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