We study the statics and dynamics of the low-temperature ferromagnetic Ising model on random d-regular graphs. Dembo and Montanari (2010) showed that the model converges locally weakly to a 50/50-mixture of the plus and minus Gibbs measures on the infinite d-regular tree, and the free energy converges to corresponding stable fixpoints of tree recursions. We obtain the large deviation rate function for the magnetization, and the local weak limit of the measure when fixed to have atypical magnetization, above the Kesten-Stigum threshold on the tree (but well into the low-temperature regime). The understanding of the free energy as a function of the magnetization also has interesting implications for phase ordering in low-temperature Glauber dynamics.
This is based mostly on joint work with W. Perkins and C. Yap.