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Extremal process of the zero-average Gaussian Free Field for $d\ge 3$

Sayan Das, Rajat Subhra Hazra

Abstract

We consider the Gaussian free field on the torus whose covariance kernel is given by the zero-average Green's function. We show that for dimension $d\ge 3$, the extremal point process associated with this field converges weakly to a Poisson random measure. As an immediate corollary, the maxima of the field converges after appropriate centering and scaling to the Gumbel distribution.

Extremal process of the zero-average Gaussian Free Field for $d\ge 3$

Abstract

We consider the Gaussian free field on the torus whose covariance kernel is given by the zero-average Green's function. We show that for dimension , the extremal point process associated with this field converges weakly to a Poisson random measure. As an immediate corollary, the maxima of the field converges after appropriate centering and scaling to the Gumbel distribution.

Paper Structure

This paper contains 7 sections, 9 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Notations and main result
  3. Main results
  4. Some known facts
  5. Outline of Proof of Theorem \ref{['zagff']}
  6. Proof of Lemma \ref{['slut']}
  7. Proof of Lemma \ref{['tech']}

Key Result

Theorem \oldthetheorem

For the sequence of point processes $\eta_n$ defined in pp1 we have that as $n \to \infty$, where $\eta$ is a Poisson random measure on $E$ with intensity measure given by $dt\otimes e^{-z} dz$ where $dt\otimes dz$ is the Lebesgue measure on $E$, and $\stackrel{d}{\to}$ is the convergence in distribution on the space $\mathcal{M}_p(E)$.

Theorems & Definitions (10)

  • Definition \oldthetheorem
  • Theorem \oldthetheorem
  • Corollary \oldthetheorem
  • Lemma \oldthetheorem: Theorem 1.5.4 of lawler2013intersections
  • Lemma \oldthetheorem: Proposition 4.6.2(a) of lawler2010random
  • Lemma \oldthetheorem: Lemma 1.3 of angelo
  • Lemma \oldthetheorem: Proposition 1.4 of angelo
  • Lemma \oldthetheorem: Theorem 2.3 of angelo
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem