Thermodynamic and Scaling Limits of the non-Gaussian Membrane Model

TL;DR

This paper investigates the large-scale behavior of a non-Gaussian membrane model, a type of random interface, establishing its thermodynamic and scaling limits across various dimensions using advanced mathematical techniques.

Contribution

It introduces a unified approach to analyze the scaling limits, Gaussian approximations, and infinite volume limits of the non-Gaussian membrane model across multiple dimensions.

Findings

Unified framework for scaling limits in dimensions 2 and 3

Gaussian approximation in negative regularity spaces for all d ≥ 2

Infinite volume limit established for d ≥ 5

Abstract

We characterize the behavior of a random discrete interface $\phi$ on $[-L,L]^d \cap \mathbb{Z}^d$ with energy $\sum V(\Delta \phi(x))$ as $L \to \infty$, where $\Delta$ is the discrete Laplacian and $V$ is a uniformly convex, symmetric, and smooth potential. The interface $\phi$ is called the non-Gaussian membrane model. By analyzing the Helffer-Sj\"ostrand representation associated to $\Delta \phi$, we provide a unified approach to continuous scaling limits of the rescaled and interpolated interface in dimensions $d=2,3$, Gaussian approximation in negative regularity spaces for all $d \geq 2$, and the infinite volume limit in $d \geq 5$. Our results generalize some of those of arXiv:1801.05663.