Variational problems with percolation: rigid spin systems

TL;DR

This paper studies the asymptotic behavior of rigid spin lattice energies, revealing a continuous interfacial limit characterized by a surface tension related to percolation theory, with implications for homogenization.

Contribution

It introduces a novel characterization of the interfacial limit energy for rigid spin systems using first-passage percolation, connecting percolation thresholds with homogenization limits.

Findings

The limit energy exhibits a non-trivial structure below the percolation threshold.

The surface tension is explicitly characterized via a percolation formula.

Homogenization of rigid spin systems is shown to be a special case of elliptic random homogenization.

Abstract

In this paper we describe the asymptotic behavior of rigid spin lattice energies by exhibiting a continuous interfacial limit energy as scaling to zero the lattice spacing. The limit is not trivial below a percolation threshold: it can be characterized by two phases separated by an interface. The macroscopic surface tension at this interface is defined through a first-passage percolation formula, related to the chemical distance on the square lattice. We also show a continuity result, that is the homogenization of rigid spin system is a limit case of the elliptic random homogenization.