi}\nThe next proposition is a consequence of standard Dirichlet form theory, see \\cite{Fukushima} and \\cite{KellerLenz12}.\n\\begin{prop}\\label{p:basic pt}\n For any $f\\in \\ell^p_m,$ $p\\in[1,\\infty],$ we have $P_t^{(D)} f\\in \\ell^p_m$ and\n $$\\|P_t^{(D)} f\\|_{\\ell^p_m}\\leq \\|f\\|_{\\ell^p_m},\\ \\ \\ \\forall t\\geq 0.$$ Moreover, $P_t^{(D)} f\\in D(L^{(D)})$ for any $f\\in \\ell^2_m.$\n\\end{prop}\nThe next property follows from the spectral theorem.\n\\begin{prop}\\label{p:exchange}\n For any $f\\in D(L^{(D)}),$ $$L^{(D)} P_t^{(D)} f=P_t^{(D)}L^{(D)} f.$$\n\\end{prop}\n\n\n\n\n\n\\subsection{Caccioppoli inequality}\nFor elliptic partial differential equations on Riemannian manifolds, the Caccioppoli inequality is well-known and yields the $L^p$ Liouville theorem for harmonic functions for $p\\in (1,\\infty),$ see Yau \\cite{Yau76}.\n\nBy adapting PDE techniques on manifolds to graphs, we obtain the Caccioppoli inequality for subsolutions to Poisson's equations.\n\\begin{lemma}\\label{l:Caccippoli}\n Let $(V,E,m,\\mu)$ be a weighted graph and $g,h:V\\to\\R$ satisfy the following $$\\Delta g\\geq h.$$ Then for any $\\eta\\in C_0(V),$\n \\begin{equation}\\label{e:caccioppoli}\n \\|\\Gamma(g) \\eta^2\\|_{\\ell^1_m}\\leq C(\\|\\Gamma(\\eta) g^2\\|_{\\ell^1_m}+\\|gh \\eta^2\\|_{\\ell^1_m}).\n \\end{equation}\n\\end{lemma}\n\\begin{proof}\nMultiplying $\\eta^2 g$ to both sides of the inequality, $\\Delta g\\geq h,$ and summing over $x\\in V$ w.r.t. the measure $m,$ we get\n\\begin{eqnarray*}\n &&\\sum_x \\eta^2gh (x) m(x)\\leq \\sum_x \\eta^2 g \\Delta g(x) m(x)\\\\\n &=&-\\frac12\\sum_{x,y}\\nabla_{xy}g\\nabla_{xy}(\\eta^2 g)\\mu_{xy}\\\\\n &=&-\\frac12\\sum_{x,y}\\nabla_{xy}g(\\nabla_{xy}g \\eta^2(x)+g(y)\\nabla_{xy}(\\eta^2))\\mu_{xy}\\\\\n &=&-\\frac12\\sum_{x,y}|\\nabla_{xy}g|^2 \\eta^2(x)\\mu_{xy}-\\frac12\\sum_{x,y}\\nabla_{xy}gg(y) (|\\nabla_{xy}\\eta|^2+2\\eta(x)\\nabla_{xy}\\eta)\\mu_{xy},\n\\end{eqnarray*} where we used Green's formula, see e.g. Lemma~\\ref{l:green formula}, in the second line since $\\eta\\in C_0(V).$ For the second term in the last line, by symmetry one has\n$$-\\frac12\\sum_{x,y}\\nabla_{xy}gg(y) |\\nabla_{xy}\\eta|^2\\mu_{xy}=-\\frac{1}{4}\\sum_{x,y}|\\nabla_{xy}g|^2 |\\nabla_{xy}\\eta|^2\\mu_{xy}\\leq 0.$$ Hence, by this observation, the previous estimate leads to\n\\begin{eqnarray*}\n &&\\frac12\\sum_{x,y}|\\nabla_{xy}g|^2 \\eta^2(x)\\mu_{xy}\\\\\n &\\leq&-\\sum_{x,y}\\nabla_{xy}gg(y) \\eta(x)\\nabla_{xy}\\eta\\mu_{xy}-\\sum_x \\eta^2gh (x) m(x)\\\\\n &\\leq& \\frac14\\sum_{x,y}|\\nabla_{xy}g|^2\\eta^2(x)\\mu_{xy}+\\sum_{x,y}|\\nabla_{xy} \\eta|^2g^2(y)\\mu_{xy}-\\sum_x \\eta^2gh (x) m(x),\n\\end{eqnarray*} where we used basic inequality $ab\\leq \\frac14a^2+b^2$ for $a,b\\in\\R.$ The lemma follows from cancelling the first term in the last line with the left hand side of the system of inequalities.\n\n\\end{proof}\n\nUsing this Caccippoli inequality, we get a uniform upper bound of the Dirichlet energy of $P_tf$ for $t>0$ and $f\\in C_0(V).$\n\\begin{lemma}\\label{lem:energy estimate}\n Let $(V,E,m,\\mu)$ be a complete graph. Then for any $f\\in C_0(V)$ and $t\\in [0,\\infty),$\n $$Q(P_tf)=\\|\\Gamma(P_t f)\\|_{\\ell^1_m}\\leq C \\|f\\|_{\\ell^2_m}\\|\\Delta f\\|_{\\ell^2_m},$$ where $C$ is a uniform constant.\n\\end{lemma}\n\\begin{proof}\n For $f\\in C_0(V),$ the local finiteness of the graph implies that $\\Delta f\\in C_0(V).$\n By the completeness of the graph, let $\\eta_k\\in C_0(V)$ satisfy \\eqref{d:complete}. Since $P_tf$ satisfies the equation $\\frac{d}{dt} P_tf=\\Delta P_t f$ for any $t>0,$ applying the Caccippoli inequality in Lemma \\ref{l:Caccippoli} with $g=P_t f,$ $h=\\frac{d}{dt} P_t f$ and $\\eta=\\eta_k,$ we have\n \\begin{eqnarray*}\\|\\Gamma(P_t f) \\eta_k^2\\|_{\\ell^1_m}&\\leq& C(\\|\\Gamma(\\eta_k) |P_t f|^2\\|_{\\ell^1_m}+\\|P_t f \\cdot\\frac{d}{dt} P_t f\\cdot \\eta_k^2\\|_{\\ell^1_m})\\\\\n &\\leq& C\\left(\\frac1k\\| P_t f\\|_{\\ell^2_m}^{2}+\\|P_t f\\|_{\\ell^2_m}\\|\\frac{d}{dt} P_t f\\|_{\\ell^2_m}\\right).\\end{eqnarray*}\nBy Proposition \\ref{p:basic pt}, $$\\|P_t f\\|_{\\ell^2_m}\\leq \\|f\\|_{\\ell^2_m}$$ and by Proposition \\ref{p:exchange} and the equation \\eqref{e:generator action},\n$$\\|\\frac{d}{dt} P_t f\\|_{\\ell^2_m}=\\|\\Delta P_t f\\|_{\\ell^2_m}=\\|P_t\\Delta f\\|_{\\ell^2_m}\\leq \\|\\Delta f\\|_{\\ell^2_m}. $$ Hence $$\\|\\Gamma(P_t f) \\eta_k^2\\|_{\\ell^1_m}\\leq C\\left(\\frac1k\\|f\\|_{\\ell^2_m}^2+\\|f\\|_{\\ell^2_m}\\|\\Delta f\\|_{\\ell^2_m}\\right).$$ By passing to the limit, $k\\to \\infty,$ the monotone convergence theorem yields the lemma.\n\\end{proof}\n\nThe following result is an improved estimate of the previous lemma which will be useful in further applications.\n\\begin{lemma}\\label{lem:max time estimate}\n Let $(V,E,m,\\mu)$ be a complete graph. Then for any $f\\in C_0(V)$ and $T>0,$ we have $\\max_{[0,T]}\\Gamma(P_t f)\\in\\ell^1_m$ and\n \\begin{equation}\\label{eq:max time}\\left\\|\\max_{[0,T]}\\Gamma(P_t f)\\right\\|_{\\ell^1_m}\\leq C_1(T,f), \\end{equation} where $C_1(T,f)$ is a constant depending on $T$ and $f.$ Moveover, $$\\max_{[0,T]}|\\Gamma(P_t f,\\frac{d}{dt} P_t f)|\\in\\ell^1_m\\quad\\mathrm{and}$$\n \\begin{equation}\\label{eq:mixed estimate}\n \\left\\|\\max_{[0,T]}|\\Gamma(P_t f,\\frac{d}{dt} P_t f)|\\right\\|_{\\ell^1_m}=\\left\\|\\max_{[0,T]}|\\Gamma(P_t f,\\Delta P_t f)|\\right\\|_{\\ell^1_m}\\leq C_2(T,f).\n \\end{equation}\n\\end{lemma}\n\\begin{proof}\nThe local finiteness yields that $\\Delta f\\in C_0(V)$ and $\\Delta^2 f\\in C_0(V)$ for $f\\in C_0(V).$\n\nFor the first assertion, the Newton-Leibniz formula yields\n \\begin{eqnarray*}\\Gamma(P_tf)&=&\\Gamma(f)+\\int_0^T\\frac{d}{ds}\\Gamma(P_sf)ds\\\\\n &=&\\Gamma(f)+2\\int_0^T\\Gamma(P_sf,\\frac{d}{ds}P_sf)ds\\\\\n &=&\\Gamma(f)+2\\int_0^T\\Gamma(P_sf,\\Delta P_sf)ds\\\\\n &=&\\Gamma(f)+2\\int_0^T\\Gamma(P_sf,P_s(\\Delta f))ds,\\end{eqnarray*} where the last equality follows from Proposition \\ref{p:exchange}. Hence by the equation \\eqref{eq:gamma} and Lemma \\ref{lem:energy estimate}\n \\begin{eqnarray*}\\left\\|\\max_{[0,T]}\\Gamma(P_tf)\\right\\|_{\\ell_m^1}&\\leq& \\|\\Gamma(f)\\|_{\\ell^1_m}+2\\left\\|\\int_0^T|\\Gamma(P_sf,P_s(\\Delta f))|ds\\right\\|_{\\ell^1_m}\\\\\n &\\leq&\\|\\Gamma(f)\\|_{\\ell^1_m}+\\int_0^T(\\|\\Gamma(P_sf)\\|_{\\ell^1_m}+\\|\\Gamma(P_s(\\Delta f))\\|_{\\ell^1_m})ds\\\\\n &\\leq&\\|\\Gamma(f)\\|_{\\ell^1_m}+CT\\|\\Delta f\\|_{\\ell^2_m}(\\|f\\|_{\\ell^2_m}+\\|\\Delta^2 f\\|_{\\ell^2_m})=:C_1(T,f). \\end{eqnarray*}\n\n The second assertion is a direct consequence of the first one. By $\\Delta f\\in C_0(V)$ and \\eqref{eq:gamma},\n \\begin{eqnarray*}\n \\left\\|\\max_{[0,T]}|\\Gamma(P_t f,\\frac{d}{dt} P_t f)|\\right\\|_{\\ell^1_m}&=&\\left\\|\\max_{[0,T]}|\\Gamma(P_t f,\\Delta P_t f)|\\right\\|_{\\ell^1_m}=\\left\\|\\max_{[0,T]}|\\Gamma(P_t f,P_t \\Delta f)|\\right\\|_{\\ell^1_m}\\\\\n &\\leq&\\frac12\\left\\|\\max_{[0,T]}\\Gamma(P_t f)\\right\\|_{\\ell^1_m}+\\frac12\\left\\|\\max_{[0,T]}\\Gamma(P_t \\Delta f)\\right\\|_{\\ell^1_m}\\\\\n &\\leq& \\frac12(C_1(T,f)+C_1(T,\\Delta f))=:C_2(T,f).\n \\end{eqnarray*}\n\n This proves the lemma.\n\\end{proof}\n\n\nNow we can show that the Dirichlet energy, $t\\mapsto Q(P_t f)$, decays in time for the semigroup $P_t$ on complete graphs.\n\\begin{prop}\\label{p:Dirichlet energy}Let $(V,E,m,\\mu)$ be a complete graph. Then for any $f\\in C_0(V),$\n $$Q(P_t f)\\leq Q(f),\\ \\ \\ \\forall t\\geq 0.$$\n Moreover, for any $f\\in D(Q),$\n $$Q(P_t f)\\leq Q(f),\\ \\ \\ \\forall t\\geq 0.$$\n\\end{prop}\n\\begin{proof}\nFor the first assertion, taking the formal derivative of time in $Q(P_tf)$ for $t>0,$ we get\n\\begin{equation}\\label{eq:eq1}\n \\frac{d}{dt}Q(P_t f)= 2\\sum_{x\\in V}\\Gamma(P_t f,\\frac{d}{dt} P_t f)(x)m(x).\n\\end{equation}\nGiven a fixed $T>t,$ note that for any $t\\in [0,T],$\n \\begin{eqnarray*}\n |\\Gamma(P_t f,\\frac{d}{dt} P_t f)(x)|&\\leq&\\max_{t\\in [0,T]}|\\Gamma(P_t f,\\frac{d}{dt} P_t f)(x)|=: g(x)\\in\\ell^1_m\n \\end{eqnarray*} which follows from \\eqref{eq:mixed estimate} in Lemma \\ref{lem:max time estimate}. Hence the absolute value of the summand on the right hand side of \\eqref{eq:eq1} is uniformly (for $t\\in [0,T]$) bounded above by a summable function $g.$ The differentiability theorem yields that $Q(P_t f)$ is differentiable in time and whose the derivative is given by \\eqref{eq:eq1}.\n\n Since $P_t f\\in D(L)$ and $\\Delta P_t f=P_t \\Delta f\\in D(Q),$ Green's formula in Lemma \\ref{l:green formula} yields \\begin{eqnarray*}\n \\frac{d}{dt}Q(P_t f)&=& 2\\sum_{x\\in V}\\Gamma(P_t f, \\Delta P_t f)(x)m(x)\\\\\n &=&-2 \\sum_{x\\in V} |\\Delta P_t f (x)|^2m(x)\\leq 0.\n\\end{eqnarray*} This proves the first assertion.\n\nFor the second assertion, set $f_k:=f\\eta_k$ for $f\\in D(Q).$ It follows from the previous result that\n$$Q(P_tf_k)\\leq Q(f_k).$$ By Lemma \\ref{l:Qdensity}, $f_k\\to f$ in the norm $\\|\\cdot\\|_{Q}.$ The monotone convergence theorem yields that $$P_tf_k\\to P_tf$$ pointwise. By Fatou's lemma, $$Q(P_tf)\\leq \\liminf_{k\\to\\infty} Q(P_t f_k)\\leq \\liminf_{k\\to\\infty} Q(f_k)=Q(f).$$ This proves the theorem.\n\\end{proof}\n\n\n\n\n\n\n\\section{Stochastic completeness}\\label{s:stochastic completeness}\n\\subsection{Gradient bounds and curvature dimension conditions}\nThe curvature dimension condition implies gradient bounds, see \\cite{BakryGentilLedoux} for the case of Markov diffusion semigroups. In fact, they are equivalent on locally finite graphs under some mild assumptions.\n\\begin{thm}\\label{thm:main}\n Let $G=(V,E,m,\\mu)$ be a complete graph with a non-degenerate measure $m$, i.e. $\\inf_{x\\in V}m(x)>0.$ Then the following are equivalent:\n \\begin{enumerate}[(a)]\n \\item $G$ satisfies $\\CCD(K,\\infty).$\n \\item For any $f\\in C_0(V),$\n $$\\Gamma(P_t f)\\leq e^{-2Kt}P_t(\\Gamma (f)).$$\n \\item For any $f\\in D(Q),$\n $$\\Gamma(P_t f)\\leq e^{-2Kt}P_t(\\Gamma (f)).$$\n \\end{enumerate}\n\\end{thm}\n\n\\begin{rem}\\label{r:difficulty}\nFor the case of finite graphs or bounded Laplacians, this result has been proven by \\cite{LiuP14,LinLiu}. To illustrate their proof strategy, we consider a finite graph $(V,E,m,\\mu)$ satisfying the $\\CCD(0,\\infty)$ condition.\n\n$(a)\\Rightarrow(b):$ For any $f:V\\to\\R,$ set $\\Lambda(s)=P_s(\\Gamma(P_{t-s}f)).$\n Then \\begin{eqnarray*}\n \\Lambda'(s)&=&\\Delta P_s(\\Gamma(P_{t-s}f))-2P_s(\\Gamma(P_{t-s}f,\\Delta P_{t-s}f))\\\\\n &=&P_s(\\Delta\\Gamma(P_{t-s}f)-2\\Gamma(P_{t-s}f,\\Delta P_{t-s}f))\\geq 0,\n \\end{eqnarray*} where the last inequality follows from the $\\CCD(0,\\infty)$ condition. However, for the case of infinite graphs, $\\Delta P_s(\\Gamma(P_{t-s}f)))=P_s\\Delta(\\Gamma(P_{t-s}f)))$ may not hold since in general we don't know whether $\\Gamma(P_{t-s}f)\\in D(L).$\n\n In addition, a strong version of gradient bounds has been proved using the following stronger curvature condition,\n see \\cite[equation 3.2.4]{BakryGentilLedoux}\n \\begin{equation}\\label{e:strong condition}\\Gamma(\\Gamma(g))\\leq 4\\Gamma(g)[\\Gamma_2(g)-K\\Gamma(g)], \\ \\ \\forall g\\in C_0(V).\\end{equation} However, this stronger curvature condition can never be fulfilled for graphs. In fact, the inequality \\eqref{e:strong condition} fails e.g. for $g=\\delta_x.$\n\\end{rem}\n\n\\subsection{Curvature dimension conditions and the properties of heat semigroups.}\n\nIn order to prove the gradient estimate under the $\\CCD(K,\\infty)$ condition, we need some lemmata. For graphs satisfying the $\\CCD(K,\\infty)$ condition, the following lemma states that $\\Gamma(P_t f)$ is a subsolution to the heat equation, a standard definition in the theory of PDEs.\n\n\\begin{lemma}\\label{l:heat subsolution}\n Let $(V,E,m,\\mu)$ be a complete graph satisfying the $\\CCD(K,\\infty)$ condition. Then for any $f\\in C_0(V)$\n $$\\frac{d}{dt}\\Gamma(P_t f)\\leq \\Delta \\Gamma(P_t f)-2K\\Gamma(P_t f).$$\n\\end{lemma}\n\\begin{proof}\n This follows from direct calculation by means of the $\\CCD(K,\\infty)$ condition and local finiteness of the graph.\n\\end{proof}\n\n\\begin{lemma}\\label{l:l1 time derivative} Let $(V,E,m,\\mu)$ be a complete graph. Then\n for any $f\\in C_0(V)$ and $t\\geq 0,$ $$\\left\\|\\frac{d}{dt}\\Gamma(P_t f)\\right\\|_{\\ell^1_m}\\leq 2\\sqrt{Q(f)Q(\\Delta f)}.$$\n\\end{lemma}\n\\begin{proof}\n This follows by the computation,\n \\begin{eqnarray*}\n &&\\left\\|\\frac{d}{dt}\\Gamma(P_t f)\\right\\|_{\\ell^1_m}=2\\sum_{x}\\left|\\Gamma(P_t f, \\frac{d}{dt}P_t f)(x)\\right|m(x)\\\\\n &=&2\\sum_{x}|\\Gamma(P_t f, \\Delta P_t f)(x)|m(x)=2\\sum_{x}|\\Gamma(P_t f, P_t \\Delta f )(x)|m(x)\\\\\n &\\leq&2 \\sqrt{\\sum_{x}\\Gamma(P_t f)m(x)\\sum_{x}\\Gamma(P_t \\Delta f)m(x)}\\\\\n &\\leq &2 \\sqrt{\\sum_{x}\\Gamma(f)m(x)\\sum_{x}\\Gamma(\\Delta f)m(x)}<\\infty,\n \\end{eqnarray*} where we used Proposition \\ref{p:exchange} for $f\\in C_0(V)$ in the third equality and Proposition \\ref{p:Dirichlet energy} for $f,\\Delta f\\in C_0(V)$ in the last one.\n\\end{proof}\n\n\n\nFor complete graphs satisfying the $\\CCD(K,\\infty)$ condition, we have higher summability of the solutions to heat equations.\n\\begin{thm}\\label{thm:w12}\n Let $G=(V,E,m,\\mu)$ be a complete graph with a non-degenerate measure $m$. If $G$ satisfies the $\\CCD(K,\\infty)$ condition, then for any $f\\in C_0(V)$ and $t\\geq 0,$\n $$\\Gamma (P_tf)\\in D(Q).$$\n\\end{thm}\n\\begin{proof}\n From the proof of Proposition \\ref{p:Dirichlet energy}, $\\Gamma(P_t f)\\in \\ell^1(V,m).$ Hence by the nondenegerancy of $m,$ $\\Gamma(P_t f)\\in \\ell^2(V,m).$ It suffices to prove that $Q(\\Gamma(P_t f))<\\infty.$\n\n Let $\\{\\eta_k\\}$ be the sequence in \\eqref{d:complete} by the completeness of the graph. Note that Lemma \\ref{l:heat subsolution} implies that $\\Gamma(P_t f)$ is a subsolution to the heat equation.\n Applying the Caccioppoli inequality \\eqref{e:caccioppoli} with $g=\\Gamma(P_t f),$ $h=\\frac{d}{dt} g+2K g$ and $\\eta=\\eta_k,$\n we get \\begin{eqnarray*}\n \\|\\Gamma(g) \\eta_k^2\\|_{\\ell^1_m}&\\leq& C(\\|\\Gamma(\\eta_k) g^2\\|_{\\ell^1_m}+\\|g(\\frac{d}{dt} g+2K g) \\eta_k^2\\|_{\\ell^1_m})\\\\\n &\\leq & C\\left(\\frac{1}{k}\\|g\\|_{\\ell^2_m}^2+\\|g\\frac{d}{dt} g\\|_{\\ell^1_m}+2|K|\\cdot\\|g\\|_{\\ell^2_m}^2\\right)\\\\\n &\\leq & C(K)(\\|g\\|_{\\ell^2_m}^2+\\|g\\frac{d}{dt} g\\|_{\\ell^1_m})\\\\\n &=&I+II,\n \\end{eqnarray*} where the constant $C(K)$ only depends on $K.$\n By the assumption that $m$ is non-degenerate, Propositions \\ref{p:nondegenerate} and\n \\ref{p:Dirichlet energy} yield that\n $$I\\leq C\\|\\Gamma(P_tf)\\|_{\\ell^1_m}^2\\leq C\\|\\Gamma(f)\\|_{\\ell^1_m}^2<\\infty.$$\n\n For the other term, noting that $\\|g\\|_{\\ell^\\infty}\\leq C\\|g\\|_{\\ell^1_m},$ by Lemma \\ref{l:l1 time derivative}, we have\n \\begin{eqnarray*}\n II&\\leq& C\\|g\\|_{\\ell^\\infty}\\|\\frac{d}{dt}g\\|_{\\ell^1_m}\\\\\n &\\leq& C \\|g\\|_{\\ell^1_m}\\|\\frac{d}{dt}g\\|_{\\ell^1_m}<\\infty.\n \\end{eqnarray*} Thus, $\\|\\Gamma(g) \\eta_k^2\\|_{\\ell^1_m}\\leq C<\\infty$ where the right hand side is independent of $k.$ By passing to the limit, $k\\to \\infty,$ Fatou's lemma yields that\n $$\\|\\Gamma(\\Gamma(P_t f))\\|_{\\ell^1_m}\\leq \\liminf_{k\\to\\infty}\\|\\Gamma(\\Gamma(P_t f)) \\eta_k^2\\|_{\\ell^1_m}\\leq C.$$ This proves the theorem.\n\n\\end{proof}\n\n\\subsection{The proofs of main theorems}\n\\begin{thm}\\label{thm:monotone}\n Let $(V,E,m,\\mu)$ be a complete graph with a non-degenerate measure $m$ and satisfying the $\\CCD(K,\\infty)$ condition. For any $f\\in C_0(V),$ $0\\leq \\zeta\\in C_0(V)$ and $t>0,$ the following function\n $$s\\mapsto G(s):=\\sum_{x\\in V}\\Gamma(P_{t-s}f)(x)P_s \\zeta(x)m(x)$$ satisfies\n $$G'(s)\\geq 2K G(s),\\quad\\quad\\quad 00.$ Taking the formal derivative of $G(s)$ in $s,$ we get\n \\begin{equation}\\label{e:Gs1}\n -2\\sum_{x}\\Gamma(P_{t-s}f,\\Delta P_{t-s}f)(x)P_s\\zeta(x)m(x)+\\sum_{x}\\Gamma(P_{t-s}f)(x)\\Delta (P_{s}\\zeta)(x)m(x)\n \\end{equation} This formal derivative is, in fact, the derivative of $G(s)$ if one can show that the absolute values of summands are uniformly (in $s$) controlled by summable functions. For the first term in \\eqref{e:Gs1}, note that $\\|P_s \\zeta\\|_{\\ell^{\\infty}}\\leq \\|\\zeta\\|_{\\ell^{\\infty}}<\\infty.