# Tightness and tails of the maximum in 3D Ising interfaces

@article{Gheissari2019TightnessAT,
title={Tightness and tails of the maximum in 3D Ising interfaces},
author={Reza Gheissari and Eyal Lubetzky},
journal={arXiv: Probability},
year={2019}
}
• Published 2019
• Mathematics, Physics
• arXiv: Probability
Consider the 3D Ising model on a box of side length $n$ with minus boundary conditions above the $xy$-plane and plus boundary conditions below it. At low temperatures, Dobrushin (1972) showed that the interface separating the predominantly plus and predominantly minus regions is localized: its height above a fixed point has exponential tails. Recently, the authors proved a law of large numbers for the maximum height $M_n$ of this interface: for every $\beta$ large, $M_n/ \log n\to c_\beta$ in… Expand
1 Citations

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#### References

SHOWING 1-10 OF 30 REFERENCES
Maximum and shape of interfaces in 3D Ising crystals.
• Mathematics, Physics
• 2019
Dobrushin (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 lowExpand
Scaling limit and cube-root fluctuations in SOS surfaces above a wall
• Mathematics, Physics
• 2013
Consider the classical $(2+1)$-dimensional Solid-On-Solid model above a hard wall on an $L\times L$ box of $\bbZ^2$. The model describes a crystal surface by assigning a non-negative integer heightExpand
Dynamics of $(2+1)$-dimensional SOS surfaces above a wall: Slow mixing induced by entropic repulsion
• Mathematics, Physics
• 2012
We study the Glauber dynamics for the $(2+1)\mathrm{D}$ Solid-On-Solid model above a hard wall and below a far away ceiling, on an $L\times L$ box of $\mathbb{Z}^2$ with zero boundary conditions, atExpand
Large deviations and continuum limit in the 2D Ising model
• Mathematics
• 1997
Summary. We study the 2D Ising model in a rectangular box ΛL of linear size O(L). We determine the exact asymptotic behaviour of the large deviations of the magnetization ∑t∈ΛLσ(t) when L→∞ forExpand
Harmonic Pinnacles in the Discrete Gaussian Model
• Mathematics, Physics
• 2014
The 2D Discrete Gaussian model gives each height function $${\eta : {\mathbb{Z}^2\to\mathbb{Z}}}$$η:Z2→Z a probability proportional to $${\exp(-\beta \mathcal{H}(\eta))}$$exp(-βH(η)), whereExpand
Exponential and double exponential tails for maximum of two-dimensional discrete Gaussian free field
We study the tail behavior for the maximum of discrete Gaussian free field on a 2D box with Dirichlet boundary condition after centering by its expectation. We show that it exhibits an exponentialExpand
Interface, Surface Tension and Reentrant Pinning Transition in the 2d Ising Model
We develop a new way to look at the high-temperature representation of the Ising model up to the critical temperature and obtain a number of interesting consequences. In the two-dimensional case, itExpand
Tightness of the recentered maximum of the two-dimensional discrete Gaussian Free Field
• Mathematics
• 2010
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 long-standing conjecture. The proofExpand
The Low-Temperature Expansion of the Wulff Crystal in the 3D Ising Model
• Mathematics
• 2001
Abstract: We compute the expansion of the surface tension of the 3D random cluster model for q≥ 1 in the limit where p goes to 1. We also compute the asymptotic shape of a plane partition of n as nExpand
Random surfaces in statistical mechanics: Roughening, rounding, wetting,...
• Physics
• 1986
In this paper we study several problems in statistical mechanics involving systems of fluctuating extended objects, such as interacting steps and domain walls. We reconsider the roughening transitionExpand