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

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 ${\mathbb{Z}}^d$, $d\ge 3$, 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.

v2: 35 pages, 3 figures. Accepted for publication in SPA

References in corpus (5)

• Quenched invariance principles for random walks with random conductances
• Invariance principle for the random conductance model with unbounded conductances
• Brownian motion on the Sierpinski carpet
• Extremes of the discrete two-dimensional Gaussian free field
• Non-Markov property of certain eigenvalue processes analogous to Dyson's model

Cited by in corpus (1)

• Branching random walk in the presence of a hard wall