Ferromagnetic Ising measures on large locally tree-like graphs.

*(English)*Zbl 1372.05032Summary: 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 |