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Extremes of the zero-average Gaussian Free Field on random regular graphs

Lisa Hartung, Andreas Klippel, Christian Mönch

TL;DR

The paper proves that the rescaled extremal statistics of the zero-average Gaussian free field on random $r$-regular graphs and on $r$-regular trees converge to a Poisson point process on $\mathds{R}$ with intensity $e^{-x}\,dx$, yielding standard Gumbel fluctuations for the maximum. The core approach combines a general Gaussian comparison principle for joint extremal convergence with sharp Green function estimates, enabling transfer of i.i.d.-like extreme value behavior to correlated GFF fields on expanders. A detailed tree-based analysis establishes the PPP limit on the regular tree, while a novel Green-function approximation on random regular graphs plus a careful near/far decomposition extends the result to graphs themselves, proving universality for $r$-regular expanders. These methods provide a robust framework for extreme value analysis of Gaussian fields on sparse, high-girth graphs and may have broader applications in level-set percolation and related probabilistic graph models.

Abstract

We study the extreme value statistics of the zero-average Gaussian free field (GFF) on random $r$-regular graphs and the Gaussian free field on $r$-regular trees. For random $r$-regular graphs of diverging size, for every fixed $r\ge3$, we show that the rescaled extremal point process of the field is asymptotically distributed, in the annealed sense, as a Poisson point process on the line with intensity $e^{-x}\,\mathrm{d}x$. The same limit behaviour is obeyed by the restriction of the GFF on $r$-regular trees to finite subsets of vertices. Our approach relies on a direct Gaussian comparison argument and precise Green function estimates.

Extremes of the zero-average Gaussian Free Field on random regular graphs

TL;DR

The paper proves that the rescaled extremal statistics of the zero-average Gaussian free field on random -regular graphs and on -regular trees converge to a Poisson point process on with intensity , yielding standard Gumbel fluctuations for the maximum. The core approach combines a general Gaussian comparison principle for joint extremal convergence with sharp Green function estimates, enabling transfer of i.i.d.-like extreme value behavior to correlated GFF fields on expanders. A detailed tree-based analysis establishes the PPP limit on the regular tree, while a novel Green-function approximation on random regular graphs plus a careful near/far decomposition extends the result to graphs themselves, proving universality for -regular expanders. These methods provide a robust framework for extreme value analysis of Gaussian fields on sparse, high-girth graphs and may have broader applications in level-set percolation and related probabilistic graph models.

Abstract

We study the extreme value statistics of the zero-average Gaussian free field (GFF) on random -regular graphs and the Gaussian free field on -regular trees. For random -regular graphs of diverging size, for every fixed , we show that the rescaled extremal point process of the field is asymptotically distributed, in the annealed sense, as a Poisson point process on the line with intensity . The same limit behaviour is obeyed by the restriction of the GFF on -regular trees to finite subsets of vertices. Our approach relies on a direct Gaussian comparison argument and precise Green function estimates.

Paper Structure

This paper contains 6 sections, 10 theorems, 72 equations.

Table of Contents

  1. Introduction and main result
  2. A Gaussian comparison principle
  3. Extrema of the Gaussian Free field on the regular tree
  4. Extrema of the Gaussian Free field on the random regular graphs
  5. The zero-average Green function on $\mathscr{G}_N$
  6. Proof of main result

Key Result

Theorem 1.1

Let $r\ge3$. Then, with the notation above, as $N\to\infty$, there exist sets of graphs $\Omega_N,\,N\in\mathds{N}$, such that $\mathbf{P}(\Omega_N^\mathsf{c})=o(1)$ and on $\Omega_N$ Consequently, $(\mathcal{P}_N,\,M_N)\ \overset{\mathds{P}_N}\Longrightarrow\ (\mathcal{P}_\infty,\,M_\infty)$, which in particular implies that $M_N$ is asymptotically Gumbel-distributed, so that

Theorems & Definitions (17)

  • Theorem 1.1: Extremal point process and maximum on the random $r$-regular graph
  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3: berestycki2018extremes
  • proof : Proof of Theorem \ref{['thm:gauss-comp-ppp']}
  • Lemma 2.4
  • proof
  • Theorem 3.1: Extremal point process and maximum on the $r$-regular tree
  • proof
  • Proposition 4.1
  • ...and 7 more