The Annals of Probability

Ferromagnetic Ising measures on large locally tree-like graphs

Anirban Basak and Amir Dembo

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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.

Article information

Ann. Probab., Volume 45, Number 2 (2017), 780-823.

Received: December 2013
Revised: October 2015
First available in Project Euclid: 31 March 2017

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Zentralblatt MATH identifier

Primary: 05C05: Trees 05C80: Random graphs [See also 60B20] 05C81: Random walks on graphs 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B26: Phase transitions (general)

Ising model random sparse graphs Gibbs measures local weak convergence


Basak, Anirban; Dembo, Amir. Ferromagnetic Ising measures on large locally tree-like graphs. Ann. Probab. 45 (2017), no. 2, 780--823. doi:10.1214/15-AOP1075.

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