Height-offset variables and pinning at infinity for gradient Gibbs measures on trees

TL;DR

This paper investigates height-offset variables for gradient Gibbs measures on trees, establishing their existence, properties, and the trade-offs involved in 'pinning at infinity' which affects structural features of the measures.

Contribution

It extends Sheffield's framework to broader classes of models, proving existence, differentiability, and concentration of HOVs, and analyzing the structural loss due to pinning at infinity.

Findings

HOVs exist as martingale limits

Lebesgue densities of HOVs are infinitely differentiable

Pinned Gibbs measures lose automorphism invariance and extremality

Abstract

Height-offset variables (HOVs) provide a mechanism, known as "pinning at infinity", to lift gradient Gibbs measures (GGMs) - describing interface increments - to proper Gibbs measures that describe absolute heights. Starting from Sheffield's seminal framework, we study HOVs for nearest-neighbor integer-valued gradient models on regular trees, under broad classes of transfer operators requiring only finite second moments and without assuming convexity. We first establish the existence of HOVs as martingale limits, prove the infinite differentiability of their Lebesgue densities, and demonstrate exponential concentration for the associated pinned Gibbs measures. Next we uncover a fundamental trade-off, as the Gibbs measures arising by "pinning at infinity" paradoxically lose several desirable structural properties. We rigorously show that they lose tree-automorphism invariance, the tree-indexed Markov chain property, and extremality within the class of Gibbs measures. Our analysis relies on martingale theory, novel past- and future-tail decompositions, and infinite product representations for moment generating functions, and it applies to free GGMs, as well as to GGMs of height-period two.