Deducing a variational principle with minimal a priori assumptions

TL;DR

This paper introduces a robust method to derive variational and large deviation principles for graph homomorphisms with minimal assumptions, relying mainly on a discrete Kirszbraun theorem and elementary tools.

Contribution

It provides a general, assumption-light framework for deducing key principles in graph homomorphism models, extending to complex and subtle cases.

Findings

Method requires only a discrete Kirszbraun theorem and basic combinatorics.

Proof avoids concentration inequalities and strict convexity assumptions.

Applicable to complex models like homogenization of limit shapes.

Abstract

We study the well-known variational and large deviation principle for graph homomorphisms from $\mathbb{Z}^m$ to $\mathbb{Z}$. We provide a robust method to deduce those principles under minimal a priori assumptions. The only ingredient specific to the model is a discrete Kirszbraun theorem i.e. an extension theorem for graph homomorphisms. All other ingredients are of a general nature not specific to the model. They include elementary combinatorics, the compactness of Lipschitz functions and a simplicial Rademacher theorem. Compared to the literature, our proof does not need any other preliminary results like e.g. concentration or strict convexity of the local surface tension. Therefore, the method is very robust and extends to more complex and subtle models, as e.g. the homogenization of limit shapes or graph-homomorphisms to a regular tree.