On the Ising model with random boundary condition

TL;DR

This paper investigates the behavior of the 2D Ising model with random boundary conditions, demonstrating that only the pure phases are typical limits and analyzing the absence of interfaces in large volumes.

Contribution

It provides a rigorous proof that the plus and minus phases are the only almost sure limit Gibbs measures under sparse boundary conditions, using multi-scale contour analysis.

Findings

Chaotic size-dependence at low temperatures.

Only '+' and '-' phases are almost sure limits.

Typical configurations contain no interfaces in large volumes.

Abstract

The infinite-volume limit behavior of the 2d Ising model under possibly strong random boundary conditions is studied. The model exhibits chaotic size-dependence at low temperatures and we prove that the `+' and `-' phases are the only almost sure limit Gibbs measures, assuming that the limit is taken along a sparse enough sequence of squares. In particular, we provide an argument to show that in a sufficiently large volume a typical spin configuration under a typical boundary condition contains no interfaces. In order to exclude mixtures as possible limit points, a detailed multi-scale contour analysis is performed.