Variational problems with percolation: rigid spin systems

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.