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 $\mathrm{\Lambda }\subset {\mathbb{Z}}^d$ at inverse temperature $\beta \ge 0$ and external field h, let ${\alpha }_1(\mathrm{\Lambda },\beta )$ be the modulus of the first zero (that closest to the origin) of its partition function (in the variable h). We prove that ${\alpha }_1(\mathrm{\Lambda },\beta )$ decreases to ${\alpha }_1({\mathbb{Z}}^d,\beta )$ as Λ increases to ${\mathbb{Z}}^d$ where $\text{◂+▸}{\alpha }_1({\mathbb{Z}}^d,\beta )\in [0,\infty )$ 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 β is strictly less than the critical inverse temperature.

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第一李杨零的热力学极限

我们完成了对 1952 年 Yang 和 Lee 提议的验证，即热力学奇点正是$\mathbb{右}$有限体积奇点$\mathbb{C}$。对于定义在有限域上的 Ising 模型$\mathrm{\Lambda }\subset {\mathbb{Z}}^d$在逆温度下$\beta \ge 0$和外场h，让${\alpha }_1（\mathrm{\Lambda },\beta ）$是其配分函数（在变量h中）的第一个零（最接近原点）的模。我们证明${\alpha }_1（\mathrm{\Lambda },\beta ）$减少到${\alpha }_1（{\mathbb{Z}}^d,\beta ）$当 Λ 增加到${\mathbb{Z}}^d$在哪里$\text{◂+▸}{\alpha }_1（{\mathbb{Z}}^d,\beta ）\epsilon [0,\mathrm{无穷大}）$是以原点为中心的最大圆盘的半径，其中热力学极限中的自由能是解析的。我们还注意到${\alpha }_1（{\mathbb{Z}}^d,\beta ）$当且仅当 β 严格小于临界逆温度时，严格为正。