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Maximum of the Gaussian interface model in random external fields

Hironobu Sakagawa

Abstract

We consider the Gaussian interface model in the presence of random external fields, that is the finite volume (random) Gibbs measure on $\mathbb{R}^{Λ_N}$, $Λ_N=[-N, N]^d\cap \mathbb{Z}^d$ with Hamiltonian $H_N(φ)= \frac{1}{4d}\sum\limits_{x\sim y}(φ(x)-φ(y))^2-\sum\limits_{x\in Λ_N}η(x)φ(x)$ and $0$-boundary conditions. $\{η(x)\}_{x\in \mathbb{Z}^d}$ 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 $η(x)$ when $d\geq 5$. In particular, we identify the leading order asymptotics of the maximum.

Maximum of the Gaussian interface model in random external fields

Abstract

We consider the Gaussian interface model in the presence of random external fields, that is the finite volume (random) Gibbs measure on , with Hamiltonian and -boundary conditions. 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 when . In particular, we identify the leading order asymptotics of the maximum.
Paper Structure (12 sections, 6 theorems, 114 equations)

This paper contains 12 sections, 6 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Model and result
  3. On the result and strategy of the proof
  4. Discussion
  5. Deviations of weighted sums of i.i.d. random variables
  6. Proof of Theorem \ref{['thm1']}
  7. Lower bound of the maximum
  8. Upper bound of the maximum
  9. Appendix
  10. The infinite volume case
  11. The lower dimensional case
  12. Entropic repulsion

Key Result

Theorem 1.1

Let $d\geq 5$ and $\eta=\{\eta(x)\}_{x\in \mathbb{Z}^d}$ be a family of i.i.d. symmetric random variables. Assume the condition $(A_\alpha)$ or $(\widetilde{A})$ and define $M^*$ as follows: where $G^*=(-\Delta)^{-1}(0, 0)$ and ${G}^{*}_{(\alpha)} =\sum\limits_{x\in \mathbb{Z}^d} ((-\Delta)^{-1}(0, x))^{\frac{\alpha}{\alpha-1}}$. Then, for every $\varepsilon, \varepsilon' >0$ we have and Moreov

Theorems & Definitions (21)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Proposition 2.1
  • Lemma 2.1
  • proof : Proof of Lemma \ref{['a1']} for the case $(A_\alpha)$ $0< \alpha <1$
  • proof : Proof of Lemma \ref{['a1']} for the case $(A_\alpha)$ $\alpha =1$
  • proof : Proof of Lemma \ref{['a1']} for the case $(A_\alpha)$ $1<\alpha \leq 2$
  • ...and 11 more