Height function localisation on trees

TL;DR

This paper investigates two models of discrete height functions on trees, demonstrating uniform localization properties and classifying extremal measures, with implications for understanding Gaussian free fields on trees.

Contribution

The paper introduces a complete classification of extremal gradient Gibbs measures and establishes uniform localization results for two models of height functions on trees.

Findings

Height variance is uniformly bounded across boundary conditions in the homomorphism model.

Complete classification of extremal gradient Gibbs measures on directed trees.

Identification of the localization-delocalization transition in the second model.

Abstract

We study two models of discrete height functions, that is, models of random integer-valued functions on the vertices of a tree. First, we consider the random homomorphism model, in which neighbours must have a height difference of exactly one. The local law is uniform by definition. We prove that the height variance of this model is bounded, uniformly over all boundary conditions (both in terms of location and boundary heights). This implies a strong notion of localisation, uniformly over all extremal Gibbs measures of the system. For the second model, we consider directed trees, in which each vertex has exactly one parent and at least two children. We consider the locally uniform law on height functions which are monotone, that is, such that the height of the parent vertex is always at least the height of the child vertex. We provide a complete classification of all extremal gradient Gibbs measures, and describe exactly the localisation-delocalisation transition for this model. Typical extremal gradient Gibbs measures are localised also in this case. Localisation in both models is consistent with the observation that the Gaussian free field is localised on trees, which is an immediate consequence of transience of the random walk.