The weak limit of Ising models on locally tree-like graphs

We consider the Ising model with inverse temperature β and without external field on sequences of graphs G _n which converge locally to the k-regular tree. We show that for such graphs the Ising measure locally weakly converges to the symmetric mixture of the Ising model with + boundary conditions and the - boundary conditions on the k-regular tree with inverse temperature β. In the case where the graphs G _n are expanders we derive a more detailed understanding by showing convergence of the Ising measure conditional on positive magnetization (sum of spins) to the + measure on the tree.