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

We consider the asymptotic normalcy of families of random variables X   which count the number of occupied sites in some large set. If P(z)=∑j=0Npjzj is the generating function associated to the random sets (i.e., there are pjpj choices of random sets with j   occupied sites), we will consider the probability measures Prob(X=m)=pmzm/P(z)Prob(X=m)=pmzm/P(z), for z   real positive. We give sufficient criteria, involving the location of the zeros of 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 Var(X)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||Λ| approaches infinity; P(z)P(z) is then the grand canonical partition function and its zeros are the Lee–Yang zeros.