Ferromagnetic Ising measures on large locally tree-like graphs

Abstract

We consider the ferromagnetic Ising model on a sequence of graphs $\mathsf{G}_{n}$ converging locally weakly to a rooted random tree. Generalizing [Probab. Theory Related Fields 152 (2012) 31–51], 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 $\mathsf{G}_{n}$ 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 countable 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.