2014/08/18 by Joel L. Lebowitz, Boris Pittel, Lebowitz, J. L. +5 · 2 citations
Mathematics · #05A16 #05C30 (Primary) #05C31 #05C80 #60C05 #60F05 #82B05 (Secondary) #82B20 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Random Matrices and Applications #Statistical Mechanics (cond-mat.stat-mech) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1408.4153
openalex publication_date 2014/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the asymptotic normalcy of families of random variables X which count the number of occupied sites in some large set. We write Prob(X=m)=pmz0m/P(z0), where P(z) is the generating function P(z)=∑j=0Npjzj and z0>0. We give sufficient criteria, involving the location of the zeros of 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) 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 approaches infinity; P(z) is then the grand canonical partition function and its zeros are the Lee-Yang zeros.