Second Order Asymptotics for the Hard Wall Probability of the 2D Harmonic Crystal

TL;DR

This paper refines the understanding of the probability that a 2D Gaussian free field remains positive within a domain, providing second-order asymptotics and revealing the tail decay behavior of the field's minimum.

Contribution

It improves previous results by deriving the second-order asymptotics for the positive probability of the field, using tail decay analysis of the field's minimum.

Findings

Derived the order of the subleading term in the probability estimate.

Established the double exponential decay of the right tail of the field's minimum.

Extended previous results to more general domain scalings.

Abstract

We estimate the probability that the discrete Gaussian free field on a planar domain with Dirichlet boundary conditions stays positive in the bulk. Improving upon the result by Bolthausen, Deuschel and Giacomin from 2001, we derive the order of the subleading term of this probability when a sequence of discretized scale-ups of given domain and compactly included smooth bulk are considered. A main ingredient in the proof is the double exponential decay of the right tail of the centered minimum of the field in the bulk, conditioned on a certain weighted average of its values to be zero.