Ferromagnetic Ising Measures on Large Locally Tree-Like Graphs

TL;DR

This paper studies the behavior of ferromagnetic Ising models on large, locally tree-like graphs, showing convergence of measures to a mixture of boundary conditions on trees and identifying conditions for positive magnetization limits.

Contribution

It generalizes previous results by establishing local weak convergence of Ising measures on graphs converging to trees, under a continuity property, and characterizes the measure conditioned on positive magnetization.

Findings

Ising measures converge to a mixture of boundary conditions on the limiting tree.

Edge-expander graphs have Ising measures conditioned on positive magnetization converging to the plus boundary condition.

The continuity property holds for all but countably many inverse temperature values within specific intervals.

Abstract

We consider the ferromagnetic Ising model on a sequence of graphs $G_n$ converging locally weakly to a rooted random tree. Generalizing [Montanari, Mossel, Sly '11], under an appropriate "continuity" property, we show that the Ising measures on these graphs converge locally weakly to a measure, which is obtained by first picking a random tree, and then the symmetric mixture of Ising measures with $+$ and $-$ boundary conditions on that tree. Under the extra assumptions that $G_n$ are edge-expanders, we show that the local weak limit of the Ising measures conditioned on positive magnetization, is the Ising measure with $+$ boundary condition on the limiting tree. The "continuity" property holds except possibly for countably many choices of $\beta$, which for limiting trees of minimum degree at least three, are all within certain explicitly specified compact interval. We further show the edge-expander property for (most of) the configuration model graphs corresponding to limiting (multi-type) Galton Watson trees.