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 d ynamics of spin models over finitely connected random graphs. We identify a threshold value for the connectivity beyond which the loop structure of th e graph becomes thermodynamically relevant. Glauber dynamics simulations sh ow that this loop structure is responsible for the onset of dynamical featu res of a local character (dynamical heterogeneities and spontaneous time sc ale separation), consistently with previous (experimental and numerical) st udies of glasses and spin glasses in their approach to the low temperature phase. [S1063-651X(99)51002-7].