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 (Formula presented.) of finite-volume singularities in (Formula presented.). For the Ising model defined on a finite (Formula presented.) at inverse temperature (Formula presented.) and external field h, let (Formula presented.) be the modulus of the first zero (that closest to the origin) of its partition function (in the variable h). We prove that (Formula presented.) decreases to (Formula presented.) as Λ increases to (Formula presented.) where (Formula presented.) 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 (Formula presented.) is strictly positive if and only if β is strictly less than the critical inverse temperature.