  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.
