Homomorphisms from the torus
In this talk, we'll present a detailed analysis of the set of weighted homomorphisms from the discrete torus $\mathbb{Z}_m^n$, where $m$ is even, to any fixed graph. We'll 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. We'll discuss some applications of this result which include solutions to conjectures of Engbers and Galvin and a conjecture of Kahn and Park. We'll also introduce some classical tools from statistical physics which form the basis of our proofs.
This is joint work with Peter Keevash.