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

Abstract

We consider the membrane model on a box $V_{N}\subset \mathbb{Z} ^{n}$ of size $(2N+1)^{n}$ with zero boundary condition in the subcritical dimensions $n=2$ and $n=3$. We show optimal estimates for the probability that the field is positive in a subset $D_{N}$ of $V_{N}$. In particular we obtain for $D_{N}=V_{N}$ that the probability to be positive on the entire domain is exponentially small and the rate is of the order of the surface area $N^{n-1}$.