A note on the Lee-Yang theorem

Abstract

The Lee-Yang theorem [l] concerns the distribution of zeros of the partition function Z and moments (a,) associated with a ferromagnetic Gibbs measure. Arising in the statistical mechanics of crystals, the ferromagnetic Gibbs measures are a class of finite measures on IR”. They depend on N real parameters h,, h2,..., h, representing an impressed magnetic field. They also depend on the choice of an even preliminary measure v on IR: the single-spin measure. Gibbs measures have natural complex continuations to complex values of the parameters h,. The partition function and moments can be continued as well, yielding functions Z(&,..., hN) and (aA; h,,...,h,) of the h k’ As generalized by Newman [2], the Lee-Yang theorem states roughly that Z@ 1,---v hN) and (ua; h 1,..., hN) have no zeros in the region (h,: Re(h,) > 0 Vk] if the moment-generating function Z(h) = f?‘, 8“ dv(o) of the preliminary measure has no zeros in {h: Re(h) > 0). However, present proofs of this theorem are somewhat indirect and complicated, resorting to multiple approximation procedures and structure theorems for analytic functions. The object of this note is to present a short, elementary proof of a strong form of the Lee-Yang theorem in the special case v(a) = S(a 1) + 6(cr + 1) of primary interest (the classical Ising ferromagnet). We use methods familiar in the study of correlation inequalities.