$ Then the equation \\eqref{eq:mixed estimate} in Lemma \\ref{lem:max time estimate} yields that for any $s\\in (\\epsilon, t-\\epsilon)$\n \\begin{eqnarray*}\n &&2|\\Gamma(P_{t-s}f,\\Delta P_{t-s}f)(x)|P_s\\zeta(x)\\leq\\sup_{s\\in (\\epsilon,t-\\epsilon)}2|\\Gamma(P_{t-s}f,\\Delta P_{t-s}f)(x)|P_s\\zeta(x)\\\\&\\leq& 2\\|\\zeta\\|_{\\ell^\\infty}\\sup_{s\\in (\\epsilon,t-\\epsilon)} |\\Gamma(P_{s}f,\\Delta P_{s}f)(x)|=:g(x)\\in\\ell^1_m.\n \\end{eqnarray*}\n For the second term in \\eqref{e:Gs1}, the equation \\eqref{eq:max time} in Lemma \\ref{lem:max time estimate} implies that for any $s\\in (\\epsilon, t-\\epsilon)$\n \\begin{eqnarray*}\n \\Gamma(P_{t-s}f)(x)|\\Delta (P_{s}\\zeta)(x)|&\\leq&\\sup_{s\\in (\\epsilon,t-\\epsilon)}\\Gamma(P_{t-s}f)(x)|\\Delta (P_{s}\\zeta)(x)|\\\\\n &=&\\sup_{s\\in (\\epsilon,t-\\epsilon)}\\Gamma(P_{t-s}f)(x)|P_{s}\\Delta \\zeta(x)|\\\\\n &\\leq& \\|\\Delta \\zeta\\|_{\\ell^\\infty}\\sup_{s\\in (\\epsilon,t-\\epsilon)}\\Gamma(P_{s}f)(x)=:h(x)\\in\\ell^1_m.\n \\end{eqnarray*} Since $g+h\\in \\ell^1_m$ which is independent of $s\\in(\\epsilon,t-\\epsilon),$ the differentiability theorem yields that $G(s)$ is differentiable and its derivative equals to \\eqref{e:Gs1}. Note that Theorem \\ref{thm:w12} and Proposition \\ref{p:basic pt} yield $\\Gamma(P_{t-s}f)\\in D(Q)$ and $P_s\\zeta\\in D(L)\\subset D(Q).$ Hence, using Green's formula \\eqref{e:green formula} in Lemma \\ref{l:green formula}, we obtain that\n \\begin{equation}\\label{e:Gs2}G'(s)=-2\\sum_{x}\\Gamma(P_{t-s}f,\\Delta P_{t-s}f)(x)P_s\\zeta(x)m(x)-\\sum_{x}\\Gamma(\\Gamma(P_{t-s}f),P_{s}\\zeta)(x)m(x).\\end{equation}\n\n We claim that for any $0\\leq h\\in D(Q),$ \\begin{eqnarray}\\label{e:Gs3}\n &&-2\\sum_{x}\\Gamma(P_{t-s}f,\\Delta P_{t-s}f)(x)h(x)m(x)-\\sum_{x}\\Gamma(\\Gamma(P_{t-s}f),h)(x)m(x)\\\\\n &\\geq& 2K\\sum_{x}\\Gamma(P_{t-s}f)h(x)m(x).\\nonumber\n \\end{eqnarray} Once this claim is verified, by applying $h=P_{s}\\zeta$ in \\eqref{e:Gs3} and the self-adjointness of operators $P_t,$ we can prove the theorem. This claim can be proved by a density argument. Firstly, the $\\CCD(K,\\infty)$ condition yields that \\eqref{e:Gs3} holds for $0\\leq h\\in C_0(V)$: In fact, by Green's formula for $h\\in C_0(V)$,\n \\begin{eqnarray*}\n &&-2\\sum_{x}\\Gamma(P_{t-s}f,\\Delta P_{t-s}f)(x)h(x)m(x)-\\sum_{x}\\Gamma(\\Gamma(P_{t-s}f),h)(x)m(x)\\\\\n &=&-2\\sum_{x}\\Gamma(P_{t-s}f,\\Delta P_{t-s}f)(x)h(x)m(x)+\\sum_{x}\\Delta(\\Gamma(P_{t-s}f))(x)h(x)m(x)\\\\\n &\\geq& 2K\\sum_{x}\\Gamma(P_{t-s}f)h(x)m(x),\n \\end{eqnarray*} where in the last inequality we used the $\\CCD(K,\\infty)$ condition.\n\nFor general $0\\leq h\\in D(Q),$ set $h_k=h\\eta_k$ where $\\{\\eta_k\\}$ is defined in \\eqref{d:complete}. It is obvious that $0\\leq h_k\\in C_0(V).$ Note that Lemma \\ref{lem:energy estimate} and Theorem~\\ref{thm:w12} yield that $\\Gamma(P_{t-s}f,\\Delta P_{t-s}f), \\Gamma(P_{t-s}f)\\in \\ell^1_m$ and $\\Gamma(P_{t-s}f)\\in D(Q).$ Hence applying \\eqref{e:Gs3} for $h_k,$ passing to the limit, $k\\to \\infty,$ we prove the theorem.\n\n\\end{proof}\n\n\n\nNow we can prove the gradient bounds of heat semigroups under the $\\CCD(K,\\infty)$ condition.\n\\begin{proof}[Proof of Theorem \\ref{thm:main}]\n $(a)\\Rightarrow (b):$ Using the same notation as in Theorem \\ref{thm:monotone}, we get\n $$G'(s)\\geq 2KG(s).$$ Hence $G(s)\\geq e^{2Ks}G(0).$ Since $P_s$ is a self-adjoint operator on $\\ell^2_m,$\n $$G(s)=\\sum_{x\\in V}P_s(\\Gamma(P_{t-s}f))(x)\\zeta(x)m(x).$$ By choosing delta functions, such as $\\zeta(x)=\\delta_y(x)$ ($y\\in V$), we prove the theorem.\n\n $(b)\\Rightarrow (a):$ Fix a vertex $x\\in V.$ By $(b),$ $$F(t):=e^{-2Kt}P_t(\\Gamma (f))(x)-\\Gamma(P_t f)(x)\\geq 0.$$ It is easy to see that $F(t)$ is differentiable and $F'(0)\\geq 0.$ Note that $$\\left.\\frac{d}{dt}\\right|_{t=0}P_t(\\Gamma (f))(x)=\\Delta P_t(\\Gamma(f))(x)|_{t=0}=\\Delta (\\Gamma (f))(x).$$\n Since the graph is locally finite, $$\\left.\\frac{d}{dt}\\right|_{t=0}\\Gamma(P_t f)(x)=2\\Gamma(P_tf,\\Delta P_tf)(x)|_{t=0}=2\\Gamma(f,\\Delta f)(x).$$\n This proves the assertion by using $F'(0)\\geq 0.$\n\n $(b)\\Leftrightarrow(c):$ This follows from a density argument.\n\\end{proof}\n\n\n\nNow we are ready to prove the analogue to Yau's result \\cite{Yau78} on graphs.\n\n \\begin{proof}[Proof of Theorem~\\ref{thm:main theorem2}]\n It suffices to prove that $P_t \\mathds{1}=\\mathds{1}$ where $\\mathds{1}$ is the constant function $1$ on $V.$ By completeness, let $\\eta_k\\in C_0(V)$ satisfy \\eqref{d:complete}. The dominated convergence theorem yields that $P_t \\eta_k\\to P_t \\mathds{1}$ pointwise. By the local finiteness of the graph, for any $x\\in V$ and $t>0,$\n \\begin{eqnarray*}\n \\Gamma(P_t\\mathds{1})(x)&=&\\lim_{k\\to \\infty} \\Gamma(P_t \\eta_k)(x)\\leq \\liminf_{k\\to\\infty}e^{-2K t}P_t (\\Gamma(\\eta_k))(x)\\\\\n &\\leq &\\liminf_{k\\to\\infty}e^{-2Kt}\\cdot\\frac{1}{k}=0.\n \\end{eqnarray*} This means that for any $t>0,$ $P_t\\mathds{1}$ is a constant function on $V.$ Since the function $P_t\\mathds{1}$ is continuous in $t$ pointwise and $P_0\\mathds{1}=\\mathds{1},$ we get $P_t\\mathds{1}=\\mathds{1}$ for any $t>0.$ This proves the theorem.\n \\end{proof}\n\n\n\n\n\n{\\bf Acknowledgements.} This work was done when the authors were visiting the Shanghai Center for Mathematical Sciences, Fudan University in Summer 2014. They acknowledge the support from SCMS.\n\nB. H. is supported by NSFC, grant no. 11401106. Y. L. is supported by NSFC, grant no. 11271011, the Fundamental Research Funds for the Central Universities and the Research Funds of Renmin University of China($11$XNI$004$).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Hydrodynamics are relevant for heavy-ion collisions}\n\nOne of the most striking lessons one may draw \\cite {review,hydro} from \nexperiments on \nheavy-ion collisions at high energy ($e.g.$ at the RHIC accelerator, Brookhaven) \nis \nthat fluid hydrodynamics seems to be relevant for understanding the dynamics of \nthe reaction. Indeed, the elliptic flow \n\\cite{JY} describing the anisotropy of the low-p$_T$ particles produced in \na collision at non zero impact parameter implies the existence of a collective \nflow of the particles following a hydrodynamical pressure gradient due to the \ninitial eccentricity in the collision. Moreover most hydrodynamical \nsimulations which are successful to describe this elliptic flow are consistent \nwith an almost ``perfect fluid'' behaviour, $i.e.$ a small ``viscosity over \nentropy'' ratio $\\eta\/s$ (see, for instance, the reviews \\cite{hydro}).\n\nThe validity of a hydrodynamical description assuming a \nquasi-perfect fluid behaviour has been nicely anticipated in \nRef.\\cite{Bjorken}. The so-called {\\it Bjorken flow} is based on the hypothesis \nof an intermediate \nstage of the reaction process, namely a boost-invariant\\footnote{The \nintroduction of hydrodynamics in the \ndescription of high-energy \nhadronic collisions has been proposed by Landau \\cite{landau}, assuming ``full \nstopping'' initial conditions which result in a non boost-invariant solution or \n$Landau\\ flow$ (see \\cite{us} for a unified description of Bjorken and Landau \nflows). We will comment later on the relevance of the Landau flow for AdS\/CFT.}\n quark-gluon plasma phase as a relativistic expanding fluid. It is formed \nafter a (quite rapid) thermalization period and finally decays into hadrons, see \nFig.~\\ref{1}. The boost-invariance can be justified in the central region \nof the collision since the observed distribution of particles is flat, in \nagreement with the prediction of hydrodynamical boost-invariance, where \n(space-time) fluid \nand (energy-momentum) particle rapidities are proved to be equal \\cite{Bjorken}, \nsee section 3.\n\n\\begin{figure}[ht]\n\\begin{center}\n\\includegraphics[width=30pc]{NoName.eps}\n\\end{center}\n\\caption{\\small{\\it Description of QGP formation in heavy ion collisions}. The \nkinematic landscape is defined by ${\\tau = \\sqrt{x_0^2-x_1^2}\\ ;\\ {\\eta=\\f 12 \n\\log \\f \n{x_0+x_1}{x_0-x_1}}\\ ;\\ {x_T\\!=\\!\\{x_2,x_3\\}}}\\ ,$\nwhere the coordinates along the light-cone are $x_0 \\pm x_1,$ the transverse \nones \nare $\\{x_2,x_3\\}$ and $\\tau$ is the proper time, $\\eta$ the ``space-time \nrapidity''.}\n\\label{1}\n\\end{figure}\n\nThe Bjorken flow was instrumental for deriving many qualitative and even \nquantitative \nfeatures of the quark-gluon plasma formation in heavy-ion \nreactions. However, as inherent to the hydrodynamic approach, it says only \nlittle on the relation with the microscopic gauge field theory, $i.e.$ Quantum \nChromodynamics (QCD). Some \nimportant questions remain unsolved, such as the reason why the fluid \nbehaves like a perfect fluid, what is the small amount of viscosity it may \nrequire, why and how fast thermalization proceeds, etc... The problem is made \neven \nmore difficult by the strong coupling regime of QCD which is very probably \nrequired, since a perturbative description leads in general to a high $\\eta\/s$. \nIndeed, the mean free path induced by the gauge theory should be small (hence \nthe coupling strong) in order to damp the near-by force transversal to the \nflow, measuring the shear viscosity. \n\nIt is thus interesting to use our modern (but still largely in progress) \nknowledge\nof non perturbative methods in quantum field theory to fill the gap between the \nmacroscopic and microscopic descriptions of the quark-gluon plasma produced in \nheavy-ion collisions. Lattice gauge theory methods are very useful to analyze \nthe static properties of the quark-gluon plasma, but there are still powerless \nto describe the plasma in collision. Hence we are led to rely upon the new \ntools offered by the Gauge\/Gravity correspondence and in particular the one \nwhich is the most studied and well-known namely the AdS\/CFT duality \n\\cite{adscft} between the \n${\\cal N}=4$ supersymmetric Yang-Mills theory and the type IIB superstring in the \nlarge $N_c$ approximation. The features of the gauge theory on the (physical) \nMinkowski space in $3+1$ dimensions at strong coupling are in one-to-one \nrelation with corresponding ones in the bulk of the target space of the 10-d \nstring and in particular in the 5-dimensional metric of the AdS space, the boundary of which \ncan be identified with the 4-dimensional Minkowski space.\n\nOne should be aware when using the AdS\/CFT tools that there does not yet exist a \ngravity dual construction \nfor QCD. However, the nice feature of the quark-gluon plasma problems is that it \nis \na deconfined phase of QCD, characterized by collective degrees of freedom and \nthus one may expect to get useful information from AdS\/CFT duality. This has \nbeen already proved when describing static geometries by an \nevaluation of $\\eta\/s$ \\cite{son}. The subject of the present lectures is the \ninvestigation of the \nGauge\/Gravity correspondence, in particular the AdS\/CFT duality, in a dynamical \nsetting corresponding to a \ncollision. \n\n\\section{{\\it Relativistic Hydrodynamics} and {\\it Bjorken Flow}}\n\\label{sechydro}\n\nOn theoretical grounds, there are quite appealing features for\napplying hydrodynamic concepts to high-energy heavy-ion reactions. Such\nconcepts have been already introduced some time ago \\cite{landau,Bjorken} and \nfind a plausible \nrealization\nnowadays. The fact that a rather dense interacting medium is created in the \nfirst stage of\nthe collision allows one to admit that the individual partonic or\nhadronic degrees of freedom are not relevant during the early evolution\nof the medium and justifies its treatment as a fluid. For the same reason local \nequilibrium is a plausible assumption. Moreover, the high\nquantum occupation numbers allow one to use a classical picture and to\nassume that the ``pieces of fluid'' may follow quasi-classical\ntrajectories in space-time, expressed as an in-out cascade\n\\cite{gl} with straight-line trajectories starting at the origin (see Fig. 2), with\n\\begin{equation}\ny=\\eta \\label{bgl}\n\\end{equation}\nwhere\n\\begin{equation}\ny=\\frac12 \\log\\left(\\frac{E+p}{E-p}\\right) \\;;\\;\\;\n\\eta=\\frac12 \\log\\left(\\frac{x_0+x_1}{x_0-x_1}\\right) \n\\end{equation}\nare respectively the rapidity and ``space-time rapidity'' of the piece of the \nfluid\\footnote{We keep the conventional notation $\\eta$, not to be confused with \n viscosity. The difference is clear enough to avoid mistakes.}.\n\\begin{center}\n\\begin{figure}[ht]\n$\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \n$\\includegraphics[width=8cm]{Rys4_bis.eps}\n\\label{2}\n\\caption{{\\it In-Out cascade.} The ``piece of fluid'' with space-time rapidity \n$\\eta$ gives rise to hadrons at rapidity $y\\equiv\\eta,$ after crossing the \n``freeze-out'' hyperbola at fixed proper-time $\\tau.$}\\end{figure} \n\\end{center} \nNote that (\\ref{bgl}) can be rewritten in the form\n\\begin{equation}\n2y=\\log u^+-\\log u^-= \\log x^+- \\log x^- \\label{uz}\n\\end{equation} where\n$u^\\pm=e^{\\pm y}$ are the light-cone components of the fluid\n(four-)velocity and $x^\\pm=x_0\\pm x_1$ are the light-cone kinematical \nvariables. \n \nTaking (\\ref{bgl}) as the starting point and using the perfect fluid \nhydrodynamics, Bjorken developped in his seminal paper \\cite{Bjorken} \na suggestive (and\nvery useful in many applications) physical picture of the \ncentral \nrapidity region of highly relativistic collisions of heavy ions. In this picture \nthe \ncondition (\\ref{bgl}) leads to a boost-invariant geometry of the expanding \nfluid \nand thus to the central\nplateau in the distribution of particles. \n \nLet us introduce the relativistic hydrodynamic equations in light-cone \nvariables. We consider the ``perfect fluid'' approximation for which the \nenergy-momentum tensor is\n\\begin{equation} \n T^{\\mu\\nu}= (\\epsilon+p)u^{\\mu}u^{\\nu} - p \\eta^{\\mu\\nu} \\label{T}\n\\end{equation} \nwhere $\\epsilon$ is the energy density, $p$ is the pressure and \n$u^{\\mu}$ is the 4-velocity. We assume that the energy density and pressure\n are related \nby the equation of state: \n\\begin{equation} \n\\epsilon = gp \n\\label{state} \n\\end{equation} \n where $1\/\\sqrt{g}$ is the sound velocity in the liquid. For the ``conformal \ncase'' $ T^{\\mu\\mu}=0$ and thus $g\\equiv 3.$ \n \nUsing \n\\begin{equation} \nu^\\pm\\equiv u^0\\pm u^1=e^{\\pm y} \n\\end{equation}\nand introducing \n\\begin{equation} \nx^\\pm= x^0\\pm x^1 =\\tau e^{\\pm \\eta}\\; \\rightarrow \\;\n(\\frac {\\d}{\\d x^0}\\pm \\f {\\d}{\\d x^1})={\\scriptstyle\\ \\frac12} \\f {\\d}{\\d \nx^\\pm} \n\\equiv{ \\scriptstyle\\ \\frac12} \\d_\\pm \n\\end{equation} \nwhere $\\tau=\\sqrt{x^+x^-}$ is the proper time and $\\eta$ is the spatial \nrapidity of the fluid, \nthe hydrodynamic equations \n \\begin{equation} \n\\d_\\mu T^{\\mu\\nu} =0 \n\\end{equation} \ntake the form \n\\begin{equation} \n\\d_\\pm T^{01}+\\frac12\\d_+(T^{11}\\pm T^{00}) \n-\\frac12 \\d_-(T^{11}\\mp T^{00})=0\\ . \n\\end{equation} \nUsing now (\\ref{T}) and the equation of state (\\ref{state}) \n we deduce from this \n\\begin{eqnarray} \ng\\d_+[\\log p]&=& -\\frac{ (1\\!+\\!g) ^2}2\\d_+y-\\frac{g^2\\!-\\!1 \n}2e^{-2y}\\d_-y \\nonumber\\\\ \ng\\d_-[\\log p]&=&\\frac{ (1\\!+\\!g) ^2}2\\d_-y +\\frac{g^2\\!-\\!1 }2e^{2y}\\d_+y \\ \n. \n\\label{ef} \n\\end{eqnarray} \nThese are two equations for two unknowns which describe \n the state of the liquid: the pressure $p$ and the \nrapidity $y$. They should be expressed in terms of the \npositions $x^+,x^-$ in the liquid. Other thermodynamic quantities can be \nobtained from the equation of state (\\ref{state}) and the standard \nthermodynamical identities: \n\\begin{equation} \np+\\epsilon = Ts\\;;\\;\\; d\\epsilon = T ds \\label{therm} \n\\end{equation} \nwhere we have assumed for simplicity vanishing chemical potential. \n \nThe result is \n\\begin{equation} \n\\epsilon =gp=\\epsilon_0 T^{g+1}\\;;\\;\\;s=s_0T^g\\;\\rightarrow \\; s\\sim \n\\epsilon^{g\/(g+1)} . \\label{esp} \n\\end{equation} \n \nThe simplest possibility to describe the expansion of the fluid \nwas suggested by Bjorken \\cite{Bjorken} who proposed to use the Ansatz \n(\\ref{bgl}) \nin the hydrodynamical context.\nIntroducing (\\ref{bgl}) into (\\ref{ef}) we obtain \n\\begin{equation} \ng\\d_+[\\log p]=-\\frac{1+g}{2x^+}\\;;\\;\\; \ng\\d_-[\\log p]=-\\frac{g+1}{2x^-} \n\\end{equation} \nfrom which we deduce \n\\begin{equation} \np=\\epsilon\\ g^{-1}= p_0\\ (x^+x^-)^{-(g+1)\/2g} = p_0\\ \\tau^{-(g+1)\/g}\\ , \n\\end{equation} \nwhere $p_0$ is a constant, and thus specifically\n \n\\begin{equation} \np=\\epsilon\/3= p_0\\ (x^+x^-)^{-2\/3} = p_0\\ \\tau^{-4\/3} \\propto T^4 \n\\label{g3}\\end{equation} \nfor the conformal case.