Delocalization of uniform graph homomorphisms from $\mathbb{Z}^2$ to $\mathbb{Z}$

TL;DR

This paper proves that in two dimensions, uniform graph homomorphisms from bZ^2 to bZ do not exhibit localization and lack translation-invariant Gibbs measures, with additional results in higher dimensions about ergodic measures.

Contribution

The paper establishes delocalization of uniform graph homomorphisms in two dimensions and characterizes ergodic Gibbs measures in higher dimensions.

Findings

No translation-invariant Gibbs measures in 2D.

Ergodic Gibbs measures in higher dimensions are extremal.

Gibbs measures are stochastically ordered.

Abstract

Graph homomorphisms from the $\mathbb{Z}^d$ lattice to $\mathbb{Z}$ are functions on $\mathbb{Z}^d$ whose gradients equal one in absolute value. These functions are the height functions corresponding to proper $3$-colorings of $\mathbb{Z}^d$ and, in two dimensions, corresponding to the $6$-vertex model (square ice). We consider the uniform model, obtained by sampling uniformly such a graph homomorphism subject to boundary conditions. Our main result is that the model delocalizes in two dimensions, having no translation-invariant Gibbs measures. Additional results are obtained in higher dimensions and include the fact that every Gibbs measure which is ergodic under even translations is extremal and that these Gibbs measures are stochastically ordered.