The weak limit of Ising models on locally tree-like graphs

TL;DR

This paper studies the behavior of the Ising model on sequences of graphs that resemble regular trees, showing that the local measure converges to a mixture of boundary-conditioned measures, with detailed results for expander graphs.

Contribution

It establishes the local weak convergence of the Ising measure on graphs converging to a regular tree and characterizes the limit as a mixture of boundary-conditioned measures, especially for expanders.

Findings

Ising measure converges to a symmetric mixture of boundary-conditioned measures on the tree.

On expander graphs, the measure conditioned on positive magnetization converges to the + boundary measure.

Provides a detailed understanding of the Ising model's local behavior on locally tree-like graphs.

Abstract

We consider the Ising model with inverse temperature beta and without external field on sequences of graphs G_n which converge locally to the k-regular tree. We show that for such graphs the Ising measure locally weak converges to the symmetric mixture of the Ising model with + boundary conditions and the - boundary conditions on the k-regular tree with inverse temperature \beta. In the case where the graphs G_n are expanders we derive a more detailed understanding by showing convergence of the Ising measure condition on positive magnetization (sum of spins) to the + measure on the tree.