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Convergence of the centered maximum of log-correlated Gaussian fields

Jian Ding, Rishideep Roy, Ofer Zeitouni

Abstract

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 suitable sense. We identify the limit as a randomly shifted Gumbel distribution, and characterize the random shift as the limit in distribution of a sequence of random variables, reminiscent of the derivative martingale in the theory of Branching Random Walk and Gaussian Chaos. We also discuss applications of the main convergence theorem and discuss examples that show that for logarithmically correlated fields, some additional structural assumptions of the type we make are needed for convergence of the centered maximum.

Convergence of the centered maximum of log-correlated Gaussian fields

Abstract

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 suitable sense. We identify the limit as a randomly shifted Gumbel distribution, and characterize the random shift as the limit in distribution of a sequence of random variables, reminiscent of the derivative martingale in the theory of Branching Random Walk and Gaussian Chaos. We also discuss applications of the main convergence theorem and discuss examples that show that for logarithmically correlated fields, some additional structural assumptions of the type we make are needed for convergence of the centered maximum.

Paper Structure

This paper contains 14 sections, 25 theorems, 174 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Expectation and tightness for the maximum
  3. Branching random walk and modified branching random walk
  4. Comparison of right tails
  5. An upper bound on the left tail
  6. Robustness of the maximum under perturbations
  7. Maximal sum over restricted pairs
  8. Proof of Lemma \ref{['lem-loc1']}
  9. Proof of Lemmas \ref{['lem-nfmax']} and \ref{['lem-epsilon-domination']}
  10. Proofs of Theorems \ref{['thm-convergence']} and \ref{['explicit-limit-law']}
  11. An approximation of the log-correlated Gaussian field
  12. Convergence in law for the centered maximum
  13. An example: the circular logarithmic REM
  14. Proof of Proposition \ref{['prop-rttailrefined']}

Key Result

Proposition 1.1

Under Assumption (A.0), there exists a constant $C = C(\alpha_0)> 0$ such that for all $N \in \mathbb{N}$ and $z \ge 1$, Furthermore, for all $z \ge 1, y \ge 0$ and $A \subseteq V_N$ we have

Figures (2)

  • Figure 1: Perturbation levels of the Gaussian field
  • Figure 2: Hierarchy of construction of the approximating Gaussian field

Theorems & Definitions (51)

  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Example 1.7
  • Example 1.8
  • Lemma 2.1
  • Lemma 2.2
  • ...and 41 more