Central limit theorems, Lee-Yang zeros, and graph-counting polynomials

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• 作者：J.L. Lebowitz^a ; ^b ; B. Pittel^c ; ^{bgp@math.ohio-state.edu" class="auth_mail" title="E-mail the corresponding author} ; D. Ruelle^a ; ^d ; E.R. Speer^a
• 关键词：Graph polynomial ; Grand canonical partition function ; Lee&ndash ; Yang ; Combinatorial ; Asymptotic enumeration ; Limit theorems
• 刊名：Journal of Combinatorial Theory, Series A
• 出版年：2016
• 出版时间：July 2016
• 年：2016
• 卷：141
• 期：Complete
• 页码：147-183
• 全文大小：635 K

文摘

We consider the asymptotic normalcy of families of random variables X   which count the number of occupied sites in some large set. If inlineImage" height="24" width="128" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0097316516000224-si1.gif">iner hidden">$P(z)={\sum }_{j=0}^N{p}_j{z}^j$ is the generating function associated to the random sets (i.e., there are p_jiner hidden">$p_j$ choices of random sets with j   occupied sites), we will consider the probability measures Prob(X=m)=p_mz^m/P(z)iner hidden">$\mathrm{Prob}(X=m)=p_m{z}^m/P(z)$, for z   real positive. We give sufficient criteria, involving the location of the zeros of P(z)iner hidden">$P(z)$, for these families to satisfy a central limit theorem (CLT) and even a local CLT (LCLT); the theorems hold in the sense of estimates valid for large N   (we assume that Var(X)iner hidden">$\mathrm{Var}(X)$ is large when N is). For example, if all the zeros lie in the closed left half plane then X is asymptotically normal, and when the zeros satisfy some additional conditions then X satisfies an LCLT. We apply these results to cases in which X counts the number of edges in the (random) set of “occupied” edges in a graph, with constraints on the number of occupied edges attached to a given vertex. Our results also apply to systems of interacting particles, with X   counting the number of particles in a box Λ whose size |Λ|iner hidden">$|\mathrm{\Lambda }|$ approaches infinity; P(z)iner hidden">$P(z)$ is then the grand canonical partition function and its zeros are the Lee–Yang zeros.