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Consider the free field on a fractal graph based on a high-dimensionalSierpinski carpet (e.g. the Menger sponge), that is, a centered Gaussian fieldwhose 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 fieldstays positive everywhere. We prove the leading-order asymptotics for the localsample mean of the free field above the hard wall on any transient Sierpinskicarpet graph, thereby extending a result of Bolthausen, Deuschel, and Zeitounifor the free field on $\mathbb{Z}^d$, $d \geq 3$, to the fractal setting. Ourproof utilizes the theory of transient regular Dirichlet forms, in conjunctionwith the relative entropy, coarse graining, and conditioning argumentsintroduced in the previous literature.

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