\n\nThus the system is \nboost-invariant: the pressure \ndoes not depend neither on $\\eta$ nor on $y$. So are $\\epsilon$, $s$ and $T$, \n given by (\\ref{esp}). \n It is interesting to note that the Landau flow corresponds asymptotically only \nto a logarithmic correction of relation \\eqref{bgl}, namely\n \n\\begin{equation}\nu^{\\pm}\\sim x^{\\pm}\\sqrt{\\log x^{\\pm}} \\ ,\\label{l}\n\\end{equation}\n which gives finally rise (as already noticed in \\cite{landau}, and for instance \n recently discussed in \\cite{us1}) to a gaussian shape in the $y$ distribution \nof the entropy, revealing a non boost-invariant picture, at least at some \ndistance from central rapidity.\n \n\\section{Interest of AdS\/QCD duality}\n\n In the previous sections we mentionned the ubiquity of hydrodynamic methods in \nthe description of QGP produced at RHIC. Yet, despite their success in \ndescribing data, we have to keep in mind that they are used as a \nphenomenological model without a real derivation from gauge theory. This is \nquite understandable since almost perfect fluid hydrodynamics is intrinsically a \nstrong coupling phenomenon - for which one lacks a purely gauge theoretical \nmethod\\footnote{Lattice QCD methods do not work well here as this would require \nanalytical continuation to Minkowski signature which is nontrivial in this \ncontext}. \n\nOn the other hand there exists a wide class of gauge theories, which can be \nstudied analytically at strong coupling. These are superconformal field theories \nwith gravity duals. String theory methods (namely AdS\/CFT correspondence) maps \ngauge theory dynamics (CFT) at strong coupling and large number of colors into \nsolving Einstein equations in asymptotically anti-de Sitter space (AdS). The \ntheories with gravity duals can differ substantially from real world QCD at zero \ntemperature. The best known example of such theory - $\\mathcal{N} = 4$ super \nYang-Mills (SYM) - is a superconformal field theory with matter in the adjoint \nrepresentation of the gauge group SU(N$_{c}$). Because of the conformal symmetry \nat the quantum level this theory does not exhibit confinement. On the other hand \ndifferences between $\\mathcal{N} = 4$ SYM and QCD are less significant above \nQCD's critical temperature, when quarks and gluons are in the deconfined phase. \nMoreover it was observed on the lattice that QCD exhibits a quasi-conformal \nwindow in the certain range of temperatures, where the equation of state is \nwell-approximated by $\\epsilon = 3 p$. The above observations together with \nexperimental results suggesting that quark-gluon plasma is a strongly coupled \nmedium is an incentive to use the AdS\/CFT correspondence as a tool to get \ninsight into the non-perturbative dynamics. \n\n\n\n\n\n\n\n\n\\section{AdS\/CFT setup}\n\nWe will now describe how to set up an AdS\/CFT computation for determining the \nspacetime behaviour of the energy-momentum tensor \\cite{US1}. This \nmethod does not make any underlying assumptions about local equilibrium or \nhydrodynamical behavior. We will obtain hydrodynamic expansion as a generic late \ntime behaviour of the expanding strongly coupled plasma. \n\nSuppose that we consider some macroscopic state of the plasma characterized by a \nspacetime profile of the energy-momentum tensor\n\\begin{equation}\nT_{\\mu\\nu}(x^\\rho)\\ .\n\\end{equation}\nThen, since the AdS\/CFT correspondence asserts the exact equivalence of gauge \nand string theory, such a state should have its counterpart on the string side \nof the correspondence. Typically it will be given by a modification of the \ngeometry of the original $AdS_5\\times S^5$ metric. This comes from the fact that \noperators in gauge theory correspond to fields in supergravity (or string \ntheory). When we consider a state with a nonzero expectation value of an \noperator, the \\emph{dual} gravity background will have the corresponding field \nmodified from its `vacuum' $AdS_5 \\times S^5$ value. In the case of the energy \nmomentum tensor the corresponding field is just the 5-dimensional metric. One then has to \nassume that the geometry is well defined i.e. it does not have a naked \nsingularity - a singularity not hidden by an event horizon -.\nThis principle will select the allowed physical spacetime profiles of \ngauge theory energy-momentum tensor. Thus together with the Einstein equations \nthis becomes the main dynamical mechanism for the strongly coupled gauge theory.\n\nThe simplest way to formulate the precise correspondence between the expectation \nvalue of the energy-momentum tensor and bulk geometry is to use the \nFefferman-Graham system of coordinates \\cite{fg} for the latter:\n\\begin{equation}\nds^2=\\f{g_{\\mu\\nu}(x^\\rho,z) dx^\\mu dx^\\nu+dz^2}{z^2}\\ .\n\\end{equation} \nThis metric has to be a solution of 5-dimensional Einstein's equation with negative \ncosmological constant\\footnote{One can show that such solutions lift to 10-dimensional \nsolutions of ten dimensional type IIB supergravity. The effective 5-dimensional negative \ncosmological constant comes from the 5-form field in 10-dimensional supergravity.}:\n\\begin{equation}\n\\label{e.einst}\nR_{\\mu\\nu}-\\f{1}{2}g_{\\mu\\nu} R - 6\\, g_{\\mu\\nu}=0\\ .\n\\end{equation}\nThe expectation value of the energy momentum tensor may be easily recovered by \nexpanding the metric near the boundary $z=0,$ following the ``holographic \nrenormalization'' procedure \\cite{Skenderis},\n\\begin{equation}\ng_{\\mu\\nu}(x^\\rho,z)=\\eta_{\\mu\\nu}+z^4 g^{(4)}_{\\mu\\nu}(x^\\rho)+\\ldots\\ .\n\\end{equation}\nThen\n\\begin{equation}\n\\cor{T_{\\mu\\nu}(x^\\rho)} = \\f{N_c^2}{2\\pi^2} \\cdot g^{(4)}_{\\mu\\nu}(x^\\rho)\\ .\n\\end{equation}\nThis relation can be used in two ways. Firstly, given a solution of Einstein \nequations we may read off the corresponding gauge theoretical energy-momentum \ntensor. Secondly, given a traceless and conserved energy-momentum profile one \nmay integrate Einstein equations into the bulk in order to obtain the dual \ngeometry\\footnote{{This can be done order by order in $z^{2}$, which is a \nnear-boundary expansion. However potential singularities are hidden deep in the \nbulk, thus this power series needs to be resummed.}}. Then the criterion of \nnonsingularity of the geometry obtained in this way will determine the allowed \nspacetime evolution of the plasma. Let us note that this formulation is in fact \nquite far away from a conventional initial value problem.\n\nBefore we move to the case of expanding plasma, it is convenient to consider the \nsimple situation of a static uniform plasma with a constant energy momentum \ntensor. Then the Einstein's equations can be solved analytically and we find \n\\cite{US1}\nthat the exact dual geometry of such a system is\n\\begin{equation}\n\\label{e.bhfef}\nds^2=-\\f{(1-z^4\/z_0^4)^2}{(1+z^4\/z_0^4)z^2}\\ dt^2\n+(1+z^4\/z_0^4)\\f{dx^2}{z^2}+ \\f{dz^2}{z^2}\\ .\n\\end{equation} \nThis metric may look at first glance unfamiliar, but a change of coordinates\n\\begin{equation}\n\\tilde{z}=\\f{z}{\\sqrt{1+\\f{z^4}{z_0^4}}}\n\\end{equation}\ntransforms it to the standard AdS Schwarzschild static black hole\n\\begin{equation}\nds^2=-\\f{1-\\tilde{z}^4\/\\tilde{z}_0^4}{\\tilde{z}^2} dt^2\n+\\f{dx^2}{\\tilde{z}^2}+\\f{1}{1-\\tilde{z}^4\/\\tilde{z}_0^4} \\f{d\\tilde{z}^2}{\\tilde{z}^2}\n\\label{standard}\\end{equation}\nwith $\\tilde{z}_0=z_0\/\\sqrt{2}$ being the location of the horizon. Before we proceed \nfurther, let us note here one crucial thing: the fact, that the dual geometry of \na gauge theory system with constant energy density is a black hole was {\\em not} \nan assumption, but rather an outcome of a computation.\n\nThe Hawking temperature\n\\begin{equation}\nT=\\f{1}{\\pi \\tilde{z}_0} \\equiv \\f{\\sqrt{2}}{\\pi z_0}\n\\end{equation}\nis then identified with the gauge theory temperature, and the entropy with the \nBekenstein-Hawking black hole entropy \n\\begin{equation}\nS=\\f{N_c^2}{2\\pi\\tilde{z}_0^3}=\\f{\\pi^2}{2} N_c^2 T^3\n\\end{equation}\nwhich is $3\/4$ of the entropy at zero coupling. To finish our discussion of the \nstatic black hole, we note that the Fefferman-Graham coordinates cover only the \npart of spacetime lying outside the horizon. \n\n\n\\section{Boost invariant flow}\n\nLet us now apply the above procedure to a generic boost-invariant flow, in view of\nmaking contact with the hydrodynamical Bjorken flow described \nin section 2. However we do not want to make any preassumptions on the dynamics, since\nwe would like to recover the hydrodynamic behaviour as an outcome of an AdS\/CFT\ncomputation. To this end let us consider the most general gauge theory energy-momentum tensor\nwhich is boost-invariant and does not depend on transverse coordinates (see Fig.~1). Then\nconservation of energy-momentum $\\partial_\\mu T^{\\mu\\nu}=0$ and tracelessness $T^\\mu_\\mu=0$ allow\nto express all nonvanishing components of $T_{\\mu\\nu}$ in terms of a single \nfunction $\\varepsilon(\\tau)$ -- the energy density at mid rapidity:\n\\begin{equation}\n\\label{e.tgen}\nT_{\\mu\\nu}\\! = \\!\n\\left(\\begin{tabular}{cccc}\n$\\varepsilon(\\tau)$ & 0 &0 & 0 \\\\\n0 & $-\\tau^3 \\f{d}{d\\tau} \\varepsilon(\\tau)\\!-\\!\\tau^2 \\varepsilon(\\tau)$ & 0 & 0 \\\\\n0 & 0 & $\\varepsilon(\\tau)\\!+\\! \\f{1}{2}\\tau \\f{d}{d\\tau} \\varepsilon(\\tau)$ & 0 \\\\\n0 & 0 & 0 & $\\varepsilon(\\tau)\\!+\\! \\f{1}{2}\\tau \\f{d}{d\\tau} \\varepsilon(\\tau)$\n\\end{tabular}\\right)\n\\end{equation}\n\\smallskip\n\nLet us concentrate, \nfollowing \\cite{US1} on the late time asymptotics of this function i.e.\n\\begin{equation}\n\\varepsilon(\\tau) \\sim \\f{1}{\\tau^s} +\\ldots\n\\end{equation}\nfor $\\tau \\to \\infty$. Energy positivity requires that $0 \\leq s \\leq 4$. We \nwill consider sharp inequalities here\\footnote{Recently the case $s=4$ has been \nconsidered in \\cite{Kajantie:2008jz}.}. The most general metric consistent with \nthe symmetry assumptions is\n\\begin{equation}\n\\label{e.ansatz}\nds^2=\\f{-e^{a(\\tau,z)} d\\tau^2 +\\tau^2 e^{b(\\tau,z)} dy^2\n +e^{c(\\tau,z)} dx^2_\\perp}{z^2} +\\f{dz^2}{z^2} \\ .\n\\end{equation}\nIn order to find the late time form of the solution corresponding to \n$\\varepsilon(\\tau)=1\/\\tau^s$ we may solve the Einstein equations in a power series for \nthe metric coefficients\n\\begin{equation}\n\\label{e.aexp}\na(\\tau,z)=\\sum_{n=0}^N a_n(\\tau) z^{4+2n}\n\\end{equation}\nwhere $a_0(\\tau)=-\\varepsilon(\\tau)=-1\/\\tau^s$. Then from each coefficient $a_n(\\tau)$ \nwe may extract the leading large $\\tau$ behaviour and neglect the subleading \nterms. It turns out that this procedure is exactly equivalent to introducing a \nscaling variable\n\\begin{equation}\nv=\\f{z}{\\tau^{\\f{s}{4}}}\n\\end{equation}\nand assuming the metric coefficients to be just functions of $v$ e.g. \n$a(z,\\tau)=a(v)$ in the large proper time limit (namely $\\tau \\to \\infty$, $z\\to \n\\infty$ with $v$ kept fixed). In this limit Einstein's equations become just \nordinary differential equations and may be solved analytically. \nThe singularity of these geometries can then be tested by computing the scalar \ncurvature invariant\n\\begin{equation}\n{\\mathfrak R^2}=R^{\\mu\\nu\\alpha\\beta}R_{\\mu\\nu\\alpha\\beta}\\ .\n\\end{equation}\nSince our solutions are defined only in the large proper time limit $\\tau \\to \n\\infty$ with the scaling variable $v$ kept fixed, we have to evaluate ${\\mathfrak R^2}$ in \nthe same manner\\footnote{{It should be stressed however, that this condition is \nreally a condition of regularity of the expansion of the curvature invariant. It \nis safe to do as long as each term in the large proper-time expansion is \nregular. On the other hand any singularity present in this expansion might be \neither a genuine curvature singularity or a singularity of the expansion, \nsee a detailed discussion in section \\ref{secsing}.}}.\n\nThis procedure is described in detail in \\cite{US1}. The result is that\n\\begin{itemize}\n\\item for generic $s$ the resulting solution is singular\n\\item the only nonsingular solution corresponds to $s=\\f{4}{3}$ which is just \nthe hydrodynamic Bjorken expansion (see \\eqref{g3}, section 2)\n\\item the resulting metric takes the form\n\\end{itemize}\n\\begin{eqnarray}\n\\label{e.flgeom}\nds^2=\\f{1}{z^2} \\left[- \\f{\\left( 1-\\f{e_0}{3}\n \\f{z^4}{\\tau^{4\/3}}\\right)^2}{1+\\f{e_0}{3}\\f{z^4}{\\tau^{4\/3}}} d\\tau^2+\n\\left( {\\textstyle 1+\\f{e_0}{3} \\f{z^4}{\\tau^{4\/3}}}\\right) (\\tau^2\n dy^2 +dx^2_\\perp)\\right] \n+ \\f{dz^2} {z^2}\n\\end{eqnarray}\nwhere we reinstated the dimensionful parameter $e_0$ so that\n\\begin{equation}\n\\varepsilon(\\tau)=e_0\/\\tau^{\\f{4}{3}}\\ .\n\\end{equation} \nLet us note some salient features of this result. The geometry (\\ref{e.flgeom}) \nbears striking resemblance to the AdS black hole geometry (\\ref{e.bhfef}) but \nwith the position of the `effective horizon' being time dependent\n\\begin{equation}\nz_0=\\sqrt[4]{\\f{3}{e_0}} \\cdot \\tau^{\\f{1}{3}}\n\\end{equation}\nThen assuming similar relations as for the black hole case one gets the Bjorken \nscaling of the temperature and entropy.\n\\begin{eqnarray}\nT &=& \\f{\\sqrt{2}}{\\pi z_0}= \\f{2^{\\f{1}{2}} e_0^{\\f{1}{4}}}{\\pi 3^{\\f{1}{4}}} \n\\tau^{-\\f{1}{3}}\\nonumber\\\\\nS &\\propto& \\frac{\\tau}{z_0^3} =const\\ .\n\\end{eqnarray}\nWe see that the `movement' of the horizon into the bulk of AdS corresponds \nphysically to cooling of the expanding gauge theory plasma system.\n\nA significant fact that has to be kept in mind is that the geometry \n(\\ref{e.flgeom}), in contrast to (\\ref{e.bhfef}), is not an exact solution of \nEinstein's equation. It is valid only for large times. For smaller times it has \nto be modified. We will now discuss this issue in more detail as it reflects \nimportant physical properties of the gauge theory plasma.\n\n\\section{Beyond perfect fluid}\n\nThe geometry (\\ref{e.flgeom}) is only a solution of Einstein equations in the \nscaling limit. However with some effort, one can get also the first subleading \ncorrections to the metric i.e.\n\\begin{equation}\n\\label{e.flexp}\na(z,\\tau)=a_0(v)+\\f{1}{\\tau^{\\f{4}{3}}}\\ a_2(v) +\\ldots\n\\end{equation}\nThen after evaluating ${\\mathfrak R^2}$, keeping track of subleading terms we find\n\\begin{equation}\n{\\mathfrak R^2}=R_0(v)+ \\f{1}{\\tau^{\\f{4}{3}}} R_2(v) +\\ldots\n\\end{equation}\nwhere $R_0(v)$ is finite, but $R_2(v)$ develops a $4^{th}$ order pole \nsingularity. The physical meaning of this behaviour is indeed quite clear. The \ngeometry (\\ref{e.flexp}) is dual to a state in gauge theory which undergoes \nexpansion according to exact {\\em perfect fluid} hydrodynamics. Yet we know that \ngauge theory plasma has nonzero viscosity and hence the perfect fluid behaviour \n\\begin{equation}\n\\varepsilon(\\tau)=\\f{1}{\\tau^{\\f{4}{3}}}\n\\end{equation}\nis not exact but, if it would be described by viscous Bjorken expansion (viscous \nhydrodynamics), it would be modified to\n\\begin{equation}\n\\varepsilon(\\tau)=\\f{1}{\\tau^{\\f{4}{3}}}\\left( 1-\\f{2\\eta_0}{\\tau^{ \\f{2}{3}}} +\\ldots \n\\right)\n\\end{equation}\nwhere $\\eta_0$ is related to the shear viscosity through $\\eta=\\eta_0\/\\tau$ \n(which follows from the scaling $\\eta \\propto T^3$).\n\nLet us show how this arises using the AdS\/CFT methods.\nWe will not presuppose a specific form of subleading correction but will start \nfrom\n\\begin{equation}\n\\varepsilon(\\tau)=\\f{1}{\\tau^{\\f{4}{3}}}\\left( 1-\\f{2\\eta_0}{\\tau^r} +\\ldots \\right)\n\\end{equation} \nwith a generic $r$. In order to verify that plasma expansion follows viscous \nhydrodynamics we will have to first show that $r=\\f{2}{3}$. The metric \ncoefficients will now have an additional piece scaling as $\\f{1}{\\tau^{r}} \na_r(v)$. It turns out that the curvature scalar ${\\mathfrak R^2}$ is always {\\em \nnonsingular} at that order\\footnote{This was first observed for $r=2\/3$ in \n\\cite{Nak1}.