Dobrushin (1972) showed that the interface of a 3D Ising model with minus boundary conditions above the @math -plane and plus below is rigid (has @math -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 @math with Dobrushin's boundary conditions, and in particular obtain a law of large numbers for @math , its maximum: if the inverse-temperature @math is large enough, then @math as @math , in probability, where @math 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 @math , it consists of a 1D spine that behaves like a random walk, in that it decomposes into a linear (in @math ) number of asymptotically-stationary weakly-dependent increments that have exponential tails. As the number @math 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 @math . These results generalize to every dimension @math .
Much progress has been made in recent years on understanding the distribution of the maximum of the 2D discrete Gaussian free field and its local geometry. It is known for instance ( @cite_14 @cite_20 @cite_18 ; see also, e.g., @cite_23 ) that this maximum is tight around an expected maximum that is asymptotically @math , and that the centered maximum has the law of a randomly shifted Gumbel random variable.
@cite_18 @cite_14 @cite_23 @cite_20
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We consider the discrete two-dimensional Gaussian free field on a box of side length @math , with Dirichlet boundary data, and prove the convergence of the law of the centered maximum of the field.© 2015 Wiley Periodicals, Inc. We consider the lattice version of the free eld in two dimensions (also called harmonic crystal). The main aim of the paper is to discuss quantitatively the entropic repulsion of the random surface in the presence of a hard wall. The basic ingredient of the proof is the analysis of the maximum of the eld which requires a multiscale analysis reducing the problem essentially to a problem on a eld with a tree structure. 2000 MSC: 60K35, 60G15, 82B41 We consider the maximum of the discrete two-dimensional Gaussian free field (GFF) in a box and prove that its maximum, centered at its mean, is tight, settling a longstanding conjecture. The proof combines a recent observation by Bolthausen, Deuschel, and Zeitouni with elements from Bramson's results on branching Brownian motion and comparison theorems for Gaussian fields. An essential part of the argument is the precise evaluation, up to an error of order 1, of the expected value of the maximum of the GFF in a box. Related Gaussian fields, such as the GFF on a two-dimensional torus, are also discussed. © 2011 Wiley Periodicals, Inc.