Time scale separation and heterogeneous off-equilibrium dynamics in spin models over random graphs

We study analytically and numerically the statics and the off-equilibrium dynamics of spin models over finitely connected random graphs. We identify a threshold value for the connectivity beyond which the loop structure of the graph becomes thermodynamically relevant. Glauber dynamics simulations show that this loop structure is responsible for the onset of dynamical features of a local character (dynamical heterogeneities and spontaneous time scale separation), consistently with previous (experimental and numerical) studies of glasses and spin glasses in their approach to the low temperature phase.