Dobrushin in 1972 showed that the interface of a 3D Ising model with minus boundary conditions above the xy-plane and plus below is rigid (has O(1) fluctuations) at every sufficiently low temperature. Since then, basic features of this interface—such as the asymptotics of its maximum—were only identified in more tractable random surface models that approximate the Ising interface at low temperatures, e.g., for the (2+1)D solid-on-solid model. Here we study the large deviations of the interface of the 3D Ising model in a cube of side length n with Dobrushin's boundary conditions, and in particular obtain a law of large numbers for M_n, its maximum: if the inverse temperature β is large enough, then M_n/logn → 2/α_β as n → ∞, in probability, where α_β is given by a large-deviation rate in infinite volume.

中文翻译：

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3D Ising 晶体中界面的最大值和形状

Dobrushin 在 1972 年表明，在xy平面上方和下方具有负边界条件的 3D Ising 模型的界面在每个足够低的温度下都是刚性的（具有O (1)波动）。从那时起，该界面的基本特征（例如其最大值的渐近线）仅在更易于处理的随机表面模型中识别出来，这些模型在低温下近似于 Ising 界面，例如，对于 (2+1)D solid-on-solid模型。在这里，我们研究了边长为n的立方体中的 3D Ising 模型的界面在 Dobrushin 边界条件下的大偏差，特别是获得了M _n的大数定律，它的最大值：如果逆温度β足够大，则M _n /log n  → 2/ α _β as n  → ∞，在概率上，其中α _β由无限体积中的大偏差率给出。