The critical Ising model on trees, concave recursions and nonlinear   capacity
  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 $n^{\alpha}b^n$ 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 $\alpha\le1/2$. Our proofs are based on a new link between nonlinear recursions on trees and $L^p$ capacities. 
Robin Pemantle, Yuval Peres