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

Abstract

We consider the asymptotic normalcy of families of random variables $X$ which count the number of occupied sites in some large set. We write $Prob(X=m)=p_mz_0^m/P(z_0)$, where $P(z)$ is the generating function $P(z)=\sum_{j=0}^{N}p_jz^j$ and $z_0>0$. 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 $\Lambda$ whose size approaches infinity; $P(z)$ is then the grand canonical partition function and its zeros are the Lee-Yang zeros.