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 ${\mathbb R}$ of finite-volume singularities in ${\mathbb C}$. For the Ising model defined on a finite $\Lambda\subset\mathbb{Z}^d$ at inverse temperature $\beta\geq0$ 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 $\alpha_1(\mathbb{Z}^d,\beta)$ as $\Lambda$ increases to $\mathbb{Z}^d$ where $\alpha_1(\mathbb{Z}^d,\beta)\in[0,\infty)$ 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 $\alpha_1(\mathbb{Z}^d,\beta)$ is strictly positive if and only if $\beta$ is strictly less than the critical inverse temperature.