}. Hence we have to go one order further i.e. find all coefficients \nappearing in the following expansion\n\\begin{equation}\na(z,\\tau)=a_0(v)+\\f{1}{\\tau^{r}} a_r(v)+\\f{1}{\\tau^{2r}} a_{2r}(v)+ \n\\f{1}{\\tau^{\\f{4}{3}}} a_2(v) +\\ldots\n\\end{equation}\nThen the curvature scalar has the form\n\\begin{equation}\n{\\mathfrak R^2}=R_0(v)+\\f{1}{\\tau^{r}} R_r(v)+\\f{1}{\\tau^{2r}} R_{2r}(v)+ \n\\f{1}{\\tau^{\\f{4}{3}}} R_2(v) +\\ldots\n\\end{equation}\nwith $R_0(v)$ and $R_r(v)$ being nonsingular, while {\\em both} $R_{2r}(v)$ and \n$R_2(v)$ turn out to have $4^{th}$ order pole singularities. In order for them \nto have a chance to cancel we have to have\n\\begin{equation}\nr=\\f{2}{3}\n\\end{equation}\nwhich is exactly the scaling of a viscosity correction to Bjorken flow. Moreover \ncancelation occurs only when the shear viscosity coefficient has the \nvalue\\footnote{We set here $e_0=1$.}\n\\begin{equation}\n\\eta_0=2^{-\\f{1}{2}} 3^{-\\f{3}{4}}\n\\end{equation}\nwhich is equivalent to $\\eta\/s=1\/4\\pi$ (for details see \\cite{RJ}). In a similar \nmanner one can go one order higher and determine a coefficient of second order \nhydrodynamics. However at that order, it turns out that there remains a leftover \nlogarithmic singularity. We will show, in section \\ref{secsing}, that the \nlogarithmic singularity arises due to a pathology of the Fefferman-Graham \nexpansion and can be avoided when one makes a different late time expansion.\n\nFinally let us comment on why it is interesting to verify the viscous \nhydrodynamic behaviour with the specific viscosity coefficient for the expanding \nplasma. Already before, there have been studies of {\\em linearized} perturbations \naround the uniform plasma which demonstrated that hydrodynamic behaviour appears \nfor small fluctuations and the value of viscosity was obtained from the Kubo \nformula \n\\cite{son}. It was interesting to verify whether hydrodynamics also applies in \nits fully \nnonlinear regime. The agreement of the resulting value of the viscosity \ncoefficient is thus a nontrivial consistency check.\n\nAnother motivation for developing an AdS\/CFT framework for studying such \ntime-dependent phenomenae is the fact that some of the most interesting and \npuzzling phenomena in heavy ion collisions are definitely very far from \nequilibruum. We will mention some examples in section \\ref{seclast}.\n\n\n\\section{Beyond boost-invariance}\n\nThe calculations presented in the previous sections were performed for systems \nwith boost invariance symmetry and full translational and rotational symmetry in \nthe transverse plane. This allowed us to perform explicit computations as the \nsymmetry assumptions effectively reduced the calculation to systems of ordinary \ndifferential equations. \nIn this manner we obtained directly the solution for gauge theory energy density \n$\\varepsilon(\\tau)$. Then, in order to find the link with hydrodynamics, we found that \nthis solution is a solution of hydrodynamic equations with specific values for \nthe transport coefficients. \n\nThis approach has both an advantage and a drawback. The advantage is that one \ndoes not presuppose any kind of dynamics and one may try to apply it in contexts \nvery far from equillibrum, where hydrodynamic description does not apply. The \ndrawback is that the appearance of hydrodynamic equations is not transparent and \nit is difficult to relax the symmetry assumptions due to the complexity of solving \nnonlinear Einstein's equations.\n\nRecently the latter drawback was addressed and it was shown in general how the \nequations of hydrodynamics arise from the gravity side \\cite{MINWALLA}. Here we \nwill briefly review this approach.\n\nLet us start from the static black hole (\\ref{standard},\\ref{e.bhfef}) but \nwritten in yet \nanother coordinate system -- the incoming Eddington-Finkelstein coordinates:\n\\begin{equation}\nds^2=-2dt dr-r^2 \\left(1- \\f{T^4}{\\pi^4 r^4} \\right)dt^2 +r^2\\eta_{ij} dx^i \ndx^j\\ .\n\\end{equation}\nHere $T$ is the temperature, $r=\\infty$ corresponds to the boundary. \n$x^\\mu=const$ are null curves going from the boundary into the black hole. The \nadvantage of this coordinate system is that it is well defined on the horizon \nand extends all the way from the boundary to the singularity at the center of \nthe black hole.\n\nThe geometry given above corresponds to a uniform plasma at rest (i.e. with the \n4-velocity $u^\\mu=(1,0,0,0)$) and given temperature $T$. We may now perform a \nboost (and perform a dilatation) to obtain the dual geometry to a moving plasma system \nwith uniform 4-velocity $u^\\mu$ and temperature $T$:\n\\begin{equation}\n\\label{e.boosted}\nds^2=-2u_\\mu dx^\\mu dr-r^2 \\left(1- \\f{T^4}{\\pi^4 r^4} \\right)u_\\mu\nu_\\nu dx^\\mu dx^\\nu +r^2(\\eta_{\\mu \\nu}+u_\\mu u_\\nu) dx^\\mu dx^\\nu\n\\end{equation}\nThe idea of ref.\\cite{MINWALLA} is to allow $u^\\mu$ and $T$ to be \n(slowly-varying) functions of the spacetime coordinates. Once this is done the \ngeometry (\\ref{e.boosted}) ceases to be an exact solution of Einstein equation \nbecause of nonvanishing gradients of the parameters $u^\\mu$ and $T$. This \nsuggests to perform an expansion of the solution in terms of gradients which has \nbeen carried out in \\cite{MINWALLA} up to second order in derivatives. The \nintegration constants arising at each order are again fixed by requiring \nregularity of the metric at the horizon. The resulting metric is expressed in \nterms of 4-velocities and temperatures and their derivatives, so when one \nextracts the energy-momentum tensor it will be given directly in terms of those \nquantities. Up to first order the expression is\n\\begin{equation}\n\\label{e.tmunumin}\nT^{\\mu\\nu}=\\f{N_c^2}{8\\pi^2} \\left\\{(\\pi T)^4 (\\eta^{\\mu\\nu}+4u^\\mu u^\\nu)- \n2(\\pi T)^3 \\sigma_{shear}^{\\mu\\nu} \\right\\} \\ .\n\\end{equation}\nThe first term is just the perfect fluid energy momentum tensor, while the \nsecond term involves the shear viscosity. \nThis result essentially demonstrates how general hydrodynamic equations arise \nfrom gravity in AdS\/CFT. Indeed, once it is shown that the general form of the gauge \ntheory energy-momentum tensor has the form (\\ref{e.tmunumin}), then conservation \nof energy momentum $\\partial_\\mu T^{\\mu\\nu}=0$ is equivalent, by definition, to conformal \nrelativistic Navier-Stokes equations. \nAs a byproduct, the above construction also gives a map from solutions of \n(viscous) hydrodynamics to gravity solutions. However this setup requires that \nthe starting point is not far off from equilibruum. For processes which do not \nadmit a hydrodynamic description (like the early stage of a heavy-ion collision) \none has to resort to different methods.\n\n\n\\section{Reduction of Singularities}\n\\label{secsing}\n\nThe leftover logarithmic singularity found in the third order of the square of \nthe Riemann tensor \\cite{Heller:2007qt} (as well as in the higher curvature \ninvariants \\cite{Benincasa:2007tp}) might be the signal of either genuine \ncurvature singularity or the singularity of the chosen expansion \nscheme\\footnote{As it was stressed before, the large-proper time expansion of \ncurvature invariants is not diffeomorphism-invariant. The encountered \nsingularities are \nphysical only if there is no coordinate transformation which removes them.}. If \nthe first is true, this means that the whole framework is inconsistent and \neither one needs to include additional degrees of freedom to cure it or the \nboost-invariant flow is unphysical\\footnote{Since it corresponds to the naked \nsingularity on the gravity side}. The results presented in \n\\cite{Benincasa:2007tp} show that no supergravity field can fix the problem, \nwhich led to conjectures, that boost-invariant flow cannot be realized within the\nsupergravity framework \\cite{Buchel:2008xr}. On the other hand, the gravity dual of \ngeneral fluid flow up to the second order in derivatives was shown to be regular \nand it was hard to imagine how possible singularities could form in the third \norder \\cite{MINWALLA, Bhattacharyya:2008xc}. The resolution of this puzzle was \npresented in \\cite{Heller:2008mb} (see also \\cite{Kinoshita:2008dq}). It turns \nout that there exists a singular coordinate transformation from Fefferman-Graham \ncoordinates to Eddington-Finkelstein ones, which yields a completely regular and \nsmooth metric from the boundary up to the standard black-brane singularity. This \nleads to regular (apart from the standard black-brane singularity) large proper-time \nexpansion of curvature invariants. The metric ansatz in Eddington-Finelstein \ncoordinates takes the form\n\n\\begin{equation}\n\\mathrm{d}s^{2} = 2 \\mathrm{d} \\tilde{\\tau} \\cdot \\mathrm{d} r - r^{2} \\tilde{A} \n\\left( \\tilde{\\tau}, r \\right) \\mathrm{d} \\tilde{\\tau}^{2} + \\left( 1 + r \\cdot \n\\tilde{\\tau} \\right)^{2} e^{\\tilde{b} \\left( \\tilde{\\tau}, r \\right)} \\mathrm{d} \ny^{2} + r^{2} e^{\\tilde{c} \\left( \\tilde{\\tau}, r\\right)} \\mathrm{d} x_{\\perp}^{2} \n\\end{equation}\n\n\\noindent and was motivated by the boosted black-brane metric (\\ref{e.boosted}) \nwith a boost and dilatation parameters $u = 1 \\cdot \\partial_{\\tilde{\\tau}}$ and \n$T \\sim \\tilde{\\tau}^{-4\/3}$. The functions $\\tilde{A} \\left( \\tilde{\\tau}, r \n\\right)$, $\\tilde{b} \\left( \\tilde{\\tau}, r \\right)$ and $\\tilde{c} \\left( \n\\tilde{\\tau}, r\\right)$ are expanded in a large-proper time expansion \nanalogously as it was in the Fefferman Graham case, i. e.\n\n\\begin{equation}\n\\tilde{A} \\left( \\tilde{\\tau}, r \\right) = \\tilde{A}_{0} \\left( r \\cdot \n\\tilde{\\tau}^{1\/3} \\right) + \\frac{1}{\\tilde{\\tau}^{2\/3}} \\tilde{A}_{1} \\left( \nr \\cdot \\tilde{\\tau}^{1\/3} \\right) + \\frac{1}{\\tilde{\\tau}^{4\/3}} \\tilde{A}_{2} \n\\left( r \\cdot \\tilde{\\tau}^{1\/3} \\right) + \\ldots\n\\end{equation}\n\n\\noindent This form of expansion can also be justified by \n\\cite{Bhattacharyya:2008ji}. The terms damped by inverse power of proper time come \nfrom the gradient expansion. The boundary metric in proper-time-rapidity \ncoordinates has non-vanishing Christoffel symbols $\\Gamma \\sim \\ttau^{-1}$, thus \nthe four velocity gradient $\\nabla u$ (which is constant in these \ncoordinates) gives a factor of $\\ttau^{-1}$. On the other hand the expansion \nparameter multiplying each term in gradient expansion is the inverse power of \nthe temperature $T$(see \\cite{MINWALLA}). Because $T \\sim \n\\ttau^{-1\/3}$, the overall damping is indeed $\\ttau^{-2\/3}$ - a fact derived in \n\\cite{Heller:2007qt} from the non-singularity argument.\n\n The \nnon-perturbative\\footnote{In the sense of large-proper time expansion} piece in \nthe metric at $\\mathrm{d} y^{2}$ introduced in \\cite{Kinoshita:2008dq} is \nresponsible for a correct limit energy density $\\rightarrow 0$. It also becomes \nimportant if one wants to solve the problem of early-time dynamics \n\\cite{Kovchegov:2007pq} using Eddington-Finkelstein coordinates.\n\nThe integration \nconstants\\footnote{Not all of them -- there is a remaining gauge freedom \n(coordinate transformation), which leaves the metric ansatz unchanged: $r \n\\rightarrow r + f\\left( \\tilde{\\tau} \\right)$, where $f\\left( \\tilde{\\tau} \n\\right)$ is an arbitrary function} are fixed by requiring the regularity of the \nmetric functions $\\tilde{A}_{i} \\left( \\tilde{v} \\right)$, $\\tilde{b}_{i} \\left( \n\\tilde{v} \\right)$ and $\\tilde{c}_{i} \\left( \\tilde{v} \\right)$ at each order \n$i$. This is justified since the Eddington-Finkelstein are valid for \n$\\tilde{\\tau} > 0$ and $0 < \\tilde{v} = r \\cdot \\tilde{\\tau} < \\infty$. The \nsingular coordinate transformation which takes the metric from \nEddington-Finkelstein coordinates to Fefferman-Graham ones is given order by \norder in the gradient expansion by\n\n\\begin{eqnarray}\n\\tilde{\\tau} \\left( \\tau, z \\right) &=& \\tau \\cdot \\Big\\{ T_{0} \\left(z \\cdot \n\\tau^{-1\/3} \\right) + \\frac{1}{\\tau^{2\/3}} T_{1} \\left(z \\cdot \\tau^{-1\/3} \n\\right) + \\ldots \\Big\\} \\ \\mathrm{,}\\\\\nr \\left( \\tau, z \\right) &=& \\frac{1}{z} \\cdot \\Big\\{ R_{0} \\left(z \\cdot \n\\tau^{-1\/3} \\right) + \\frac{1}{\\tau^{2\/3}} R_{1} \\left(z \\cdot \\tau^{-1\/3} \n\\right) + \\ldots \\Big\\}\\ \\mathrm{.}\n\\end{eqnarray}\nThe leading-order solutions (corresponding to the perfect fluid on the gauge \ntheory side) \n are related by\n\\begin{eqnarray}\n\\label{coordtrafo}\n\\ttau &\\rightarrow& \\tau \\left\\{ 1 - \\frac{1}{\\tau^{2\/3}} \\left[ \\frac{3^{1\/4}\n \\pi}{4 \\sqrt{2}} + \\frac{3^{1\/4}}{2 \\sqrt{2}}\n \\tan^{-1}{\\left(\\frac{3^{1\/4}}{\\sqrt{2}} r \\cdot \\tau^{1\/3}\\right)}\\ + \n\\right.\\right. \\nonumber\\\\ &&\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ + \\left.\\left.\\frac{3^{1\/4}}{4 \n\\sqrt{2}} \\log {\\left(\\frac{r \\cdot \\tau^{1\/3} -\n \\frac{\\sqrt{2}}{3^{1\/4}}}{ r \\cdot \\tau^{1\/3} +\n \\frac{\\sqrt{2}}{3^{1\/4}}}\\right)} \\right] \\right\\}\\ \\mathrm{,}\n\\nonumber \\\\ \nr &\\rightarrow& \\frac{1}{z} \\cdot \\sqrt{1 + \\frac{z^{4}}{3 \\cdot \\tau^{4\/3}}}\n\\mathrm{.} \n\\end{eqnarray}\nThe transformation is singular at $z = 3^{1\/4} \\tau^{1\/3}$, which is precisely \nthe locus of the logarithmic singularity encountered in \\cite{Heller:2007qt}. \nFormulas for higher order transformation coefficients are too long to be \npresented here and can be found in \\cite{Math}. \nThe energy-momentum tensor extracted from the solution in Eddington-Finkelstein \ncoordinates reproduces the energy momentum tensor obtained in \n\\cite{Heller:2007qt}.\n\n\n\n\\section{Beyond hydrodynamics}\n\n\\label{seclast}\n\nGauge-gravity duality has already proven to be an invaluable tool in describing \nproperties of static or near-equilibrium (hydrodynamics) strongly coupled gauge \ntheory systems. Noticeable achievements in that direction are the viscosity \nevaluation bound \n\\cite{son} and the consistent formulation of the second order conformal \nhydrodynamics \\cite{Baier:2007ix, MINWALLA}. These successes came \nfrom the holographic understanding of hydrodynamics. On the other hand there is \nmuch more interesting and nontrivial dynamics than hydro. Far from equilibrium \nbehavior of gauge theories is a fascinating and pretty much open problem of \nexperimental importance, like the early universe or initial stages of heavy ion \ncollisions\\footnote{In the late stages of heavy ion collision, strongly coupled \nquark-gluon plasma forms and holographic technics at strong gauge coupling are \nbetter justified then just after the collision (running of the coupling). \nNevertheless applying AdS\/CFT correspondence to describe far from equilibrium \nprocesses in gauge theories is an interesting problem even from a purely \ntheoretical point of view.}. The AdS\/CFT correspondence is surely capable to \nshed new light on these problems, or even be understood as a formulation of far \nfrom equilibrium gauge theory.\n\nIn the context of heavy-ion collisions the most important and probably the most \ndifficult questions concern the issues of early time dynamics \n\\cite{Kovchegov:2007pq} and the transition to an isotropic \\cite{Janik:2008tc} \nand \nthermalized medium. One of the puzzles here is the short time in which nuclear \nmatter approach local equilibrium. Experimental data fitting well with \nhydrodynamical simulations with small viscosity justified applications of the \nAdS\/CFT correspondence at strong coupling for the late stages of heavy-ion \ncollisions. It is not clear to what extent early time dynamics is driven by \nnon-perturbative effects and whether the lessons learned from AdS\/CFT might be \ndirectly applied to the nuclear matter in the early stages of the evolution. \nApproaching equilibrium is also of an interest from the General Relativity point \nof view. Isotropic \nand thermalized matter on the gauge theory side corresponds to a black hole in \nAdS, whereas thermalization and approach to local equilibrium should be governed \nby the dynamics of gravitational collapse.\n\nPerhaps some of these questions might be answered by studying collisions of \nshock-waves in AdS. The geometry corresponding to a projectile in 3+1 dimensions \nwas constructed in \\cite{US1} using holographic renormalization. The \nmetric\n\\begin{equation}\n\\mathrm{d} s^{2} = \\frac1{z^{2}}\\left\\{- 2 \\mathrm{d} x^{+} \\mathrm{d} x^{-} + \nf\\left( x^{-} \n\\right) \\cdot z^{4} \\left( \\mathrm{d} x^{-} \\right)^{2} + \\mathrm{d} \nx_{\\perp}^{2} + \\mathrm{d} z^{2}\\right\\}\n\\end{equation}\nwith an arbitrary function $f \\left( x^{-} \\right)$ corresponds to the situation \nwhen\n\n\\begin{itemize}\n\\item the dynamics is one-dimensional (i.e. no dependence on transverse coordinates),\n\\item the energy-momentum tensor depends only on a single light-cone variable (here \nchosen to be $x^{-} = x^{0} - x^{1}$).