# Extreme values for two-dimensional discrete Gaussian free field

@article{Ding2012ExtremeVF, title={Extreme values for two-dimensional discrete Gaussian free field}, author={Jian Ding and Ofer Zeitouni}, journal={arXiv: Probability}, year={2012} }

We consider in this paper the collection of near maxima of the discrete, two dimensional Gaussian free field in a box with Dirichlet boundary conditions. We provide a rough description of the geometry of the set of near maxima, estimates on the gap between the two largest maxima, and an estimate for the right tail up to a multiplicative constant on the law of the centered maximum.

## 38 Citations

Convergence in Law of the Maximum of the Two‐Dimensional Discrete Gaussian Free Field

- Mathematics
- 2013

We consider the discrete two‐dimensional Gaussian free field on a box of side length $N$, with Dirichlet boundary data, and prove the convergence of the law of the centered maximum of the field.©…

Ballot Theorems for the Two-Dimensional Discrete Gaussian Free Field

- Mathematics
- 2020

We provide uniform bounds and asymptotics for the probability that a two-dimensional discrete Gaussian free field on an annulus-like domain and with Dirichlet boundary conditions stays negative as…

Extreme value statistics of 2d Gaussian Free Field: effect of finite domains

- Mathematics
- 2015

We study minima statistics of the 2d Gaussian Free Field on circles in the unit disk with Dirichlet boundary condition. Free energy distributions of the associated Random Energy models are exactly…

Thick points of high-dimensional Gaussian free fields

- Computer Science, MathematicsAnnales de l'Institut Henri Poincaré, Probabilités et Statistiques
- 2018

This work adopts a sphere averaging regularization to study polynomial-correlated Gaussian Free Fields in higher-than-two dimensions and introduces the definition of thick points which, heuristically speaking, are points where the value of theGaussian Free Field is unusually large.

Poisson-Dirichlet Statistics for the extremes of the two-dimensional discrete Gaussian free field

- Mathematics
- 2013

In a previous paper, the authors introduced an approach to prove that the statistics of the extremes of a log-correlated Gaussian field converge to a Poisson-Dirichlet variable at the level of the…

Tightness of the recentered maximum of log-correlated Gaussian fields

- Mathematics, Computer Science
- 2013

This work proves tightness of the recentered maximum of the Gaussian fields and provides exponentially decaying bounds on the right and left tails and applies this result to a version of the two-dimensional continuous Gaussian free field.

Extremes of the 2d scale-inhomogeneous discrete Gaussian free field: Sub-leading order and tightness

- Mathematics
- 2019

This is the first of a three paper series in which we present a comprehensive study of the extreme value theory of the scale-inhomogeneous discrete Gaussian free field. This model was introduced by…

Extrema of the two-dimensional Discrete Gaussian Free Field

- Mathematics
- 2017

These lecture notes offer a gentle introduction to the two-dimensional Discrete Gaussian Free Field with particular attention paid to the scaling limits of the level sets at heights proportional to…

Full extremal process, cluster law and freezing for the two-dimensional discrete Gaussian Free Field

- Mathematics
- 2016

Convergence of the centered maximum of log-correlated Gaussian fields

- Mathematics
- 2015

We show that the centered maximum of a sequence of log-correlated Gaussian fields in any dimension converges in distribution, under the assumption that the covariances of the fields converge in a…

## References

SHOWING 1-10 OF 41 REFERENCES

Extremes of the discrete two-dimensional Gaussian free field

- Mathematics
- 2006

We consider the lattice version of the free field in two dimensions and study the fractal structure of the sets where the field is unusually high (or low). We then extend some of our computations to…

Recursions and tightness for the maximum of the discrete, two dimensional Gaussian Free Field

- Mathematics
- 2010

We consider the maximum of the discrete two dimensional Gaussian free field in a box, and prove the existence of a (dense) deterministic subsequence along which the maximum, centered at its mean, is…

Entropic repulsion and the maximum of the two-dimensional harmonic crystal

- Mathematics
- 2001

We consider the lattice version of the free field in two dimensions (also called harmonic crystal). The main aim of the paper is to discuss quantitatively the entropic repulsion of the random surface…

Exponential and double exponential tails for maximum of two-dimensional discrete Gaussian free field

- Mathematics
- 2011

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 exponential…

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 longstanding conjecture. The proof…

Thick Points of the Gaussian Free Field

- Mathematics
- 2009

Xia Hua The d − dimensional Gaussian free field (GFF) is a natural d − dimensional dimensional time analog of Brownian motion. It places an important role in statistical physics and the theory of…

Statistical mechanics of logarithmic REM: duality, freezing and extreme value statistics of 1/f noises generated by Gaussian free fields

- Mathematics
- 2009

We compute the distribution of the partition functions for a class of one-dimensional random energy models with logarithmically correlated random potential, above and at the glass transition…

The one-sided barrier problem for Gaussian noise

- Mathematics
- 1962

This paper is concerned with the probability, P[T, r(τ)], that a stationary Gaussian process with mean zero and covariance function r(τ) be nonnegative throughout a given interval of duration T.…

Is the critical percolation probability local?

- Mathematics
- 2009

We show that the critical probability for percolation on a d-regular non-amenable graph of large girth is close to the critical probability for percolation on an infinite d-regular tree. We also…

Cover times for Brownian motion and random walks in two dimensions

- Mathematics
- 2001

LetT (x;") denote the rst hitting time of the disc of radius " centered at x for Brownian motion on the two dimensional torus T 2 . We prove that sup x2T2T (x;")=j log"j 2 ! 2= as " ! 0. The same…