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 16000224&_mathId=si1.gif&_user=111111111&_pii=S0097316516000224&_rdoc=1&_issn=00973165&md5=cbc4b929d5d6cd420e67c84af5c7f994">16000224-si1.gif">$P(z)={\sum }_{j=0}^N{p}_j{z}^j$ is the generating function associated to the random sets (i.e., there are 16000224&_mathId=si2.gif&_user=111111111&_pii=S0097316516000224&_rdoc=1&_issn=00973165&md5=2752b617fe18a03aacfd6a38b0fc26a5" title="Click to view the MathML source">p_j$p_j$ choices of random sets with j   occupied sites), we will consider the probability measures 16000224&_mathId=si3.gif&_user=111111111&_pii=S0097316516000224&_rdoc=1&_issn=00973165&md5=589cc7813f1cd9ca9a90a7faf3056f8f" title="Click to view the MathML source">Prob(X=m)=p_mz^m/P(z)$\mathrm{Prob}(X=m)=p_m{z}^m/P(z)$, for z   real positive. We give sufficient criteria, involving the location of the zeros of 16000224&_mathId=si250.gif&_user=111111111&_pii=S0097316516000224&_rdoc=1&_issn=00973165&md5=d41aa953269b2589e41a33df4ef6b299" title="Click to view the MathML source">P(z)$P(z)$, 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 16000224&_mathId=si5.gif&_user=111111111&_pii=S0097316516000224&_rdoc=1&_issn=00973165&md5=71e248c2bf18b9e3f6c6815dae2254ea" title="Click to view the MathML source">Var(X)$\mathrm{Var}(X)$ 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 16000224&_mathId=si6.gif&_user=111111111&_pii=S0097316516000224&_rdoc=1&_issn=00973165&md5=0e557c7ebce5e640ef67de3772bbfb50" title="Click to view the MathML source">|Λ|$|\mathrm{\Lambda }|$ approaches infinity; 16000224&_mathId=si250.gif&_user=111111111&_pii=S0097316516000224&_rdoc=1&_issn=00973165&md5=d41aa953269b2589e41a33df4ef6b299" title="Click to view the MathML source">P(z)$P(z)$ is then the grand canonical partition function and its zeros are the Lee–Yang zeros.