\n\\end{itemize}\n\nTraceless and conserved energy momentum tensor satisfying the above assumptions \ntakes a particularly simple form -- its only non-zero component is $T^{- -} = f \n\\left( x^{-} \\right)$. Choosing $f \\left( x^{-} \\right) = M \\delta \\left( x^{-} \n\\right) $ leads to a shock-wave -- infinitely thin plane of matter moving at the \nspeed of light, which is a toy-model for highly boosted nucleus. The idea is to \ncollide two such projectiles and single out the physical behavior of the plasma \nfrom the regularity of the dual geometry. This is a difficult problem, because \nof the broken boost-invariance, which leads to solving Einstein equations in 3 \nvariables ($x^{+}$, $x^{-}$ and $z$ or equivalently $\\tau$, $y$ and $z$). The \ngeometry before the collision ($x^{+} + x^{-} < 0$) is known -- it is simply \nthe superposition of two incoming shock-waves\n\n\\begin{equation}\n\\mathrm{d} s^{2} = \\frac1{z^{2}}\\left\\{- 2 \\mathrm{d} x^{+} \\mathrm{d} x^{-}\\! + \n\\! M \\delta \n\\left( x^{-} \\right) z^{4} \\left( \\mathrm{d} x^{-} \\right)^{2} \\!+\\! M \n \\delta \\left( x^{+} \\right) z^{4} \\left( \\mathrm{d} x^{+} \n\\right)^{2} \\! +\\! \\mathrm{d} x_{\\perp}^{2} \\!+ \\!\\mathrm{d} z^{2}\\right\\} \n\\mathrm{.}\n\\end{equation}\n\nShock-waves collide at $x^{+} = x^{-} = 0$ and from now on the dynamics of the \nsystem must be deduced from Einstein equations. The first attempt to address \nthis issue in \\cite{Kajantie:2008rx}\nfocused on the simpler setup than presented so far -- a shock-waves collision in \n1+1 dimensions. The energy-momentum tensor for such a system before the \ncollision is given by $T_{+ +} = f \\left( x^{-} \\right) $ and $T_{- -} = g \n\\left( x^{+} \\right) $ with vanishing off-diagonal components. This is at the \nsame time the most general form of the energy-momentum tensor for a 1 + 1 \ndimensional CFT. A nice feature is that the dual geometry for the {\\em whole} \ncollision process can be constructed here exactly. However in this low \ndimensional context, the projectiles pass each other unaffected (or propelled \nback-to-back \\cite{Kajantie:2008rx}), so the physics \nof plasma production and thermalization is absent here. On the \nother hand the problem of genuine interest -- collision of shock-waves in 3+1 \ndimensions -- requires some approximation scheme in which Einstein equations \nbecome tractable. Up to now, two proposals have been made. The first one \n\\cite{Grumiller:2008va} treats proper time as a small parameter but suffers from \na negative energy density in some regions due to the conditions imposed at the \nlight-cone. The second one \\cite{Albacete:2008vs} solves Einstein equations \nperturbatively in $M$ leading to the prediction that shock-waves stop almost \nimmediately after the collision (reminding of the {\\it full stopping} condition \nof the Landau flow, see section \\ref{sechydro}.). This problems surely deserve further \nstudies. \n\nThere are also other studies of dynamical processes in an evolving plasma system \nwhich go beyond hydrodynamics. One example is the problem of thermalization of \nsmall perturbations around the expanding plasma (some first investigations has \nbeen performed in \\cite{US1}). Another use of the evolving geometries \nis to study other physical processes in the presence of the evolving plasma \nsystem like e.g. the physics of mesons and flavours studied through embedding D7 \nbranes in the time-dependent geometries \\cite{flavors}. Finally one may study \nisotropisation of anisotropic plasma. The first investigations have been \nperformed in \\cite{Janik:2008tc} (see the talk by P. Witaszczyk in the same \nproceedings).\n\n\\section{Summary}\nThe Gauge\/Gravity approach to the formation and evolution of a quark-gluon \nplasma in heavy-ion collisions described above has the interest of casting an \nexploratory bridge between the rigorous results of string theory and some \npending questions raised by the experiments on quark-gluon plasma. These \nquestions cannot yet be raised in the framework of strongly coupled QCD, for \nwhich we do not possess the adequate tools, but they can be addressed for the \nfirst time in a quantitative and rigorous way in the supersymmetric case of the \nAdS\/CFT correspondence. It is thus a novel and valuable approach and can serve \nas a model for further studies.\n\nLet us summarize some aspects of this approach, being aware (and with apologies \nfor those not quoted or mentioned) that this subject is in constant development which\nwill force us to mention only a few of them.\n\nStarting with the experimental evidence that hydrodynamics is relevant in the \nformation and evolution of a quark-gluon plasma in heavy-ion collisions and in \nparticular of the ``Bjorken flow'' description, we present the AdS\/CFT setup \nallowing to describe the dynamics of the plasma (in the AdS\/CFT case). We show \nthat it is possible to derive the geometry dual to the asymptotic evolution of \nthe plasma in terms of an expansion in a scaling variable. the nonsingularity \nrequirement on the Gravity side gives a set of ``selection rules'' on the Gauge \nside: the perfect fluid at first order, the $\\eta\/s=1\/4\\pi$ property and other \ntransport coefficients at higher orders...\n\nBeyond the boost-invariant Bjorken flow, there exists an intriguing but rigorous \none-to-one correspondence between the complete (and even completed using \nAdS\/CFT!) hydrodynamic equations and the solutions of the Einstein equations in \nthe bulk of the 5-dimensional space. Some apparent obstacles, such as the appearance of \nlogarithmic singularities in the asymptotic expansion of the 5-dimensional metric, have been \nshown to be a mere artefact of the choice of the expansion parameter and have \nbeen cured.\n\nBeyond the hydrodynamical description of the transient plasma, one goal is now \nto exoplore the dynamical aspects far from equilibrium. We describe some very \nrecent attempts, which even though not conclusive yet, show the interest of the \nextension of Gauge\/Gravity correspondence to attack some down-to-earth \nproblems, such as the short thermalization time, probably observed by the \nheavy-ion phenomenology and more generally the effect of the initial conditions \non the whole process.\n\nIt is quite interesting to see that some complex aspects of Gauge field theory \ndynamics can find unexpected answers from Gravity. It would be intringuing that \nsome nontrivial aspects of Gravity (such as the dynamics of moving black holes) \nalso could gain some new insight from the correspondence with some aspects of \nheavy-ion collisions.\n\n\\section*{Acknowledgements}\nThis investigation was partly supported by the \nMEiN research grants 1 P03B 04529 (2005-2008) (RP), 1 P03B 04029 (2005-2008) (RAJ and MPH) and N202 247135 (2008-2010) (MPH), by 6 Program of European\nUnion ``Marie Curie Transfer of Knowledge'' Project: Correlations in Complex\nSystems ``COCOS'' MTKD-CT-2004-517186 and the RTN network ENRAGE MRTN-CT-2004-005616. MPH also acknowledges the support from the British Council within the Polish-British Young Scientists Programme and hospitality of Durham University while these notes were being completed.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction and main results.}\\label{intro}\n\nWe consider a {\\it first-come-first-served}\nmulti-server system with $s$ identical servers.\nLet $\\tau$ be a typical interarrival time and\n$\\sigma$ a typical service time.\nIndependent identically distributed sequences\nof interarrival times $\\{\\tau_n\\}$ with mean\n$a={\\mathbb E}\\tau$ and service times $\\{\\sigma_n\\}$\nwith mean $b={\\mathbb E}\\sigma$ are assumed to be mutually independent.\nWe also assume throughout the paper that the distribution\nof $\\sigma$ has unbounded support, i.e.\n$B(x):={\\mathbb P}\\{\\sigma\\le x\\}<1$ for all $x$, and that the\nsystem is {\\it stable}, i.e. $\\rho := b\/a < s$.\n\nThere are two equivalent ways to describe\nthe dynamics of a multi-server system.\nFirst, we may assume that customers form a single\nqueue in front of all servers, and that the first customer\nin the queue moves immediately to a server which\nbecomes idle. Second, we may assume that customers\nform $s$ individual queues (lines) -- one queue for\neach server, service times of customers become\nknown upon their arrival, and each arriving\ncustomer is directed to the line with a minimal\ntotal workload (we also assume that queues are\nnumbered, and if there are more than one minimal\nworkloads, then a customer chooses the one with the\nminimal number). In the rest of the paper,\nwe mostly follow the second description of the model.\n\nFor $n=1$, $2$, \\ldots, let $\\vv V_n=(V_{n1},\\ldots,V_{ns})$\nbe the vector of residual workloads in lines $1$, \\ldots, $s$\nwhich are observed by the $n$-th customer\nupon its arrival into the system.\nThe value of $D_n:=\\min\\{V_{nj},j\\le s\\}$ is the waiting time,\nor the delay which customer $n$ experiences.\nThe $n$-th customer joins the $i_n$-th line. Then\n$$\ni_n=\\min\\{i:V_{ni}=D_n\\}\n$$\nand\n\\begin{eqnarray*}\nV_{n+1,i} &=& \\left\\{\n\\begin{array}{ll}\n(V_{ni}+\\sigma_n-\\tau_{n+1})^+ &\\mbox{ if } i=i_n,\\\\\n(V_{ni}-\\tau_{n+1})^+ &\\mbox{ if } i\\not=i_n.\n\\end{array}\n\\right.\n\\end{eqnarray*}\n\nLet $R(\\vv w)=(R_1(\\vv w),\\ldots,R_s(\\vv w))$\nbe the operator on ${\\mathbb R}^s$ which orders\nthe coordinates of $\\vv w\\in{\\mathbb R}^s$\nin the non-descending order, i.e.,\n$R_1(\\vv w)\\le\\cdots\\le R_s(\\vv w)$.\nFor $n=1$, $2$, \\ldots, put $\\vv W_n=R\\vv V_n$.\nThen $D_n=W_{n1}$ and the vectors $\\{\\vv W_n\\}$\nsatisfy the\nKiefer--Wolfowitz \\cite{KW55} recursion:\n\\begin{equation}\\label{KiW}\n\\vv W_{n+1} = R((W_{n1}+\\sigma_n-\\tau_{n+1})^+,\n(W_{n2}-\\tau_{n+1})^+,\\ldots, (W_{ns}-\\tau_{n+1})^+).\n\\end{equation}\n\nIn a stable system,\nthere exists a unique\nstationary distribution for the Kiefer-Wolfowitz vectors\n$\\vv W_n$, and the distribution of $\\vv W_n$\nconverges to the stationary distribution\nin the total variation norm, as $n\\to\\infty$.\nIn particular, the same holds for the $D_n$:\nthere exists a unique\ndistribution of the stationary waiting time (delay)\n$D$, and the distribution of $D_n$ converges to that\nof $D$ in the total variation norm.\n\nIn a single server queue ($s=1$),\nthe waiting times $D_n$\nsatisfy the Lindley recursion \\cite{Lindley}:\n$$\nD_{n+1}=(D_n+\\sigma_n-\\tau_{n+1})^+.\n$$\nRecall that, given $D_1=0$,\n$D_{n+1}$ coincides in distribution with\n$\\max(S_k,k\\le n)$ where $S_0=0$ and\n$S_n=\\sum_{k=1}^n(\\sigma_k-\\tau_{k+1})$, for $n\\ge 1$.\nIt is well known\n(see, for example, \\cite{P,Ver,APQ}) that\nthe tail of stationary waiting\ntime $D$ is related to the service time distribution tail\n$\\overline B(x)={\\mathbb P}\\{\\sigma>x\\}$ via the equivalence\n\\begin{eqnarray}\\label{W.single}\n{\\mathbb P}\\{D>x\\} &\\sim& \\frac{\\rho}{1-\\rho}\n\\overline B_r(x) \\quad\\mbox{ as }x\\to\\infty,\n\\end{eqnarray}\nprovided the {\\it subexponentiality} of the\n{\\it residual service time distribution} $B_r$ defined by its tail\n\\begin{eqnarray*}\n\\overline B_r (x) &:=& \\frac{1}{b}\n\\int_x^\\infty \\overline B(y)\\,dy,\\ \\ x>0\n\\end{eqnarray*}\nis guaranteed.\nRecall that a distribution $G$ on ${\\mathbb R}^+$\nis {\\it subexponential},\n $G\\in {\\cal S}$,\nif $\\overline{G*G}(x)\\sim2\\overline G(x)$ as $x\\to\\infty$.\n\nIt is also well-known that, in a single server queue,\nfor any $\\gamma>0$,\n$D$ has a finite $\\gamma$th moment, ${\\mathbb E} D^{\\gamma}<\\infty$\nif and only if\n${\\mathbb E}\\sigma^{\\gamma+1}<\\infty$, see \\cite{KW56}.\nEquivalently, ${\\mathbb E}D^\\gamma<\\infty$ if and only if\n\\begin{eqnarray*}\n{\\mathbb E} \\sigma_{r,1}^\\gamma &<& \\infty\n\\end{eqnarray*}\nwhere random variable $\\sigma_{r,1}$ has distribution $B_r$.\n\nLess is known about the stationary delay $D$ in\nthe multi-server queue. It is well understood\nthat the heaviness of the stationary waiting time tail\ndistribution\ndepends substantially on the load $\\rho$ on the system\n(see, for example, the conjecture on tail equivalence\nby Whitt in \\cite{W};\nexistence results for moments in \\cite{S,SS,SV,SV2011};\nasymptotic results for fluid queues fed by heavy-tailed\non-off flows in \\cite{BMZ,BZ}).\nMore precisely, the tail distribution depends on $\\rho$ via the value\nof its integer part $k = [\\rho]\\in\\{0,1,\\ldots,s-1\\}$\n\nFor a $GI\/GI\/s$ system, a heuristic idea on a probable way for the large deviations\nto occur may be described as follows.\nTake $N=x\\frac{k}{b-ka}$, for a very large $x$.\nLet all service times\n$\\sigma_{n-N-s+k}$, \\ldots, $\\sigma_{n-N-1}$\nbe big enough, say $\\sigma_{n-N-i}>x+Na$,\n$i=1$, \\ldots, $s-k$.\nThen the other $k$ servers form an unstable\n$GI\/GI\/k$ queue system, because the cumulative\ndrift of the corresponding workloads\napproximately equals $b-ka>0$.\nIn time $N$ all workloads of these queues\nwill exceed level $x$ (again approximately).\nIn this way, at time $N$, all $s$ workloads\nbecome greater than $x$ with probability\nwhich is asymptotically not less than\n$\\overline B^{s-k}(x+Na)\\approx\n\\overline B^{s-k}\\bigl(x\\frac{b}{b-ka}\\bigr)\n=\\overline B^{s-k}\\bigl(x\\frac{\\rho}{\\rho-k}\\bigr)$.\nWe use these heuristic arguments below in Section\n\\ref{lower.bound} to derive a lower bound.\nWe follow more precise calculations\nto obtain a better lower bound\nof order $\\overline B_r^{s-k}\\bigl(x\\frac{\\rho}{\\rho-k}\\bigr)$.\n\nWe recall now a few basic properties of heavy-tailed distributions and relations\nbetween them. A distribution function $F$ is\n\\begin{itemize}\n\\item\n{\\it long-tailed}, $F\\in {\\cal L}$, if\n$\\overline{F}(x+1) \\sim \\overline{F}(x)$, as $x\\to\\infty$;\n\\item\n{\\it dominated varying}, $F\\in {\\cal D}$,\nif $\\overline{F}(2x)\\ge c\\overline{F}(x)$, for some $c>0$ and for all\n$x$;\n\\item\n{\\it intermediate regularly varying}, $F\\in {\\cal IRV}$,\nif\n$$\n\\lim_{\\varepsilon \\downarrow 0} \\liminf_{x\\to\\infty}\n\\overline{F}(x(1+\\varepsilon ))\/\\overline{F}(x) =1;\n$$\n\\item\n{\\it regularly varying},\n$F\\in {\\cal RV}$, if $\\overline{F}(x) = l(x)x^{-\\alpha}$\nfor $x>0$ where $\\alpha \\ge 0$ is the {\\it index} of regular variation\nand $l(x)$ is a {\\it slowly varying at infinity}\nfunction, i.e. $l(cx)\\sim l(x)$ as $x\\to\\infty$.\n\\end{itemize}\nThe following relations are known:\n\\begin{equation}\\label{relations}\n{\\cal RV} \\subset {\\cal IRV} \\subset {\\cal L} \\cap {\\cal D} \\subset {\\cal S},\n\\end{equation}\nsee e.g. \\cite{FKZ}, pp. 33 and 54.\n\nIn \\cite{FK}, we treated the case $s=2$ in detail\nand found the {\\it exact asymptotics} for ${\\mathbb P}\\{D>x\\}$.\nWe also described the {\\it most probable way\nfor the occurrence of the large deviations}.\nThat means that, for the stationary waiting time to be large,\ntwo large service times have to be large if $\\rho <1$ and\n$B_r$ is a subexponential distribution\n(see\n\\cite[Theorem 1]{FK}) and one service time has to be\nlarge if $1<\\rho <2$ and if $B$ is\n{\\it long-tailed} and $B_r$ is\n{\\it intermediate\nregularly varying}\n(see \\cite[Theorem 2]{FK}).\nWe also obtained a number of simple bounds. First,\nTheorem 1 in \\cite{FK} yields the following\n\n\\begin{theorem}\\label{co.2.max}\nLet $s=2$, $\\rho<1$, and let the residual time\ndistribution $B_r$ be subexponential.\nThen the tail of the stationary waiting time\nsatisfies the asymptotic relation, as $x\\to\\infty$,\n\\begin{eqnarray*}\n{\\mathbb P}\\{D>x\\} &\\sim&\n\\frac{\\rho^2}{2-\\rho}\\Bigl[(\\overline B_r (x))^2\n+\\int_0^\\infty \\overline B_r(x+ya)\\overline B(x+y(a-b))dy\\Bigr].\n\\end{eqnarray*}\nAs a corollary, one can obtain the following\nbounds for the stationary waiting time,\nas $x\\to\\infty$:\n\\begin{eqnarray*}\n\\Bigl(\\frac{\\rho^2(2+\\rho)}{2(2-\\rho)}+o(1)\\Bigr)\n\\overline B_r^2 (x)\n\\le {\\mathbb P}\\{D>x\\}\n\\le \\Bigl(\\frac{\\rho^2}{2(1-\\rho)}+o(1)\\Bigr)\n\\overline B_r^2 (x).