Thermodynamic limit of the first Lee-Yang zero

We complete the verification of the 1952 Yang and Lee proposal that thermodynamic singularities are exactly the limits in $R$ of finite-volume singularities in $C$. For the Ising model defined on a finite $\Lambda \subset Z^d$ at inverse temperature $\beta \ge 0$ and external field h, let ${\alpha }_1(\Lambda ,\beta )$ be the modulus of the first zero (that closest to the origin) of its partition function (in the variable h). We prove that ${\alpha }_1(\Lambda ,\beta )$ decreases to ${\alpha }_1(Z^d,\beta )$ as Λ increases to $Z^d$ where ${\alpha }_1(Z^d,\beta )\in [0,\infty )$ 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 ${\alpha }_1(Z^d,\beta )$ is strictly positive if and only if β is strictly less than the critical inverse temperature.