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Near-maxima of the two-dimensional Discrete Gaussian Free Field

Marek Biskup, Stephan Gufler, Oren Louidor

Abstract

We consider the Discrete Gaussian Free Field (DGFF) in domains $D_N\subseteq\mathbb Z^2$ arising, via scaling by $N$, from nice domains $D\subseteq\mathbb R^2$. We study the statistics of the values order $\sqrt{\log N}$ below the absolute maximum. Encoded as a point process on $D\times\mathbb R$, the scaled spatial distribution of these near-extremal level sets in $D_N$ and the field values (in units of $\sqrt{\log N}$ below the absolute maximum) tends, as $N\to\infty$, in law to the product of the critical Liouville Quantum Gravity (cLQG) $Z^D$ and the Rayleigh law. The convergence holds jointly with the extremal process, for which $Z^D$ enters as the intensity measure of the limiting Poisson point process, and that of the DGFF itself; the cLQG defined by the limit field then coincides with $Z^D$. While the limit near-extremal process is measurable with respect to the limit continuum GFF, the limit extremal process is not. Our results explain why the various ways to "norm" the lattice cLQG measure lead to the same limit object, modulo overall normalization.

Near-maxima of the two-dimensional Discrete Gaussian Free Field

Abstract

We consider the Discrete Gaussian Free Field (DGFF) in domains arising, via scaling by , from nice domains . We study the statistics of the values order below the absolute maximum. Encoded as a point process on , the scaled spatial distribution of these near-extremal level sets in and the field values (in units of below the absolute maximum) tends, as , in law to the product of the critical Liouville Quantum Gravity (cLQG) and the Rayleigh law. The convergence holds jointly with the extremal process, for which enters as the intensity measure of the limiting Poisson point process, and that of the DGFF itself; the cLQG defined by the limit field then coincides with . While the limit near-extremal process is measurable with respect to the limit continuum GFF, the limit extremal process is not. Our results explain why the various ways to "norm" the lattice cLQG measure lead to the same limit object, modulo overall normalization.

Paper Structure

This paper contains 15 sections, 31 theorems, 179 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Preliminaries
  4. Green function
  5. Gibbs-Markov property
  6. Ballot estimates
  7. Characterizations of cLQG measure
  8. Moment calculations and estimates
  9. Strategy and key lemmas
  10. Truncated first moment
  11. Truncated second moment
  12. Proof of Theorem \ref{['thm-A']}
  13. Near-extremal process convergence
  14. Adding the extremal process
  15. Including the field itself

Key Result

Theorem \oldthetheorem

Let $D\in\mathfrak D$ and let $\{D_N\}_{N\ge1}$ be defined by E:2.1. Given a sample $h^{D_N}$ of the DGFF in $D_N$, define $\zeta_N^D$ by e:1.3 with $\alpha:=2/\!\sqrt{g}$ and regard $h^{D_N}$ as an element of $\mathbb R^{\mathcal{M}_{{\rm c}}(D)}$ via Let $\eta^D_N$ be the measure on the left of E:1.3. Then, relative to the vague topology on the space of Radon measures on $\overline D\times\over

Figures (2)

  • Figure 1: An illustration of event $E_j$ in the proof of Lemma \ref{['lemma-3.2a']}. A path of the simple random walk started at the center point winds around the annulus $A_j$ before exiting through the outer perimeter of $A_j$. The path necessarily crosses the shaded area which represents the complement of $D_N$.
  • Figure 2: An illustration of the sets defined in \ref{['E:5.42']} and \ref{['E:5.43']}. The square boxes $B_1$ and $B_2$ are centered at vertices $x$ and $y$, respectively. The shaded area marks the set $W'$ underlying the definitions of $B_1'$, $B_2'$ and $B_3'$.

Theorems & Definitions (65)

  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Theorem \oldthetheorem: Near-extremal process convergence
  • Remark \oldthetheorem
  • Corollary \oldthetheorem
  • Lemma \oldthetheorem: Green function asymptotic
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem: Gibbs-Markov property
  • ...and 55 more