Full extremal process in four-dimensional membrane model

TL;DR

This paper studies the extremal process of a four-dimensional membrane model, establishing its structure, uniqueness, and invariance properties, extending methods from the two-dimensional Gaussian free field to higher dimensions.

Contribution

It introduces a detailed analysis of the extremal process in 4D membrane models, including explicit formulas and a novel approach adapting 2D GFF techniques to 4D.

Findings

Cluster-like geometry of extreme points

Existence and uniqueness of the extremal process

Poisson structure with explicit formulas

Abstract

We investigate the extremal process of four-dimensional membrane models as the size of the lattice $N$ tends to infinity. We prove the cluster-like geometry of the extreme points and the existence as well as the uniqueness of the extremal process. The extremal process is characterized by a distributional invariance property and a Poisson structure with explicit formulas. Our approach follows the same philosophy as the two-dimensional Gaussian free field (2D GFF) via the comparison with the modified branching random walk. The proofs leverage the ``Dysonization'' technique and a careful treatment of the correlation structure, which is more intricate than the 2D GFF case. As a by-product, we also obtain a simpler proof of a crucial sprinkling lemma.