Maximum of the Gaussian interface model in random external fields

TL;DR

This paper investigates how random external fields influence the maximum height of a Gaussian interface in high dimensions, revealing that the tail behavior of the disorder significantly alters the asymptotic maximum.

Contribution

It provides the first analysis of the maximum of Gaussian interfaces under quenched disorder, identifying the asymptotic behavior based on the tail properties of the external fields.

Findings

Maximum height asymptotics depend on the tail behavior of the disorder.

In dimensions d ≥ 5, the asymptotic maximum is characterized explicitly.

Disorder can either raise or lower the typical maximum height depending on its distribution.

Abstract

We consider the Gaussian interface model in the presence of random external fields, that is the finite volume (random) Gibbs measure on $\mathbb{R}^{\Lambda_N}$, $\Lambda_N=[-N, N]^d\cap \mathbb{Z}^d$ with Hamiltonian $H_N(\phi)= \frac{1}{4d}\sum\limits_{x\sim y}(\phi(x)-\phi(y))^2-\sum\limits_{x\in \Lambda_N}\eta(x)\phi(x)$ and $0$-boundary conditions. $\{\eta(x)\}_{x\in \mathbb{Z}^d}$ is a family of i.i.d. symmetric random variables. We study how the typical maximal height of a random interface is modified by the addition of quenched bulk disorder. We show that the asymptotic behavior of the maximum changes depending on the tail behavior of the random variable $\eta(x)$ when $d\geq 5$. In particular, we identify the leading order asymptotics of the maximum.