The critical Ising model on trees, concave recursions and nonlinear capacity

Abstract

We consider the Ising model on a general tree under various boundary conditions: all plus, free and spin-glass. In each case, we determine when the root is influenced by the boundary values in the limit as the boundary recedes to infinity. We obtain exact capacity criteria that govern behavior at critical temperatures. For plus boundary conditions, an L 3 capacity arises. In particular, on a spherically symmetric tree that has n α b n vertices at level n (up to bounded factors), we prove that there is a unique Gibbs measure for the ferromagnetic Ising model at the relevant critical temperature if and only if α ≤ 1/2. Our proofs are based on a new link between nonlinear recursions on trees and L p capacities.

Mixing Time of Critical Ising Model on Trees is Polynomial in the Height

“…Unfortunately, in our case we have an arbitrary boundary condition, imposed by the block-dynamics. This eliminates the symmetry of the system, which was a crucial part of the arguments of [31]. The most delicate step in the proof of Theorem 1 is the extension of these results of [31] to any boundary condition.…”

“…This eliminates the symmetry of the system, which was a crucial part of the arguments of [31]. The most delicate step in the proof of Theorem 1 is the extension of these results of [31] to any boundary condition. This is achieved by carefully tracking down the effect of the boundary on the expected reconstruction result in each site, combined with correlation inequalities and an analytical study of the corresponding log-likelihood-ratio function.…”

“…This quantity was studied by [31] in the free-boundary case, where it was related to capacity-type parameters of the tree (see [15] for a related result corresponding to the high temperature regime). Unfortunately, in our case we have an arbitrary boundary condition, imposed by the block-dynamics.…”

“…The authors of [31] studied certain spatial mixing properties of the Ising model on trees (with free or all-plus boundary conditions), and related them to the L p -capacity of the underlying tree. In Sect.…”

“…The important case of the Ising model on a regular tree, known as the Bethe lattice, has been intensively studied (e.g., [3][4][5][6][7]15,17,18,23,27,31]). On this canonical example of a non-amenable graph (one whose boundary is proportional to its volume), the model exhibits a rich behavior.…”