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March 2017 Ferromagnetic Ising measures on large locally tree-like graphs
Anirban Basak, Amir Dembo
Ann. Probab. 45(2): 780-823 (March 2017). DOI: 10.1214/15-AOP1075


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.


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Anirban Basak. Amir Dembo. "Ferromagnetic Ising measures on large locally tree-like graphs." Ann. Probab. 45 (2) 780 - 823, March 2017.


Received: 1 December 2013; Revised: 1 October 2015; Published: March 2017
First available in Project Euclid: 31 March 2017

zbMATH: 1372.05032
MathSciNet: MR3630287
Digital Object Identifier: 10.1214/15-AOP1075

Primary: 05C05 , 05C80 , 05C81 , 60J80 , 82B20 , 82B26

Keywords: Gibbs measures , Ising model , Local weak convergence , random sparse graphs

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.45 • No. 2 • March 2017
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