Homomorphisms from the torus

TL;DR

This paper analyzes weighted homomorphisms from the discrete torus to any graph, revealing their probabilistic structure and deriving sharp asymptotics for combinatorial counts like independent sets and colorings.

Contribution

It provides a detailed probabilistic and structural analysis of homomorphisms from the torus, solving several longstanding conjectures and applying methods from physics and combinatorics.

Findings

Distribution close to a constructed dominant phase

Sharp asymptotics for independent sets and colorings

Disproved a conjecture of Kahn and Lawrenz

Abstract

We present a detailed probabilistic and structural analysis of the set of weighted homomorphisms from the discrete torus $\mathbb{Z}_m^n$, where $m$ is even, to any fixed graph: we show that the corresponding probability distribution on such homomorphisms is close to a distribution defined constructively as a certain random perturbation of some dominant phase. This has several consequences, including solutions (in a strong form) to conjectures of Engbers and Galvin and a conjecture of Kahn and Park. Special cases include sharp asymptotics for the number of independent sets and the number of proper $q$-colourings of $\mathbb{Z}_m^n$ (so in particular, the discrete hypercube). We give further applications to the study of height functions and (generalised) rank functions on the discrete hypercube and disprove a conjecture of Kahn and Lawrenz. For the proof we combine methods from statistical physics, entropy and graph containers and exploit isoperimetric and algebraic properties of the torus.