Ballot Theorems for the Two-Dimensional Discrete Gaussian Free Field

TL;DR

This paper derives bounds and asymptotic probabilities for a 2D discrete Gaussian free field remaining negative within an annulus, revealing how these probabilities behave as the domain size increases.

Contribution

It provides the first uniform bounds and asymptotic analysis for the probability of the 2D discrete Gaussian free field staying negative in annular domains.

Findings

Established uniform bounds for the negative probability

Derived asymptotic behavior as domain size grows

Analyzed the influence of boundary conditions on the field

Abstract

We provide uniform bounds and asymptotics for the probability that a two-dimensional discrete Gaussian free field on an annulus-like domain and with Dirichlet boundary conditions stays negative as the ratio of the radii of the inner and the outer boundary tends to infinity.