Entropic repulsion of Gaussian free field on high-dimensional Sierpinski carpet graphs

TL;DR

This paper studies the behavior of a Gaussian free field on high-dimensional Sierpinski carpet graphs with a hard wall, extending known results from Euclidean lattices to fractal structures using advanced probabilistic methods.

Contribution

It extends the asymptotic analysis of entropic repulsion for Gaussian free fields from Euclidean spaces to fractal graphs, specifically high-dimensional Sierpinski carpets.

Findings

Established leading-order asymptotics for the local sample mean of the free field.

Extended previous Euclidean lattice results to fractal graph settings.

Utilized Dirichlet forms, entropy, and conditioning techniques in the proof.

Abstract

Consider the free field on a fractal graph based on a high-dimensional Sierpinski carpet (e.g. the Menger sponge), that is, a centered Gaussian field whose covariance is the Green's function for simple random walk on the graph. Moreover assume that a "hard wall" is placed at height zero so that the field stays positive everywhere. We prove the leading-order asymptotics for the local sample mean of the free field above the hard wall on any transient Sierpinski carpet graph, thereby extending a result of Bolthausen, Deuschel, and Zeitouni for the free field on $\mathbb{Z}^d$, $d \geq 3$, to the fractal setting. Our proof utilizes the theory of transient regular Dirichlet forms, in conjunction with the relative entropy, coarse graining, and conditioning arguments introduced in the previous literature.