\n\\end{eqnarray*}\nAnother corollary is: if, in addition, the distribution\n$B$ is regularly varying with index $\\gamma>1$,\nthen, as $x\\to\\infty$:\n\\begin{eqnarray*}\n{\\mathbb P}\\{D>x\\} &\\sim& c(\\overline B_r(x))^2,\n\\end{eqnarray*}\nwhere\n\\begin{eqnarray*}\nc &=& \\frac{\\rho^2}{2-\\rho}\\Bigl[1+\\frac{\\rho}{\\gamma-1}\n\\int_0^\\infty\\frac{dz}{(1+z)^{\\gamma-1}(1+z(1-\\rho))^\\gamma}\\Bigr].\n\\end{eqnarray*}\n\\end{theorem}\n\nFor the case $\\rho>1$, we also proved in \\cite{FK}\n\n\\begin{theorem}\\label{th.2.min.lower}\nLet $s=2$, $1<\\rho<2$, and let both $B$ and $B_r$ be\nsubexponential distributions.\nThen the tail of the stationary waiting time\nsatisfies the following inequalities:\n\\begin{eqnarray*}\n\\limsup_{x\\to\\infty}\\frac{{\\mathbb P}\\{D>x\\}}\n{\\overline{B}_r(2x)}\n&\\le& \\frac{\\rho}{2-\\rho},\n\\end{eqnarray*}\nand, for any fixed $\\delta>0$,\n\\begin{eqnarray*}\n\\liminf_{x\\to\\infty}\\frac{{\\mathbb P}\\{D>x\\}}\n{\\overline B_r\\bigl(\\frac{\\rho+\\delta}{\\rho-1}x\\bigr)}\n&\\ge& \\frac{\\rho}{2-\\rho}.\n\\end{eqnarray*}\nIf, in particular, $B$ is subexponential and $B_r$\nis\nintermediate\nregularly varying,\nthen\n\\begin{eqnarray*}\n{\\mathbb P}\\{D>x\\} &\\sim& \\frac{\\rho}{2-\\rho}\n\\overline B_r\\Bigl(\\frac{\\rho}{\\rho-1}x\\Bigr)\n\\quad\\mbox{ as }x\\to\\infty.\n\\end{eqnarray*}\n\\end{theorem}\n\nFor an arbitrary $s\\ge 2$ number of servers,\nthe best result on the existence of moments\nwas obtained in \\cite[Theorem 4.1]{SV}\n(here ${\\cal L}_1^\\gamma$ is a specific class of distributions\nintroduced in \\cite{SV}):\n\n\\begin{theorem}\\label{mom.SV}\nLet $k<\\rho0$, ${\\mathbb E}D^\\gamma$\nis finite if and only if\n\\begin{equation}\\label{min.fin}\n{\\mathbb E}(\\min(\\sigma_{r,1},\\ldots,\\sigma_{r,s-k}))^\\gamma\n<\\infty,\n\\end{equation}\n\\end{theorem}\nsee Section \\ref{proof} for the proof.\nActually, this result (which is sharper than Theorem \\ref{mom.SV})\nmay be deduced from the results of \\cite{SV},\nbut was not stated there. The corresponding proof\nin \\cite{SV} involves a comparison with the so-called\nsemi-cyclic service discipline.\nTo the best of our knowledge, the latter approach does not allow\none to obtain upper bounds for the tail distribution\nof $D$.\n\nThe main aim of the present paper is to introduce\na novel approach for constructing upper bounds for\nthe stationary waiting time in multi-server queues\n(see Section \\ref{sec.upper.bound} below).\nThis allows us to derive estimates for the tail probabilities of the distribution of\nthe stationary waiting time if the common distribution of service times is of\nsupexponential type, and, further, to establish the {\\it principle of big jumps}\nin a particular case of intermediate varying distributions.\nAlso, based on the new approach, we will obtain\na direct proof of Theorem\n\\ref{exist.mom} (see Section \\ref{proof}).\n\nThe most explicit bounds are obtained for the case\n$\\rho<1$.\n\n\\begin{theorem}\\label{co.s.max}\nLet $\\rho=b\/a<1$ and let the residual time\ndistribution $B_r$ be subexponential.\nThen the tail distribution of the stationary\nwaiting time admits the following bounds:\n\\begin{eqnarray*}\n\\frac{\\rho^s}{s!} \\le\n\\liminf_{x\\to\\infty}\n\\frac{{\\mathbb P}\\{D>x\\}}{\\overline B_r^s(x)}\n&\\le& \\limsup_{x\\to\\infty}\n\\frac{{\\mathbb P}\\{D>x\\}}{\\overline B_r^s (x)}\n\\le \\Bigl(\\frac{\\rho}{1-\\rho}\\Bigr)^s.\n\\end{eqnarray*}\n\\end{theorem}\n\nWe present here the lower and upper bounds only.\nAs it was described in \\cite{FK},\nthe only case where the tail asymptotics are available\nwith an explicit constant multiplier is the case\nof regularly varying service time distribution.\nThe corresponding calculations\nare rather involved\nand deal with the law of large numbers\nand a summation over a specific $s$-dimensional domain\nwith planar boundaries. These calculations have been\ncarried out\nin \\cite{FK} in the case of\n$s=2$ servers.\n\nThe proof of Theorem \\ref{co.s.max} (see Section\n\\ref{k=0}) is based on a simple argument which\ncannot be applied if $\\rho>1$. For an arbitrary $\\rho$,\nwe have the following result.\n\n\\begin{theorem}\\label{th:subexp.bounds}\nLet $k\\in\\{0,\\ldots,s-1\\}$ and $\\delta>0$.\nIf $\\rho>k$, then\n\\begin{eqnarray*}\n{\\mathbb P}\\{D>x\\}\n&\\ge& \\frac{\\rho^{s-k}+o(1)}{(s-k)!}\n\\overline{B_r}^{s-k}\\Bigl(\\frac{\\rho+\\delta}{\\rho-k}x\\Bigr)\\ \\mbox{ as }x\\to\\infty.\n\\end{eqnarray*}\nIf $\\rhox\\}\n&\\le& \\Bigl(\\begin{array}{c}s\\\\k\\end{array}\\Bigr)\n\\Bigl(\\frac{(k+1)\\rho}{k+1-\\rho}+o(1)\\Bigr)^{s-k}\n\\overline B_r^{s-k}(x(1-\\delta))\\ \\mbox{ as }x\\to\\infty.\n\\end{eqnarray*}\n\\end{theorem}\n\nThe lower bound follows from Theorem \\ref{th.lower}\nin Section \\ref{lower.bound}.\nThe proof of the upper bound\nmay be found in Section \\ref{sec.upper.bound}.\nIt is based on\nresults from\nSection \\ref{majorant},\nwhere we present a novel construction of a\nconsistent majorant for $D_n$.\n\n\nNote that the lower and the upper bounds in\nTheorem \\ref{th:subexp.bounds} are not necessarily\nof the same order. In particular, if distribution\n$B$ is of Weibull type then the ratio of the upper\nand the lower bounds tends to infinity, as $x$ increases.\nIn this case we do not have any ideas about how\ncorrect\/exact\/sharp bounds would look like.\nBut if, in particular, the residual\n service time distribution belongs to the class ${\\cal D} \\cap\n{\\cal L}$, \nthen\nthese bounds differ by a multiplicative constant only.\n\nNote that in Theorem \\ref{th:subexp.bounds}\nwe require conditions on the residual distribution $B_r$\nand not on the distribion $B$ itself. This is in line\nwith the key results on subexponentiality like (\\ref{W.single}).\n\n\\begin{corollary}\\label{regularly.case}\nLet the residual service time distribution\n$B_r$ be long-tailed and dominated varying.\nLet $k<\\rhox\\} \\le c_2\\overline B_r^{s-k}(x).\n\\end{equation}\n\\end{corollary}\n\nThe result follows directly from Theorem \\ref{th:subexp.bounds},\nthe last inclusion in (\\ref{relations})\nand the definition of the dominated variation.\n\n\nIn the particular case of distributions of intermediate variation,\nwe will use \\cite[Theorem 7]{BaF} to complement\n Corollary \\ref{regularly.case} by establishing\nthe ``principle of $s-k$ big jumps'':\nthe main cause of the value of $D$ to be big is to have\n$s-k$ big service times, see Section \\ref{sec.upper.bound} for\nthe precise statement.\n\n\n\n\n\\section{Comparison of systems with different\ninter-arrival times.}\\label{comparison}\n\nHere we present results which, in particular,\nallow us to obtain lower and upper bounds for the stationary\ndelay in a\ngeneral $GI\/GI\/s$ system in terms of a simpler $D\/GI\/s$\nsystem with deterministic\ninterarrival times. We use the following partial ordering:\nfor two vectors $\\vv x = (x_1,\\ldots ,x_s)$ and\n$\\vv y = (y_1,\\ldots ,y_s)$, we write\n$\\vv x \\le \\vv y$ if $x_j\\le y_j$\nfor all $j=1$, \\ldots, $s$.\n\nConsider two $GI\/GI\/s$ systems,\nsay $\\widetilde{\\vv V}$ and $\\widehat{\\vv V}$,\nwith service times $\\sigma_n$\nand with interarrival times\n$\\widetilde\\tau_n$ and $\\widehat\\tau_n$ respectively.\nLet $\\widetilde D_n$ and $\\widehat D_n$\nbe the corresponding waiting times in these systems. Let\n$\\xi_n=\\widehat\\tau_{n+1}-\\widetilde\\tau_{n+1}$.\nWe obtain an upper bound for delay $\\widetilde D_n$\nin terms of delay $\\widehat D_n$\nand the sequence $\\xi_n$.\n\n\\begin{lemma}\\label{lem:upper.bound}\nFor all $n\\ge1$,\n$\\widetilde D_n \\le \\widehat D_n+M_{n-1}$,\nwhere $M_0=0$ and $M_n=(M_{n-1}+\\xi_n)^+$.\n\\end{lemma}\n\n\\begin{proof}\nPut $\\vv e_1=(1,0,\\ldots,0)$ and $\\vv 1=(1,\\ldots,1)$.\nIt suffices to prove the inequality\n\\begin{eqnarray}\\label{upper.via.M}\n\\widetilde{\\vv W}_n &\\le& \\widehat{\\vv W}_n+\\vv 1M_{n-1}\n\\quad\\mbox{ a.s.}\n\\end{eqnarray}\nWe proceed by induction. For $n=1$\nwe have $\\vv 0\\le\\vv 0+\\vv 1M_0$. Assume inequality\n(\\ref{upper.via.M}) to hold\nfor some $n$ and prove it for $n+1$.\nWe have\n\\begin{eqnarray*}\n\\widetilde{\\vv W}_{n+1} &=&\nR(\\widetilde{\\vv W}_n+\\vv e_1\\sigma_n\n-\\vv 1\\widetilde\\tau_{n+1})^+\\\\\n&\\le& R(\\widehat{\\vv W}_n+\\vv 1M_{n-1}\n+\\vv e_1\\sigma_n-\\vv 1\\widetilde\\tau_{n+1})^+\\\\\n&=& R(\\widehat{\\vv W}_n+\\vv e_1\\sigma_n\n-\\vv 1\\widehat\\tau_{n+1}\n+\\vv 1(M_{n-1}+\\xi_n))^+.\n\\end{eqnarray*}\nSince $(u+v)^+\\le u^++v^+$,\n\\begin{eqnarray*}\n\\widetilde{\\vv W}_{n+1}\n&\\le& R(\\widehat{\\vv W}_n+\\vv e_1\\sigma_n\n-\\vv 1\\widehat\\tau_{n+1})^++\\vv 1(M_{n-1}+\\xi_n)^+\n\\equiv \\widehat{\\vv W}_{n+1}+\\vv 1M_n,\n\\end{eqnarray*}\nand the proof of (\\ref{upper.via.M}) is complete.\n\\end{proof}\n\nThe following corollary\nwill be used to obtain lower bounds.\nIt is similar to Lemma 2 in \\cite{FK}.\n\n\\begin{corollary}\\label{cor:lower.bound.D}\nLet $\\vv W_n'$ be a stable $s$-server\nqueue system with the same service times $\\sigma_n$\nas in $\\vv W_n$ and with the constant interarrival times $a'$.\nIf $a'>a={\\mathbb E}\\tau$, then, for any $\\varepsilon>0$,\nthere exists $x_0$ such that\n\\begin{eqnarray*}\n{\\mathbb P}\\{D>x\\} &\\ge&\n(1-\\varepsilon){\\mathbb P}\\{D'>x+x_0\\}\\ \\mbox{ for all }x.\n\\end{eqnarray*}\nOne can take $x_0$ such that\n$$\n{\\mathbb P}\\Bigl\\{\\sup_{n\\ge 0}\n\\sum_{i=1}^n(\\tau_i-a')\\le x_0\\Bigr\\}\\ge1-\\varepsilon.\n$$\n\\end{corollary}\n\n\\begin{proof}\nTake $\\widetilde\\tau_n=a'$ and $\\widehat\\tau_n=\\tau_n$\nin Lemma \\ref{lem:upper.bound},\nthen $\\xi_n=\\tau_n-a'$.\nA weak limit, $M$, of the sequence $M_n$ exists (since\n${\\mathbb E}\\xi_1=a-a'<0$)\nand has the same distribution as\n\\begin{eqnarray*}\nM &=_{\\rm st}& \\max\\{0,\\ \\xi_1,\\\n\\xi_1+\\xi_2,\\ \\ldots,\\ \\xi_1+\\cdots+\\xi_n,\\ \\ldots\\}.\n\\end{eqnarray*}\nBy Lemma \\ref{lem:upper.bound}, $D_n'\\le D_n+M_{n-1}$.\nHence, $D_n\\ge D'_n-M_{n-1}$\nSince $D'_n$ does not depend on $\\tau$'s,\n$D'_n$ and $M_{n-1}$ are independent. Therefore,\n\\begin{eqnarray*}\n{\\mathbb P}\\{D_n>x\\} &\\ge&\n{\\mathbb P}\\{M_{n-1}\\le x_0\\}{\\mathbb P}\\{D'_n>x+x_0\\}.\n\\end{eqnarray*}\nLetting $n$ go to infinity,\nwe obtain the desired bound.\n\\end{proof}\n\n\n\n\n\n\\section{The case $\\rho<1$, proof of Theorem \\ref{co.s.max}.}\n\\label{k=0}\n\nThe lower bound in Theorem \\ref{co.s.max} follows\nfrom Lemma \\ref{l.lower.k=0} below which also\ngeneralises Theorem \\ref{th.lower}\n(see Section \\ref{lower.bound}) in the case $k=0$.\n\n\\begin{lemma}\\label{l.lower.k=0}\nLet $\\rho>0$. Then, for any function\n$h(x)\\to\\infty$ as $x\\to\\infty$,\n$$\n{\\mathbb P}\\{D>x\\} \\ge \\frac{\\rho^s+o(1)}{s!}\n\\overline{B_r}^s(x+h(x)).\n$$\nIn particular, if the residual time\ndistribution $B_r$ is long-tailed (that is,\n$\\overline B_r(x+1) \\sim \\overline B_r(x)$\nas $x\\to\\infty$), then\n$$\n{\\mathbb P}\\{D>x\\} \\ge \\frac{\\rho^s+o(1)}{s!}\n\\overline{B_r}^s(x)\\ \\mbox{ as }x\\to\\infty.\n$$\n\\end{lemma}\n\nWe start with an auxiliary result.\n\n\\begin{lemma}\\label{q.k}\nLet $\\{q_i\\}_{i\\ge 1}$ be a non-increasing\nsequence of positive numbers. Then, for any $s\\ge 1$,\n\\begin{eqnarray*}\n\\sum_{1\\le i_1<\\ldotsa$.\nFor $\\vv i=(i_1,\\ldots,i_s)$,\n$1\\le i_1<\\ldotsx+(n-i_1)a', \\ldots,\n\\sigma_{i_s}>x+(n-i_s)a'\\}\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\nC_n(\\vv i)\n&=& \\bigcap_{i\\le n,i\\ne i_1,\\ldots,i_s}\n\\{\\sigma_i\\le x+(n-i)a'\\}.\n\\end{eqnarray*}\nSince the mean ${\\mathbb E}\\sigma$ exists, we have that,\nuniformly in $n$ and $\\vv i$,\n\\begin{eqnarray*}\n{\\mathbb P}\\{C_n(\\vv i)\\}\n=1-{\\mathbb P}\\{\\overline{C_n(\\vv i)}\\}\n&\\ge& 1-\\sum_{i=0}^\\infty {\\mathbb P}\\{\\sigma_1>x+ia'\\}\n\\to 1 \\quad\\mbox{ as }x\\to\\infty.\n\\end{eqnarray*}\nFor each vector $\\vv i$, events $A_{n}(\\vv i)$ and\n$C_{n}(\\vv i)$ are independent. Further,\nevents $A_n(\\vv i)\\cap C_n(\\vv i)$\nare disjoint for distinct vectors $\\vv i$.\nThese observations together yield\n\\begin{eqnarray}\\label{union.of.An}\n{\\mathbb P}\\Bigl\\{\\bigcup_{\\vv i}A_n(\\vv i)\n\\cap C_n(\\vv i)\\Bigr\\}\n&=& \\sum_{\\vv i}{\\mathbb P}\\{A_n(\\vv i)\\}{\\mathbb P}\\{C_n(\\vv i)\\}\n\\ge (1-o(1))\\sum_{\\vv i}{\\mathbb P}\\{A_n(\\vv i)\\}\n\\end{eqnarray}\nas $x\\to\\infty$, uniformly in $n$.\nThe event $A_n(\\vv i)$ implies that $D'_n>x$. Therefore,\n\\begin{eqnarray*}\n{\\mathbb P}\\{D'_n>x\\} &\\ge& (1-o(1))\n\\sum_{\\vv i}{\\mathbb P}\\{A_n(\\vv i)\\}\n\\end{eqnarray*}\nas $x\\to\\infty$, uniformly in $n$. We now prove that\n\\begin{eqnarray}\\label{Sigma.k=0}\n\\lim_{n\\to\\infty}\n\\sum_{\\vv i} {\\mathbb P}\\{A_n(\\vv i)\\}\n&\\ge& \\frac{(b\/a')^s}{s!}\\overline{B_r}^s(x+sa').\n\\end{eqnarray}\nIndeed, by the independence of the $\\sigma$'s,\n\\begin{eqnarray*}\n\\sum_{1\\le i_1<\\ldots0$ there exists $x_0$ such that\n\\begin{eqnarray*}\n{\\mathbb P}\\{D>x\\} &\\ge&\n(1-\\varepsilon){\\mathbb P}\\{D'>x+x_0\\}\\\\\n&\\ge& (1-\\varepsilon-o(1))\\frac{(b\/a')^s}{s!}\n\\overline{B_r}^s(x+sa'+x_0).\n\\end{eqnarray*}\nBy the arbitrary choice of $a'>a$ and $\\varepsilon>0$,\nthe proof of Lemma \\ref{l.lower.k=0} is complete.\n\\end{proofof}\n\n\\begin{proofof}{the upper bound in Theorem \\ref{co.s.max}}\nWe start with the case of deterministic $\\tau$,\ni.e., $\\tau_n\\equiv a$. We follow the lines from \\cite{FK}\nwhere, for $\\rho<1$ (that is for $bx\\} &\\le& {\\mathbb P}^s\\{U_1>x\\},\n\\end{eqnarray*}\nand we can apply known results for the single\nserver queue:\nfrom (\\ref{W.single}),\n\\begin{eqnarray*}\n{\\mathbb P}\\{U_1>x\\} &\\sim&\n\\frac{\\rho}{1-\\rho}\\overline B_r(x),\n\\end{eqnarray*}\nwhich gives us the upper bound\nin Theorem \\ref{co.s.max} if interarrival times are deterministic.\n Now the proof in the general case follows\nfrom \\cite[Lemma 1]{FK}.\n\\end{proofof}\n\n\n\n\n\n\\section{Auxiliary results.}\\label{lln}\n\nIn this Section we collect a number\nof auxiliary facts related to monotonicity and\nto\nthe strong law of large numbers for\nunstable multi-server systems.\nThe results seem not to be new, so we\nprovide only short sketches of proofs for self-containedness.\n\nLet $\\vv W_n$ be a sequence satisfying\nthe Kiefer-Wolfowitz recursion (\\ref{KiW}),\nwith initial value $\\vv W_1 \\ge 0$.\n\n\\begin{lemma}\\label{monotonicity}\n(1)\nFor any $n$, $\\vv W_n$ is a non-decreasing function\nof the initial value and of service times and a non-increasing function of interarrival\ntimes. This means that if $\\widetilde{\\vv W}_n$\nis a sequence satisfying another Kiefer-Wolfowitz recursion with initial\nvalue $\\widetilde{\\vv W}_1$ and with interarrival times $\\{\\widetilde{\\tau}_n\\}$ and service\ntimes $\\{\\widetilde{\\sigma}_n\\}$ and if\n$\\vv W_1 \\le \\widetilde{\\vv W}_1$ (coordinate-wise), $\\sigma_j\\le \\widetilde{\\sigma}_j$,\nand $\\tau_j \\ge \\widetilde{\\tau}_j$, for $j=1,\\ldots ,n-1$,\nthen\n$\\vv W_n \\le \\widetilde{\\vv W}_n$.\\\\\n(2)\nFor any $n\\ge 2$, the difference $\\sum_{i=1}^s (W_{ni}\n-W_{n-1,i})$ is a non-increasing function of the initial\nvalue $\\vv W_1$: if ${\\vv W}_1 \\le \\widetilde{\\vv W}_1$,\nthen $\\sum_{i=1}^s (W_{ni}\n-W_{n-1,i}) \\ge \\sum_{i=1}^s (\\widetilde{W}_{ni}\n-\\widetilde{W}_{n-1,i}).$\n\\end{lemma}\nThe first monotonicity property holds because both operators $R$ and\n$\\max (0,\\cdot )$ are monotone. The second property follows\nsince function $(x+y)^+-x$ is non-increasing in $x$, for any\nfixed $y$.\n\n\\begin{lemma}\\label{slln.for.unstable}\nLet $b>sa$, so the $s$-server system with workload vectors ${\\vv W}_n$ is unstable.\nThen,\n\\begin{equation}\\label{SLLN1}\n\\frac{W_{n1}}{n}\\to \\frac{b-sa}{s}\n\\quad \\mbox{and}\\quad\n\\frac{W_{ns}}{n}\\to \\frac{b-sa}{s}\n\\quad\\mbox{ as }n\\to\\infty,\n\\end{equation}\nboth with probability $1$ and in mean.\n\\end{lemma}\n\n\\begin{proof}\nNote that, for any $n=1,2,\\ldots$,\n\\begin{equation}\\label{ind1}\nW_{n+1,s}-W_{n+1,1} \\le \\max (W_{1,s}-W_{1,1}, \\sigma_1,\\ldots , \\sigma_n).\n\\end{equation}\nIndeed, if $W_{ns}-W_{n1}>\\sigma_n$, then $W_{n+1,s}-W_{n+1,1}\\le W_{ns}-W_{n1}$,\nand if $W_{ns}-W_{n1}\\le \\sigma_n$, then $W_{n+1,s}-W_{n+1,1}\\le \\sigma_n$, so the\ninduction argument completes the proof of (\\ref{ind1}). Next,\n\\begin{equation}\\label{ind2}\n\\max (W_{1s}-W_{11}, \\sigma_1,\\ldots , \\sigma_n )\/n \\to 0 \\quad \\mbox{a.s.}\n\\end{equation}\nbecause $(W_{1s}-W_{11})\/n\\to0$ and, since ${\\vv E}\\sigma$ is\nfinite, events $\\{ \\sigma_k\/k > \\varepsilon \\}$ occur only finitely often,\nfor any $\\varepsilon >0$.