Calculus of Variations and Geometric Measure Theory

G. Scilla

Variational problems with percolation: rigid spin systems

created by scilla on 26 Nov 2012
modified on 17 Feb 2014


Published Paper

Inserted: 26 nov 2012
Last Updated: 17 feb 2014

Journal: Adv. Math. Sci. Appl.
Volume: 23
Number: 1
Pages: 187-207
Year: 2013


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 de fined through a first-passage percolation formula, related to the chemical distance on the lattice $\mathbb{Z}^2$. We also show a continuity result, that is the homogenization of rigid spin system is a limit case of the elliptic random homogenization.

Keywords: Gamma-convergence, Variational problems, lattice energies, first-passage percolation, rigid spins, chemical distance