Gaussian free field on the tree subject to a hard wall II: Asymptotics

TL;DR

This paper provides a detailed asymptotic analysis of the Gaussian free field on a binary tree conditioned to be positive at leaves, revealing the mutual singularity of conditioned and unconditioned laws.

Contribution

It offers a comprehensive, sharp asymptotic description of the field's law under the hard-wall constraint, extending previous bounds to detailed local and global statistics.

Findings

Asymptotics for local field statistics near a vertex

Asymptotics for global statistics like minimum, maximum, and empirical mean

Conditional and unconditional laws are asymptotically mutually singular

Abstract

This is the second 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 the first work ("Gaussian free field on the tree subject to a hard wall I: Bounds") we identified the repulsion profile followed by the field in order to fulfill this "hard-wall constraint" event. In this work, we use these findings to obtain a comprehensive, sharp asymptotic description of the law of the field under this conditioning. We provide asymptotics for both local statistics, namely the (conditional) law of the field in a neighborhood of a vertex, as well as global statistics, including the (conditional) law of the minimum, maximum, empirical population mean and all subcritical exponential martingales. We conclude that the laws of the conditional and unconditional fields are asymptotically mutually singular with respect to each other.