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

TL;DR

This paper investigates the probability that the membrane model in dimensions 2 and 3 remains positive over a domain, revealing that this probability diminishes exponentially with the surface area of the domain.

Contribution

It provides optimal estimates for positivity probabilities of the membrane model in low dimensions, highlighting the exponential decay rate related to surface area.

Findings

Probability to be positive on the entire domain is exponentially small.

Decay rate of positivity probability is proportional to the surface area.

Results are optimal estimates 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 $N^{n-1}$.