A variational principle for a non-integrable model

TL;DR

This paper introduces a new method to establish variational principles for non-integrable discrete systems, demonstrated through graph homomorphisms to regular trees, revealing a continuum of ergodic measures.

Contribution

It presents the first non-trivial variational principle for a non-integrable model using a novel technique based on a discrete Kirszbraun theorem and concentration inequalities.

Findings

Existence of a variational principle for graph homomorphisms to regular trees.

Discovery of a continuum of shift-invariant ergodic gradient Gibbs measures.

Development of a robust technique applicable to non-integrable systems.

Abstract

We develop a new robust technique to deduce variance principles for non-integrable discrete systems. To illustrate this technique, we show the existence of a variational principle for graph homomorphisms from $\Z^m$ to a $d$-regular tree. This seems to be the first non-trivial example of a variational principle in a non-integrable model. Instead of relying on integrability, the technique is based on a discrete Kirszbraun theorem and a concentration inequality obtained through the dynamic of the model. As a consequence of this result, we obtain the existence of a continuum of shift-invariant ergodic gradient Gibbs measures for graph homomorphisms from $\Z^m$ to a regular tree.