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Ferromagnetic Ising measures on large locally tree-like graphs. (English) Zbl 1372.05032
Summary: We consider the ferromagnetic Ising model on a sequence of graphs $$G_{n}$$ converging locally weakly to a rooted random tree. Generalizing [A. Montanari, Probab. Theory Relat. Fields 152, No. 1–2, 31–51 (2012; Zbl 1242.82012)], 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 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.

##### MSC:
 05C05 Trees 05C80 Random graphs (graph-theoretic aspects) 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 arising in equilibrium statistical mechanics 82B26 Phase transitions (general) in equilibrium statistical mechanics
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