TY - JOUR

T1 - Maximum and Shape of Interfaces in 3D Ising Crystals

AU - Gheissari, Reza

AU - Lubetzky, Eyal

N1 - Publisher Copyright:
© 2020 Wiley Periodicals LLC

PY - 2020

Y1 - 2020

N2 - 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 Mn, its maximum: if the inverse temperature β is large enough, then Mn/logn → 2/αβ as n → ∞, in probability, where αβ is given by a large-deviation rate in infinite volume. We further show that, on the large-deviation event that the interface connects the origin to height h, it consists of a 1D spine that behaves like a random walk in that it decomposes into a linear (in h) number of asymptotically stationary, weakly dependent increments that have exponential tails. As the number T of increments diverges, properties of the interface such as its surface area, volume, and the location of its tip, all obey CLTs with variances linear in T. These results generalize to every dimension d ≥ 3.

AB - 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 Mn, its maximum: if the inverse temperature β is large enough, then Mn/logn → 2/αβ as n → ∞, in probability, where αβ is given by a large-deviation rate in infinite volume. We further show that, on the large-deviation event that the interface connects the origin to height h, it consists of a 1D spine that behaves like a random walk in that it decomposes into a linear (in h) number of asymptotically stationary, weakly dependent increments that have exponential tails. As the number T of increments diverges, properties of the interface such as its surface area, volume, and the location of its tip, all obey CLTs with variances linear in T. These results generalize to every dimension d ≥ 3.

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U2 - 10.1002/cpa.21960

DO - 10.1002/cpa.21960

M3 - Article

AN - SCOPUS:85094659202

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

ER -