Thermodynamic limit of the first Lee-Yang zero

Abstract

We complete the verification of the 1952 Yang and Lee proposal that thermodynamic singularities are exactly the limits in ${\mathbb R}$ of finite-volume singularities in ${\mathbb C}$. For the Ising model defined on a finite $Λ\subset\mathbb{Z}^d$ at inverse temperature $β\geq0$ and external field $h$, let $α_1(Λ,β)$ be the modulus of the first zero (that closest to the origin) of its partition function (in the variable $h$). We prove that $α_1(Λ,β)$ decreases to $α_1(\mathbb{Z}^d,β)$ as $Λ$ increases to $\mathbb{Z}^d$ where $α_1(\mathbb{Z}^d,β)\in[0,\infty)$ is the radius of the largest disk centered at the origin in which the free energy in the thermodynamic limit is analytic. We also note that $α_1(\mathbb{Z}^d,β)$ is strictly positive if and only if $β$ is strictly less than the critical inverse temperature.