The scaling limit of the membrane model

Probability to be positive for the membrane model in dimensions 2 and 3

“…If ψ was C 0, 4−n 2 -Hölder continuous (with Hölder constant 1), this event would have a positive probability uniformly in N and L. Now ψ is only C 0, 4−n 2 −ε -Hölder continuous (see [MS17], [CDH18]), so we cannot expect a lower bound independent of N . Instead, we prove in Subsection 3.2 that the probability of Ω VN−L,∞ is bounded below by e −c N n−1 (L+1) n−1 .…”

“…Corollary 1.2 easily implies that conditioning on Ω δN,+ does not change the order of the maximum of the field. Indeed the Hölder continuity results from [CDH18] (see Corollary 2.2 there) imply that N − 4−n 2 max x∈VN ψ x converges in distribution to a non-concentrated random variable M . By the Borell-TIS inequality the random variables N − 4−n 2 max x∈VN ψ x have sub-Gaussian tails uniformly in N and therefore…”

“…In [MS17], Müller and Schweiger obtained very precise estimates on the Green's function of the discrete Bilaplacian in the subcritical dimensions 2 and 3. These results in particular imply that the membrane model is Hölder continous, [MS17,CDH18]. Here we use the estimates to provide bounds for the probability of the interface to be positive on certain subsets of its domain.…”