\n\nFurther,\n$$\n\\frac{1}{n}\\sum_{i=1}^s W_{ni} \\ge \\frac{1}{n}\\sum_{j=1}^{n-1} (\\sigma_j-s\\tau_{j+1})\n\\to b-sa >0 \\quad \\mbox{a.s.,}\n$$\nso $\\liminf_{n\\to\\infty}W_{ns}\/n \\ge (b-sa)\/s$, and, from (\\ref{ind1})-(\\ref{ind2}),\nthere exists an a.s. finite random variable $\\nu$ such that $W_{n1}>0$, for all $n\\ge\\nu$.\nSo, for $n\\ge\\nu$,\n\\begin{equation}\\label{ind3}\n\\frac{1}{n}\\sum_{i=1}^s W_{ni}\n=\\frac{1}{n}\\sum_{i=1}^s W_{\\nu i} + \\frac{1}{n} \\sum_{j=\\nu}^n (\\sigma_j-s\\tau_{j+1})\n\\to b-sa \\quad \\mbox{a.s.,}\n\\end{equation}\nand (\\ref{ind1})-(\\ref{ind3}) lead to convergence a.s. in (\\ref{SLLN1}).\nFinally, since $0\\le W_{ns}\/n \\le W_{1s}\/n + \\sum_{j=1}^{n-1} \\sigma_j\/n$\nand since random variables $\\sum_{j=1}^{n-1} \\sigma_j\/n$ are uniformly integrable,\nconvergence in mean also follows.\n\\end{proof}\n\n\\begin{lemma}\\label{slln.for.s-1}\nAssume $b>(s-1)a$. For any $\\varepsilon>0$,\nthere exist $A<\\infty$ and an integer $d\\ge 1$ such that,\nfor any initial value ${\\vv W}_1$ with $W_{1s}\\ge A$,\n$$\n{\\mathbb E}\\{W_{1+d,1}+\\ldots+W_{1+d,s}\n-W_{11}-\\ldots-W_{1s} \\} \n\\le d(b-sa+\\varepsilon).\n$$\n\\end{lemma}\n\n\\begin{proof}\nBy property (2) of Lemma \\ref{monotonicity}, it is enough to prove the\nresult for initial value $W_{11}=\\ldots =W_{s-1,1}=0$, $W_{s1}=A$ only.\n\nChoose $C$ such that ${\\mathbb E} \\min (\\tau, C) \\ge a-\\varepsilon \/2$.\nBy property (1) of Lemma \\ref{monotonicity}, we may prove the lemma\nwith interarrival times\n$\\min (\\tau_j ,C)$ in place of $\\tau_j$.\n\nConsider an auxiliary unstable $GI\/GI\/(s-1)$ queue $\\widehat{\\vv W}_n$\nwith initial zero value\nand, by applying the previous lemma, find $d$ such that\n${\\mathbb E} \\sum_{i=1}^{s-1}\\widehat{W}_{1+d,i} \\le d(b-(s-1)a+\\varepsilon\/2)$.\nThen return to the $s$-server queue and take $A=(d+1)C$.\nWe will prove that\n\\begin{equation}\\label{L43}\n\\sum_{i=1}^{s}W_{1+d,i} = \\sum_{i=1}^{s-1}\\widehat{W}_{1+d,i} +\nA - \\sum_{j=1}^{d} \\min (\\tau_{j},C) \\quad \\mbox{a.s.,}\n\\end{equation}\nthen the result will follow.\n\nConsider vectors ${\\vv V}_n$ and numbers $i_n$ as in the Introduction, with initial\nvalues $V_{1,1}=\\ldots =V_{1,s-1}=0$ and $V_{1,s}=A$. Note that $V_{n,s}\\ge A-(n-1)C>0$,\nfor all $n=1,2,\\ldots,d+1$.\n\nLet $\\mu = \\min ( d+1, \\min \\{n\\ge 1 \\ : \\ i_n=s \\} ).$ Then\n$R(V_{\\mu,1},\\ldots,V_{\\mu,s-1})=(\\widehat{W}_{\\mu,1}\\ldots,\\widehat{W}_{\\mu,s-1})$\nand\n\\begin{equation}\\label{LL43}\n\\sum_{i=1}^s W_{\\mu,i} = \\sum_{i=1}^s V_{\\mu,i}=\n\\sum_{i=1}^{s-1} V_{\\mu,i}+V_{\\mu,s} = \\sum_{i=1}^{s-1}\\widehat{W}_{\\mu,i}+A-\\sum_{j=1}^{\\mu -1}\n\\min (\\tau_j,C).\n\\end{equation}\nThis ends the proof of (\\ref{L43}) if $\\mu = d+1$.\n\nIn the case $\\mu 0$ and $\\widehat{W}_{n,i}>0$, for all $\\mu \\le n \\le d+1$ and $i=1,\\ldots,s$.\nThen, from (\\ref{LL43}),\n\\begin{eqnarray*}\n\\sum_{i=1}^s W_{d+1,i}\n&=&\n\\sum_{i=1}^s W_{\\mu,i}+\\sum_{j=\\mu}^d (\\sigma_j -s\\min (\\tau_j,C))\\\\\n&=&\n\\sum_{i=1}^{s-1}\\widehat{W}_{\\mu,i} +\\sum_{j=\\mu}^d (\\sigma_j- (s-1)\\min (\\tau_j,C))\n+ A-\\sum_{j=1}^d \\min (\\tau_j,C)\n\\end{eqnarray*}\nwhich coincides again with the right side of (\\ref{L43}).\n\n\\end{proof}\n\n\n\n\n\n\\section{Lower Bound.}\\label{lower.bound}\n\nThe following result holds without any restrictions\non the service time\ndistribution $B$\n(a similar result was formulated\nand proved in \\cite[Theorem 3.1]{SV}).\n\n\\begin{theorem}\\label{th.lower}\nLet $k\\in\\{0,1,\\ldots,s-1\\}$ be such that $\\rho>k$.\nThen, for any fixed $\\delta>0$,\n$$\n{\\mathbb P}\\{D>x\\} \\ge \\frac{\\rho^{s-k}+o(1)}{(s-k)!}\n\\overline{B_r}^{s-k}\\Bigl(\\frac{\\rho+\\delta}{\\rho-k}x\\Bigr)\n\\quad\\mbox{as }x\\to\\infty.\n$$\n\\end{theorem}\n\n\\begin{proof}\nWe exploit the technique of $s-k$ big jumps.\nFollowing Corollary \\ref{cor:lower.bound.D},\nwe consider only deterministic\ninterarrival times, $\\tau\\equiv a$.\n\nThe case $k=0$ was considered in Lemma \\ref{l.lower.k=0}.\nSo now let $k\\ge 1$.\nLet $\\widetilde{\\vv W}_n=(\\widetilde W_{n1},\\ldots,\\widetilde W_{nk})$\nbe the residual workload vector in the $GI\/GI\/k$ system\nwith the same interarrival and service times as in the original\nsystem,\nand with $k$ servers.\nSince $\\rho>k$, the $k$-server system is unstable.\nHence, by Lemma \\ref{slln.for.unstable},\nboth the minimal coordinate $\\widetilde W_{n1}$\nand the maximal coordinate $\\widetilde W_{nk}$\ndrift to infinity as $n\\to\\infty$ with probability 1,\nwith the same rate\n$(b-ka)\/k$. Then\n\\begin{eqnarray*}\n{\\mathbb P}\\Bigl\\{\\widetilde W_{N1}>N\\Bigl(\\frac{b-ka}{k}-\\delta\\Bigr),\n\\ \\widetilde W_{ik}\\le N\\Bigl(\\frac{b-ka}{k}+\\delta\\Bigr)\n\\mbox{ for all }i\\le N\\Bigr\\} &\\to& 1\n\\ \\mbox{ as }N\\to\\infty.\n\\end{eqnarray*}\n\nIf we assume that there are initially big workloads at $s-k$ servers\nwhile the $k$ other queues are empty,\nthen, with high probability, the $k$ smallest workloads\nevolve like the $k$-server system with workloads $\\widetilde{\\vv W}_n$,\nfor a long while. This observation implies that\n\\begin{eqnarray*}\n\\lefteqn{{\\mathbb P}\\Bigl\\{\nW_{N1}>N\\Bigl(\\frac{b-ka}{k}-\\delta\\Bigr),\n\\ W_{ik}\\le N\\Bigl(\\frac{b-ka}{k}+\\delta\\Bigr),\nW_{i,k+1}>N\\Bigl(\\frac{b-ka}{k}+\\delta\\Bigr)\n\\mbox{ for all }i\\le N\\Big|}\\\\\n&&\\hspace{40mm}\nW_{1k}=0, W_{1,k+1}>\nN\\Bigl(\\frac{b-ka}{k}+\\delta\\Bigr)+Na\\Bigr\\}\n\\to 1\\ \\mbox{ as }N\\to\\infty.\n\\end{eqnarray*}\nTake $c$ such that\n\\begin{eqnarray}\\label{def.c}\n\\frac{b-ka}{k}+\\delta\n&\\le& (1+c\\delta)\n\\Bigl(\\frac{b-ka}{k}-\\delta\\Bigr)\n\\end{eqnarray}\nfor all sufficiently small $\\delta>0$, and let\n\\begin{eqnarray}\\label{def.N}\nx=N\\Bigl(\\frac{b-ka}{k}-\\delta\\Bigr).\n\\end{eqnarray}\nThen\n\\begin{eqnarray*}\n{\\mathbb P}\\{D_N>x\\mid W_{1k}=0,\nW_{1,k+1}>x(1+c\\delta)+Na\\}\n&\\to& 1\\ \\mbox{ as }x\\to\\infty.\n\\end{eqnarray*}\nBy the monotonicity of the $s$-server\nqueueing system in its initial state (see Lemma \\ref{monotonicity}), we obtain\n\\begin{eqnarray}\\label{slln.double}\n{\\mathbb P}\\{D_N>x\\mid W_{1,k+1}>x(1+c\\delta)+Na\\}\n&\\to& 1\\ \\mbox{ as }x\\to\\infty.\n\\end{eqnarray}\n\nFor $\\vv i=(i_1,\\ldots,i_{s-k})$,\n$1\\le i_1<\\ldotsy+(n-i_1)a, \\ldots,\n\\sigma_{i_{s-k}}>y+(n-i_{s-k})a\\}.\n\\end{eqnarray*}\nAgain like in (\\ref{union.of.An}) we have\n\\begin{eqnarray}\\label{union.An.N}\n{\\mathbb P}\\Bigl\\{\\bigcup_{\\vv i:i_{s-k}x(1+c\\delta)+(n-j)a$,\nfor all $j\\in[i_{s-k}+1,n]$.\nConsider $\\vv i$ such that $i_{s-k}x(1+c\\delta)+Na$.\nTogether with (\\ref{slln.double}) it yields\n\\begin{eqnarray}\\label{cond.DA}\n{\\mathbb P}\\Bigl\\{D_n>x\\Big|\\bigcup_{\\vv i:i_{s-k}x\\}\n&\\ge& \\liminf_{n\\to\\infty}\n{\\mathbb P}\\Bigl\\{W_{n1}>x\\Big|\n\\bigcup_{\\vv i:i_{s-k}x\\}\n&\\ge& \\frac{\\rho^{s-k}-o(1)}{(s-k)!}\n\\overline{B_r}^{s-k}\\Bigl(x\\Bigl(\n1+c\\delta+\\frac{ka}{b-ka-k\\delta}\\Bigr)+(s-k)a\\Bigr).\n\\end{eqnarray*}\nHence, for every $\\delta>0$,\n\\begin{eqnarray*}\n{\\mathbb P}\\{D>x\\}\n&\\ge& \\frac{\\rho^{s-k}+o(1)}{(s-k)!}\n\\overline{B_r}^{s-k} \\Bigl(x\n\\Bigl(1+\\frac{ka+\\delta}{b-ka}\\Bigr)\\Bigr)\\\\\n&=& \\frac{\\rho^{s-k}+o(1)}{(s-k)!}\n\\overline{B_r}^{s-k} \\Bigl(x\n\\frac{b+\\delta}{b-ka}\\Bigr)\n\\ \\mbox{ as }x\\to\\infty.\n\\end{eqnarray*}\nThe proof of Theorem \\ref{th.lower} is complete.\n\\end{proof}\n\n\n\n\n\\section{New Majorant.}\\label{majorant}\n\nIn the proof of the upper bound in Theorem \\ref{co.s.max}\n(see Section \\ref{k=0}), we introduced $s$ parallel\nsingle server queues that provide a suitable\nmajorant in the case $\\rho<1$.\nIf $\\rho\\ge 1$ then the single server system with\nservice time distribution $B$ is unstable and\nthe above scheme does not work.\nFor an arbitrary $\\rho$, we need a more\ncomplex procedure to obtain a majorant.\nHereinafter let $k=[b\/a]$ be the integer part of $b\/a$.\nWe continue to assume constant interarrival times $\\tau \\equiv a$.\n\nAgain let $\\sigma_{ni}$, $n\\ge1$, $i\\le s$, be\nindependent random variables with common distribution $B$.\nDefine service times $\\sigma_n$ in the original\n$D\/GI\/s$ system as in Section \\ref{k=0}.\nConsider again $s$ auxiliary single server queues $D\/GI\/1$,\nbut now with different deterministic arrival times\n$T_n = n(k+1)(a-h)$ where\n\\begin{eqnarray}\\label{choice.h}\n\\frac{k}{k+1}\\Bigl(a-\\frac{b}{k+1}\\Bigr)b$, so that\nthe queues are stable.\nLet $U_i$ be a stationary waiting time in\nthe $i$th auxiliary queue. Since, for each $n$,\n\\begin{eqnarray}\\label{max_upper_independence}\n\\mbox{the sequences }\n\\{U_{n1},n\\ge 1\\},\\ldots,\\{U_{ns},n\\ge 1\\}\n\\mbox{ are mutually independent},\n\\end{eqnarray}\nthe limiting vector $(U_{1},\\ldots,U_{s})$ consists\nof independent identically distributed coordinates too.\n\nIn contrast to the case $\\rho <1$,\nit may not be true in general that, say, $V_{n1}$ is smaller than $U_{n1}$.\nNevertheless, for $\\rho < k+1$, we can prove\nthat, for any set $I$ of $k+1$ indices,\n$\\sum_{i\\in I} V_{ni}\\le \\sum_{i\\in I}U_{ni}+\\eta_I$\nwhere $\\eta_I$ has a light-tailed distribution,\nthis is Lemma \\ref{l.sum.of.W.le.U} below.\nBut first we state the main result of the Section which is\nan analogue of (\\ref{D.min})\nfor the general $\\rho$.\n\n\\begin{lemma}\\label{l.D.le.orderU}\nThere exists a number $\\beta>0$ and a random variable $\\eta$ such that\n${\\mathbb E}e^{\\beta\\eta}<\\infty$ and, for all $n$, with probability\n1,\n\\begin{eqnarray*}\nD_n &\\le& U_{n,(k+1)}+\\eta,\n\\end{eqnarray*}\nwhere $U_{n,(k+1)}$ is the $(k+1)$th order\nstatistic of vector $(U_{n1},\\ldots,U_{ns})$.\n\\end{lemma}\n\nNow we formulate and prove the following result.\nBased on it, we give the proof of Lemma \\ref{l.D.le.orderU}\nat the end of the section.\n\n\\begin{lemma}\\label{l.sum.of.W.le.U}\nThere exists $\\beta>0$ such that, for any set of $k+1$\nindices $I=\\{i(1),\\ldots,i(k+1)\\}$, there is a random\nvariable $\\eta_I$ such that\n${\\mathbb E}e^{\\beta\\eta_I}<\\infty$ and, for any $n$,\nwith probability 1,\n\\begin{eqnarray*}\n\\sum_{i\\in I} V_{ni} &\\le&\n\\sum_{i\\in I}U_{ni}+\\eta_I.\n\\end{eqnarray*}\n\\end{lemma}\n\n\\begin{proof}\nFix some $i'\\in I$.\nConsider an auxiliary $GI\/GI\/k+1$ system\n$\\vv V'_n=(V'_{ni},i\\in I)$ with the same interarrival\ntimes equal to $\\tau_n$, but whose service times $\\sigma'_n$\nare chosen in a special manner.\nAt any time $n$, if $i_n\\in I$ ($i_n$ is defined in the proof\nof Theorem \\ref{co.s.max}, see Section \\ref{k=0})\nthen put $\\sigma'_n=\\sigma_{n,i_n}$ and $i'_n=i_n$.\nIf $i_n\\not\\in I$ then put $\\sigma'_n=\\sigma_{ni'}$\nand $i'_n=i'$.\nApplying property (1) of Lemma \\ref{monotonicity}, we\nget that\n$R(V_{ni},i\\in I)\\le R\\vv V_n'$\ncoordinatewise, for any $n$. Therefore,\n\\begin{eqnarray*}\n\\sum_{i\\in I} V_{ni}\n&\\le& \\sum_{i\\in I} V'_{ni}.\n\\end{eqnarray*}\nHence, it suffices to prove that\n\\begin{eqnarray}\\label{reform.V}\n\\sum_{i\\in I} V'_{ni} &\\le&\n\\sum_{i\\in I} U_{ni}+\\eta_I.\n\\end{eqnarray}\n\nFor every $i\\in I$ and for any $n$,\n\\begin{eqnarray*}\nU_{n+1,i}-U_{ni}\n&\\ge& \\sigma_{ni}-(k+1)(a-h),\n\\end{eqnarray*}\nand hence\n\\begin{eqnarray}\\label{increment.of.U}\n\\sum_{i\\in I}U_{n+1,i}-\\sum_{i\\in I}U_{ni}\n&\\ge& \\sigma_n'-(k+1)(a-h)+\n\\sum_{i\\in I,i\\not=i'_n}(\\sigma_{in}-(k+1)(a-h))\\nonumber\\\\\n&=& \\sigma_n'-(k+1)a+\\zeta_{I,n},\n\\end{eqnarray}\nwhere the independent identically distributed\nrandom variables\n$$\n\\zeta_{I,n}:=\\sum_{i\\in I,i\\not=i'_n}(\\sigma_{in}-b)\n+(k+1)^2h+k(b-(k+1)a)\n$$\nhave a positive mean, by the left inequality\nin (\\ref{choice.h}).\n\nTake $\\varepsilon\\in(0,{\\mathbb E}\\zeta_{I,n})$.\nLet $d$ and $A$ be defined by Lemma \\ref{slln.for.s-1}\napplied to the system $(V'_{1+n,i},i\\in I)$.\nConsider $d$-skeleton of $\\vv V'$, that is,\nthe sequence $\\vv V'_{1+nd}$.\nFor every $n$, if the maximal coordinate of\n$(V'_{1+nd,i},i\\in I)$ is not bigger than $A$,\nthen\n\\begin{eqnarray*}\n\\sum_{i\\in I}V'_{1+nd+d,i}-\\sum_{i\\in I}V'_{1+nd,i}\n&\\le& \\sigma'_{nd+1}+\\ldots+\\sigma'_{nd+d},\n\\end{eqnarray*}\nwhich together with (\\ref{increment.of.U}) implies that\n\\begin{eqnarray*}\n\\sum_{i\\in I}V'_{1+nd+d,i}-\\sum_{i\\in I}V'_{1+nd,i}\n&\\le& \\sum_{i\\in I}U_{1+nd+d,i}-\\sum_{i\\in I}U_{1+nd,i}\n+d(k+1)a-\\zeta_{I,1+nd}-\\ldots-\\zeta_{I,nd+d}\\\\\n&\\le& \\sum_{i\\in I}U_{1+nd+d,i}-\\sum_{i\\in I}U_{1+nd,i}\n+d((k+1)a+b),\n\\end{eqnarray*}\nsince $\\zeta_{I,n}\\ge -b$. To conclude, if\nthe maximal coordinate of $(V'_{1+nd,i},i\\in I)$\nis not bigger than $A$, then\n\\begin{eqnarray*}\n\\sum_{i\\in I}V'_{1+nd,i} &\\le& (k+1)A,\n\\end{eqnarray*}\nso that\n\\begin{eqnarray}\\label{increment.of.Vprime.1}\n\\sum_{i\\in I}V'_{1+nd+d,i}\n&\\le& \\sum_{i\\in I}U_{1+nd+d,i}+d((k+1)a+b)+(k+1)A\\nonumber\\\\\n&:=& \\sum_{i\\in I}U_{1+nd+d,i}+C,\n\\end{eqnarray}\n\nFurther, if the maximal coordinate of\n$(V'_{1+nd,i},i\\in I)$ is bigger than $A$,\nthen, by Lemma \\ref{slln.for.s-1}, we have\n\\begin{eqnarray*}\n\\sum_{i\\in I}V'_{1+nd+d,i}-\\sum_{i\\in I}V'_{1+nd,i}\n&\\le& \\theta_{I,1+nd},\n\\end{eqnarray*}\nwhere $\\theta_{I,1+nd}$ are independent identically\ndistributed random variables with mean\n${\\mathbb E}\\theta_{I,1+nd}\\le d(b-(k+1)a+\\varepsilon)$\nand such that\n\\begin{eqnarray*}\n\\theta_{I,1+nd} &\\le& \\sigma'_{nd+1}+\\ldots+\\sigma'_{nd+d}.\n\\end{eqnarray*}\nIn this case, by (\\ref{increment.of.U}),\n\\begin{eqnarray}\\label{increment.of.Vprime.2}\n\\sum_{i\\in I}V'_{1+nd+d,i}-\\sum_{i\\in I}V'_{1+nd,i}\n&\\le& \\sum_{i\\in I}U_{1+nd+d,i}-\\sum_{i\\in I}U_{1+nd,i}\n+\\theta_{I,1+nd}-(\\sigma'_{nd}+\\ldots+\\sigma'_{nd+d})\\nonumber\\\\\n&&\\hspace{20mm}+d(k+1)a-\\zeta_{I,1+nd}-\\ldots-\\zeta_{I,nd+d}\n\\nonumber\\\\\n&:=& \\sum_{i\\in I}U_{1+nd+d,i}-\\sum_{i\\in I}U_{1+nd,i}\n+\\widetilde\\theta_{I,1+nd}.\n\\end{eqnarray}\n\nInequalities (\\ref{increment.of.Vprime.1}) and\n(\\ref{increment.of.Vprime.2}) imply that always\n\\begin{eqnarray}\\label{diff.rough}\n\\sum_{i\\in I} V'_{1+nd+d,i}\n&\\le& \\sum_{i\\in I} U_{1+nd+d,i}+C\n+\\max_n\\sum_{j=0}^n\\widetilde\\theta_{I,1+jd}.\n\\end{eqnarray}\nHere $\\widetilde{\\theta}_{I,1+nd}, n=1,2,\\ldots$ are\nindependent identically distributed random variables that\nare bounded from above and have a negative mean. Indeed,\n\\begin{eqnarray*}\n\\widetilde\\theta_{I,1+nd}\n&=& \\theta_{I,1+nd}-(\\sigma'_{nd+1}+\\ldots+\\sigma'_{nd+d})\n+d(k+1)a-\\zeta_{I,1+nd}-\\ldots-\\zeta_{I,nd+d}\\\\\n&\\le& d(k+1)a-\\zeta_{I,1+nd}-\\ldots-\\zeta_{I,nd+d}\\\\\n&\\le& d((k+1)a+b)\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\n{\\mathbb E}\\widetilde\\theta_{I,1+nd}\n&=& {\\mathbb E}\\theta_{I,1+nd}-bd\n+d(k+1)a-d{\\mathbb E}\\zeta_{I,1+nd}\\\\\n&\\le& d(b-(k+1)a+\\varepsilon)-bd+d(k+1)a-d{\\mathbb E}\\zeta_{I,1+nd}\\\\\n&=& d(\\varepsilon-{\\mathbb E}\\zeta_{I,1+nd})<0,\n\\end{eqnarray*}\nby the choice of $\\varepsilon$.\nTherefore, there exists $\\beta>0$ such that\n${\\mathbb E}e^{\\beta\\widetilde\\theta_{I,1+nd}}<1$\nand then the following estimate holds:\n$$\n{\\mathbb E}\\exp\\Bigl\\{\\beta\n\\max_n\\sum_{j=0}^n\\widetilde\\theta_{I,1+jd}\\Bigr\\}\n\\le \\sum_n\n({\\mathbb E}e^{\\beta\\widetilde\\theta_{I,1+nd}})^n\n<\\infty.\n$$\nLet\n$$\n\\widetilde\\eta_I:=C+\\max_n\\sum_{j=0}^n\\widetilde\\theta_{I,1+jd},\n$$\nthen ${\\mathbb E}e^{\\beta\\widetilde\\eta_I}<\\infty$ and,\nby the upper bound (\\ref{diff.rough}), for all $n$,\n\\begin{eqnarray}\\label{V.U.int}\n\\sum_{i\\in I} V'_{1+nd+d,i}\n&\\le& \\sum_{i\\in I} U_{1+nd+d,i}+\\widetilde\\eta_I.\n\\end{eqnarray}\nSince, in addition, for every $l\\in[1,d]$,\n$$\n\\sum_{i\\in I} V'_{1+nd+d+l,i}-\\sum_{i\\in I} V'_{1+nd+d+l-1,i}\n\\le \\sum_{i\\in I} U_{1+nd+d+l,i}-\\sum_{i\\in I} U_{1+nd+d+l-1,i}+(k+1)a,\n$$\nwe conclude from (\\ref{V.U.int}) that, for every $l\\in[1,d]$,\n\\begin{eqnarray*}\n\\sum_{i\\in I} V'_{1+nd+d+l,i}\n&\\le& \\sum_{i\\in I} U_{1+nd+d+l,i}+\\widetilde\\eta_I+l(k+1)a,\n\\end{eqnarray*}\nso that, for every $n$,\n\\begin{eqnarray*}\n\\sum_{i\\in I} V'_{ni}\n&\\le& \\sum_{i\\in I} U_{ni}+\\widetilde\\eta_I+d(k+1)a,\n\\end{eqnarray*}\nwhich completes the proof of Lemma \\ref{l.sum.of.W.le.U}\nwith $\\eta_I:=\\widetilde\\eta_I+d(k+1)a$.\n\\end{proof}\n\n\\begin{proofof}{Lemma \\ref{l.D.le.orderU}}\nFor every collection $I$ of $k+1$ coordinates\n\\begin{eqnarray*}\nD_n &\\le& \\frac{1}{k+1}\\sum_{i\\in I} V_{ni},\n\\end{eqnarray*}\nsince $D_n$ is the minimal coordinate.