Gaussian free field on the tree subject to a hard wall I: Bounds

TL;DR

This paper analyzes the behavior of the discrete Gaussian free field on a binary tree with a positivity constraint at the leaves, providing sharp probability asymptotics and insights into the field's localization and fluctuations.

Contribution

It offers the first sharp asymptotics for the probability of the positivity constraint and characterizes the field's profile and fluctuations under this conditioning.

Findings

Sharp asymptotics for the positivity event probability

Identification of the field's repulsion profile

Estimates of mean, fluctuations, and covariances

Abstract

This is the first in a series of two works which study the discrete Gaussian free field on the binary tree when all leaves are conditioned to be positive. In this work, we obtain sharp asymptotics for the probability of this "hard-wall constraint" event, and identify the repulsion profile followed by the field in order to achieve it. We also provide estimates for the mean, fluctuations and covariances of the field under the conditioning, which show that in the first log-many generations the field is localized around its mean. These results are used in the sequel work ("Gaussian free field on the tree subject to a hard wall II: Asymptotics") to obtain a comprehensive asymptotic description of the law of the field under the conditioning.