\nThen it follows from Lemma \\ref{l.sum.of.W.le.U} that\n\\begin{eqnarray}\\label{dn.le.sum}\nD_n &\\le& \\frac{1}{k+1}\\sum_{i\\in I} U_{ni}+\\eta_I.\n\\end{eqnarray}\nTake $\\eta:=\\displaystyle\\max_{I:|I|=k+1}\\eta_I$.\nThen ${\\mathbb E}e^{\\beta\\eta}<\\infty$ and\n\\begin{eqnarray}\\label{dn.le.sum1}\nD_n &\\le& \\frac{1}{k+1}\\sum_{i\\in I} U_{ni}+\\eta.\n\\end{eqnarray}\nTake $I$ such that $\\{U_{ni},i\\in I\\}$\nare the $k+1$ smallest coordinates of vector $(U_{n1},\\ldots,U_{ns})$.\nThen $U_{ni}\\le U_{n,(k+1)}$ for every $i\\in I$.\nTogether with (\\ref{dn.le.sum1})\nit yields the inequality of the lemma.\n\\end{proofof}\n\n\n\n\n\n\\section{Upper Bound and the Principle of $s-k$ Big Jumps.}\\label{sec.upper.bound}\n\nNow we turn to the upper bound.\nLemma \\ref{l.D.le.orderU}\nallows to prove the following general result.\n\n\\begin{theorem}\\label{th.deter.upper}\nLet $\\rho0$ such that\n\\begin{eqnarray*}\n{\\mathbb P}\\{D>x+y\\}\n&\\le& \\Big(\\begin{array}{c}s\\\\k\\end{array}\\Big)\n(\\overline F(x))^{s-k}+const\\cdot e^{-\\beta y}\n\\ \\mbox{ for all }x,y>0,\n\\end{eqnarray*}\nwhere $F$ is the distribution of random variable\n$$\nM:=\\sup\\Bigl(0,\\ \\sum_{j=1}^n\n(\\sigma_j-(k+1)(a-h)),\\ n\\ge 1\\Bigr).\n$$\n\\end{theorem}\n\n\\begin{proof}\nFirst, by inequality\n$b\\equiv{\\mathbb E}\\sigma<(k+1)(a-h)$\nand by the strong law of large numbers,\nthe maximum $M$ is finite with probability 1.\nBy Lemma \\ref{l.D.le.orderU},\n\\begin{eqnarray*}\n{\\mathbb P}\\{D_n>x+y\\}\n&\\le& {\\mathbb P}\\{U_{n,(k+1)}+\\eta>x+y\\}\\\\\n&\\le& {\\mathbb P}\\{U_{n,(k+1)}>x\\}+\n{\\mathbb P}\\{\\eta>y\\}.\n\\end{eqnarray*}\nTaking into account the independence of the $U$'s in\n(\\ref{max_upper_independence}), we obtain\n\\begin{eqnarray*}\n{\\mathbb P}\\{D_n>x+y\\}\n&\\le& \\Big(\\begin{array}{c}s\\\\k\\end{array}\\Big)\n{\\mathbb P}^{s-k}\\{U_{n1}>x\\}+\n{\\mathbb P}\\{\\eta>y\\}.\n\\end{eqnarray*}\nLetting $n\\to\\infty$ and taking into account\nthe duality between the single server system\nand the maximum of the corresponding random walk,\nwe arrive at the following inequality:\n\\begin{eqnarray}\\label{Dprime}\n{\\mathbb P}\\{D>x+y\\}\n&\\le& \\Big(\\begin{array}{c}s\\\\k\\end{array}\\Big)\n(\\overline F(x))^{s-k}+\n{\\mathbb P}\\{\\eta>y\\}.\n\\end{eqnarray}\nSince $\\eta$ has a finite exponential\nmoment, we obtain the statement of the theorem.\n\\end{proof}\n\n\\begin{proofof}{the upper bound in Theorem \\ref{th:subexp.bounds}}\nIt follows from Theorem \\ref{th.deter.upper} that\n\\begin{eqnarray*}\n{\\mathbb P}\\{D>x\\}\n&\\le& \\Big(\\begin{array}{c}s\\\\k\\end{array}\\Big)\n\\overline F^{s-k}(x(1-\\delta))\n+const\\cdot e^{-\\beta\\delta x}.\n\\end{eqnarray*}\nDue to the subexponentiality of $B_r$ we obtain\nfrom the analogue of (\\ref{W.single}) for the maximum of\na random walk (see, e.g., \\cite[Theorem 5.2]{FKZ})\nthat, as $x\\to\\infty$,\n$$\n\\overline F(x)\\sim\\frac{b}{(k+1)(a-h)-b}\\overline B_r(x).\n$$\nTaking $h$ in (\\ref{choice.h}) close to its minimal value,\nsay $h=\\frac{k}{k+1}\\Bigl(a-\\frac{b}{k+1}\\Big)+\\varepsilon$,\n$\\varepsilon>0$, we arrive at the following estimate:\n$$\n\\overline F(x)\\sim\\frac{b}{a-b\/(k+1)-(k+1)\\varepsilon}\\overline B_r(x)\n=\\frac{(k+1)\\rho}{k+1-\\rho-(k+1)^2\\varepsilon\/a}\\overline B_r(x).\n$$\nIn addition,\n$\\overline B_r(x(1-\\delta))\\cdot e^{\\beta\\delta x}\n\\to \\infty$ as $x\\to\\infty$. All these facts and\narbitrarity of choice of $\\varepsilon>0$ imply\nthe desired bound.\n\\end{proofof}\n\nIt what follows, for two families of events $A_x$ and $B_x$ of positive probabilities\nindexed by $x$, we write $A_x \\sim B_x$ if ${\\mathbb P} \\{A_x\\setminus B_x\\} = o({\\mathbb P}\\{A_x\\})$\nand ${\\mathbb P} \\{B_x\\setminus A_x\\} = o({\\mathbb P}\\{A_x\\})$ as $x\\to\\infty$. Note that\n$A_x\\sim B_x$ implies ${\\mathbb P}\\{A_x\\}\\sim {\\mathbb P}\\{B_x\\}$, but not vice versa.\n\nWe establish now the principle of $s-k$ big jumps in the case of\nintermediate\nregularly varying\ndistributions.\nFor simplicity, we do it again for $D\/GI\/s$ system with deterministic\ninter-arrival times.\nFor this, we consider the representation of the stationary workload\nin the backward time (the so-called ``Loynes scheme'').\nWe again use the joint representation of $s$ individual queues and of\nthe $s$-server system given in the previous section and assume that\nall queues run for a long time, from time $-\\infty$, and that\n$U_i$ is the stationary waiting time of the customer that arrives at the $i$th\nqueue at time $0$. Then\n$$\nU_i = \\sup (0, \\xi_{-1,i}, \\xi_{-1,i}+\\xi_{-2,i}, \\ldots ,\n\\xi_{-1,i}+\\ldots +\\xi_{-n,i}, \\ldots )\n$$\nwhere $\\xi_{j,i} =\\sigma_{j,i}-\\widehat{a}$, for $i=1,\\ldots,s$ and $j=-1,-2,\\ldots$,\nand $\\widehat{a}=(k+1)(a-h)$.\nFurther, ${\\vv W}_n$ are stationary vectors for $n\\le 0$, and\n$$\n{\\vv W}_{n+1}= R((W_{n1}+\\sigma_n-a)^+, (W_{n2}-a)^+,\\ldots,(W_{n,s}-a)^+),\n$$\nfor all $n<0$. Here again $i_n = \\min \\{i : \\ V_{ni}=D_n \\}$ and $\\sigma_n=\\sigma_{n,i_n}$.\nThen, for any $x>0$, by Lemma \\ref{l.D.le.orderU},\n$$\n\\{D_0 >x\\} \\subset \\bigcup_J \\{ \\min_{i\\in J} U_i + \\eta >x \\}\n$$\nwhere $D_0$ is the stationary waiting time in the $D\/GI\/s$ queue,\ni.e. the minimal coordinate of vector ${\\vv W}_0$,\n$J$'s are subsets of $\\{1,2,\\ldots,s\\}$ of cardinality $s-k$, and $\\eta$ is\na random variable with light-tailed distribution. Then\n$$\n\\{D_0 > x\\} = \\bigcup_J \\{D_0>x, \\min_{i\\in J} U_i + \\eta >x \\}.\n$$\n\n\nAssume that the\nresidual\ndistribution function $B_r$ of service times is\nintermediate\nregularly varying\n(for that, it is sufficient for $B$ to be intermediate varying).\nThen, clearly, each random variable $U_i$ has an intermediate regularly varying\ndistribution since ${\\mathbb P} (U_i>x) \\sim c \\overline{B}_r(x)$.\nSince\nthe random variables $U_i$ are mutually independent, the distribution\nof $\\min_{i\\in J} U_i$ is also intermediate regularly varying,\n${\\mathbb P} (\\min U_i >x) \\sim c^{s-k} \\left(\\overline{B}_r(x)\n\\right)^{s-k}$.\n\nIt is well-known, see e.g. \\cite[Ch 5]{FKZ}, that\n$$\n\\{ U_i>x \\} \\sim \\bigcup_{n\\ge 1} \\{\\sigma_{-n,i}>x+n\\widehat{a} \\}\n$$\nand therefore, for any set $J \\subset \\{1,2,\\ldots,s\\}$,\n$$\n\\{\\min_{i\\in J} U_i >x\\} \\sim \\bigcap_{i\\in J} \\left(\\bigcup_{n>0}\n\\{\\sigma_{-n,i}>x+n\\widehat{a} \\}\\right).\n$$\n\nWe use the following property of intermediate regularly varying distributions\n(its proof is postponed until the end of the section;\na similar result for equivalence of probabilities\nmay be found in \\cite{AAK}):\n\n\\begin{lemma}\\label{irv3}\nIf $X$ and $Y$ are two random variables such that $X$\nhas an intermediate regularly varying distribution and\n${\\mathbb P}\\{|Y|>x\\}=o({\\mathbb P}\\{X>x\\})$ as\n$x\\to\\infty$, then $\\{X+Y>x\\} \\sim \\{X>x\\}$,\nfor any joint distribution of $X$ and $Y$.\n\\end{lemma}\n\nApplying Lemma \\ref{irv3} with $X=\\min_{i\\in J} U_i$\nand $Y=\\eta$, for all $J$ of cardinality $s-k$, we get\n\\begin{eqnarray*}\n\\{D_0>x\\}\n&\\sim &\n\\bigcup_J \\{D_0>x, \\min_{i\\in J} U_i >x\\}\\\\\n&\\sim &\n\\bigcup_J \\bigcap_{i\\in J} \\left(\\bigcup_{n>0}\n\\{D_0>x, \\sigma_{-n,i}>x+n\\widehat{a} \\}\\right),\n\\end{eqnarray*}\nsince the upper and the lower bounds for\n${\\mathbb P}\\{D_0>x\\}$ are of the same order,\nsee \\cite[Theorem 7]{BaF} or \\cite{FK} for further arguments.\nNow represent any event on the right in the latter equation as a union\nof two events\n$$\n\\{D_0>x, \\sigma_{-n,i}>x+n\\widehat{a} , i_{-n}=i\\} \\cup\n\\{D_0>x, \\sigma_{-n,i}>x+n\\widehat{a} , i_{-n}\\neq i\\}\n$$\nwhere\n\\begin{eqnarray*}\n{\\mathbb P}\\{D_0>x, \\sigma_{-n,i}>x+n\\widehat{a}, i_{-n}\\neq i\\}\n&=& {\\mathbb P}\\{D_0>x, i_{-n}\\neq i\\} {\\mathbb P}\\{\\sigma_{-n,i}>x+n\\widehat{a}\\}\\\\\n&\\le& {\\mathbb P}\\{D_0>x\\}{\\mathbb P}\\{\\sigma_{-n,i}>x+n\\widehat{a}\\}.\n\\end{eqnarray*}\nSo, for any set $J$, the union of events\n$$\n\\bigcap_{i\\in J} \\left(\\bigcup_{n>0}\n\\{D_0>x, \\sigma_{-n,i}>x+n\\widehat{a}, i\\neq i_{-n}\\}\\right)\n$$\nhas probability $O({\\mathbb P} \\{D_0>x\\} \\overline{B}_r (x))\n= o({\\mathbb P} \\{D_0>x\\})$.\nSince there is only a finite number of sets $J$,\nwe obtain the following result.\n\n\\begin{theorem}\\label{bigjumps}\nAssume that $\\rho \\in (k,k+1)$ and that the distribution of service\ntimes\nis intermediate regularly varying.\nAs $x\\to\\infty$,\n\\begin{equation}\\label{bigj1}\n{\\mathbb P} \\{D_0>x\\} \\sim {\\mathbb P} \\Bigl\\{ D_0>x,\n\\bigcup_{0 x+n_j \\widehat{a} \\}\n\\Bigr\\}.\n\\end{equation}\n\\end{theorem}\n\n{\\bf Remark.}\nIn the proof of Theorem \\ref{bigjumps},\nwe followed the scheme introduced in \\cite{FK},\nsee also \\cite{BaF}, \\cite{BMZ}, and \\cite{BZ}\nfor similar constructions.\nTheorem \\ref{bigjumps} is not the final statement.\nWe may go further and obtain the following result.\nAssume that $B$ is a regularly varying distribution.\nThen,\nfor some positive and finite constant $C$ and as $x\\to\\infty$,\n\\begin{equation}\\label{equivalence1}\n{\\mathbb P} \\{D_0>x\\} \\sim C \\overline{B}^{s-k}_{r}(x).\n\\end{equation}\nThe result seems to be correct, but its proof would be very lengthy and\nwould require a\nscrupulous calculation, so we decided not to proceed further in this\ndirection.\n\nWe provide a hint for a plausible proof\ninstead. First, one may consider an auxiliary\ndeterministic model with $(n-s)$ very big service times\n$y_1$, \\ldots, $y_{s-k}$ that occur\nat time instants $-n_1>-n_2>\\ldots >-n_{s-k}$ and replace all\nother service times by their mean $b$. We also assume that, before\nthe first jump, the workload vector is zero.\nFor this model, we may find conditions on the $y$'s for the\nminimal coordinate of the workload vector at time 0 to be not\nsmaller than $x$. Then repeat the same for all the other times\nof jumps $-n_{1}>-n_{2}>\\ldots >-n_{s-k}$. The union of these\nregions may be represented as a combination of unions and differences\nof a finite number of truncated half-spaces of dimension $s-k$.\nSummation of tail probabilities over each such set gives the\nprobability of order $\\overline{B}^{s-k}_{r}(x)$, thanks to the properties\nof regularly varying functions. So a finite combination of sums and\ndifferences of these probabilities gives a probability of the same\norder. It cannot be of a lower order, due to the lower\nbound.\n\n\\begin{proofof}{Lemma \\ref{irv3}}\nFrom Theorem 2.47 in \\cite{FKZ}, if $X$ has\nan intermediate regularly varying distribution, then\n$$\n{\\mathbb P}\\{X>x+h(x)\\}\\sim {\\mathbb P}\\{X>x\\}\n\\sim {\\mathbb P}\\{X>x-h(x)\\}\n$$\nas $x\\to\\infty$, for any function $h(x)\\to\\infty$ such\nthat $h(x)=o(x)$. Hence, by the monotonicity arguments,\n$$\n\\{X>x+h(x)\\}\\sim \\{X>x\\}\\sim \\{X>x-h(x)\\}.\n$$\nSince the distribution of $X$ is intermediate regularly\nvarying and since ${\\mathbb P}\\{|Y|>x\\}=o({\\mathbb P}\\{X>x\\})$\nas $x\\to\\infty$, we have \n${\\mathbb P}\\{|Y|>\\varepsilon x\\}=o({\\mathbb P}\\{X>x\\})$\nas $x\\to\\infty$, for every $\\varepsilon>0$. Then \nthere exists $h(x)=o(x)$ such that\n$$\n{\\mathbb P}\\{|Y|>h(x)\\}=o({\\mathbb P}\\{X>x\\}).\n$$\nTherefore, as $x\\to\\infty$,\n\\begin{eqnarray*}\n\\{X>x\\} &\\sim& \\{X>x+h(x)\\}\\setminus\\{Y<-h(x)\\}\\\\\n&=& \\{X>x+h(x),Y\\ge-h(x)\\} \\subseteq \\{X+Y>x\\}\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\n\\{X>x\\} &\\sim& \\{X>x-h(x)\\}\\cup\\{Y>h(x)\\}\n\\supseteq \\{X+Y>x\\}, \n\\end{eqnarray*}\nwhich justifies the events equivalence,\n$\\{X+Y>x\\} \\sim \\{X>x\\}$.\n\\end{proofof}\n\n\\section{Existence of moments: proof of Theorem 4.}\\label{proof}\n\nSince the tail distribution of $\\min(\\sigma_{r,1},\\ldots,\\sigma_{r,s-k})$\nis equal to $(\\overline B_r(x))^{s-k}$, we obtain from\nTheorem \\ref{th.lower}\n$$\n{\\mathbb P}\\{D>x\\} \\ge\nc_1{\\mathbb P}\\{\\min(\\sigma_{r,1},\\ldots,\\sigma_{r,s-k})\n>c_2x\\}.\n$$\nSince, for any non-negative random variable $\\eta$,\n$$\n{\\mathbb E}\\eta^\\gamma = \\gamma\\int_0^\\infty x^{\\gamma-1}\n{\\mathbb P}\\{\\eta>x\\}dx,\n$$\nwe have\n$$\n{\\mathbb E}D^\\gamma \\ge \\frac{c_1}{c_2}\n{\\mathbb E}(\\min(\\sigma_{r,1},\\ldots,\\sigma_{r,s-k}))^\\gamma.\n$$\nand the existence of the moment of order $\\gamma$\nfor the delay $D$ implies with necessity (\\ref{min.fin}).\n\nNow assume (\\ref{min.fin}). Consider $s-k$\nindependent copies $M_1$, \\ldots, $M_{s-k}$ of the\nrandom variable $M$ introduced in Theorem \\ref{th.deter.upper}.\nThen the assertion of Theorem \\ref{th.deter.upper}\ncan be rewritten in the following way:\n$$\n{\\mathbb P}\\{D>x+y\\} \\le\n\\Big(\\begin{array}{c}s\\\\k\\end{array}\\Big)\n{\\mathbb P}\\{\\min(M_1,\\ldots,M_{s-k})>x\\}\n+const\\cdot e^{-\\beta y}.\n$$\nTake $y=x$. Then ${\\mathbb E}D^\\gamma<\\infty$ follows\nif we prove that\n\\begin{equation}\\label{M1.Ms-k}\n{\\mathbb E}(\\min(M_1,\\ldots,M_{s-k}))^\\gamma < \\infty.\n\\end{equation}\nIn order to do it, we explore the ladder height construction\nfor the maximum $M$ of a random walk $S_n=X_1+\\ldots+X_n$\nwhere $X_j=\\sigma_j-b-\\varepsilon$.\nSince this random walk has a negative drift,\nthe first ladder epoch and the first ladder height\n$$\n\\theta=\\min(n\\ge 1:S_n>0), \\quad \\widetilde\\chi=S_{\\theta},\n$$\nboth are degenerate random variables;\n$$\np\\equiv {\\mathbb P}\\{\\theta<\\infty\\}={\\mathbb P}\\{M>0\\}<1.\n$$\nDenote by $\\chi$ a random variable with distribution\n$$\n{\\mathbb P}\\{\\chi\\in B\\}={\\mathbb P}\\{\\widetilde\\chi\\in B\\}\/p.\n$$\nLet $\\chi_j$ be independent copies of $\\chi$.\nIf $\\eta$ is an independent counting random variable with\ndistribution ${\\mathbb P}\\{\\eta=j\\}=(1-p)p^j$, $j=0$, $1$, \\ldots,\nthen $M$ is equal in distribution to $\\chi_1+\\ldots+\\chi_\\eta$.\n\nLet $\\chi_{i,j}$ be again independent copies of $\\chi$\nand $\\eta_j$ be independent copies of $\\eta$.\nThen $\\min(M_1,\\ldots,M_{s-k})$ is equal in distribution to\n$$\n\\min\\biggl(\\sum_{j=1}^{\\eta_1}\\chi_{1,j},\\ldots,\n\\sum_{j=1}^{\\eta_{s-k}}\\chi_{s-k,j}\\biggr).\n$$\nThe latter\nminimum does not exceed\n$$\n\\sum_{j_1=1}^{\\eta_1}\\ldots\\sum_{j_{s-k}=1}^{\\eta_{s-k}}\n\\min(\\chi_{1,j_1},\\ldots,\\chi_{s-k,j_{s-k}}).\n$$\nTaking into account that for non-negative arguments\n$$\n(x_1+\\ldots+x_N)^\\gamma\\le N^\\gamma(x_1^\\gamma+\\ldots+x_N^\\gamma),\n$$\nwe get the following estimate:\n\\begin{eqnarray*}\n\\min\\biggl(\\sum_{j=1}^{\\eta_1}\\chi_{1,j},\\ldots,\n\\sum_{j=1}^{\\eta_{s-k}}\\chi_{s-k,j}\\biggr)^\\gamma\n&\\le& (\\eta_1+\\ldots+\\eta_{s-k})^\\gamma\n\\sum_{j_1=1}^{\\eta_1}\\ldots\\sum_{j_{s-k}=1}^{\\eta_{s-k}}\n\\min(\\chi_{1,j_1},\\ldots,\\chi_{s-k,j_{s-k}})^\\gamma.\n\\end{eqnarray*}\nIn particular, the mean of the term in the left side of the equality above\nis not larger than\n\\begin{eqnarray*}\n\\lefteqn{\\sum_{j_1=1}^\\infty\\ldots\\sum_{j_{s-k}=1}^\\infty\n(1-p)p^{j_1+\\ldots+j_{s-k}}(j_1+\\ldots+j_{s-k})^\\gamma\nj_1\\cdot\\ldots\\cdot j_{s-k}\n{\\mathbb E}\\min(\\chi_{1,1},\\ldots,\\chi_{s-k,1})^\\gamma}\\\\\n&&\\hspace{40mm} = {\\mathbb E}(\\eta_1+\\ldots+\\eta_{s-k})^\\gamma\n\\eta_1\\ldots\\eta_{s-k}\n{\\mathbb E}\\min(\\chi_{1,1},\\ldots,\\chi_{s-k,1})^\\gamma.\n\\end{eqnarray*}\nSince the $\\eta$'s have finite exponential moments,\nthe first mean on the right is finite.\nNow we show finiteness of the second mean. First,\n$$\n{\\mathbb P}\\{\\chi>x\\} =\n\\int_{-\\infty}^0 \\overline B(x-y)\\mu(dy),\n$$\nwhere the measure $\\mu$ is defined by\n\\begin{eqnarray*}\n\\mu(dy) &=& \\sum_n {\\mathbb P}\\{S_n\\in dy, S_k\\le 0\n\\mbox{ for all }k\\le n-1\\}\\\\\n&\\le& \\sum_n {\\mathbb P}\\{S_n\\in dy\\}.\n\\end{eqnarray*}\nThen, by the key renewal theorem,\n$$\nc\\equiv \\sup_{y\\le 0}\\mu(y-1,y] < \\infty,\n$$\nwhich yields\n$$\n{\\mathbb P}\\{\\chi>x\\} \\le c \\sum_{j=0}^\\infty\n\\overline B(x+j)\\le c\\overline B_r(x-1).\n$$\nTherefore, due to condition (\\ref{min.fin}),\n$$\n{\\mathbb E}\\min(\\chi_{1,1},\\ldots,\\chi_{s-k,1})^\\gamma<\\infty,\n$$\nwhich completes the proof.\n\n\n\n\n\n\\acknowledgments{This research was supported\nby EPSRC grant No.~R58765\/01\nand RFBR grant No.~10-01-00161.}\nThe authors thank Bert Zwart for stimulating discussions, \nand Arcady Shemyakin and James Cruise for stylistic comments.\n\n\n\n\n\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}