Sherrington-Kirkpatrick Spin-Glass Dynamics

In this second part we want to describe the Glauber dynamics approach for the Sherrington-Kirkpatrick spin-glass dynamics. The physical motivation we have in mind is the same as described by Ben Arous and Guionnet in the proceedings for the “continuous setting.” Glauber dynamics means that we want to study a reversible Markov process directly for the Sherrington-Kirkpatrick (SK) Gibbs measures on the (discrete) state space {−1, 1}N, that is, in the “hard” spin picture. To formulate such dynamics, we have to use jump processes. The use of jump processes is really more natural in the SK model than the use of diffusions, although diffusions are closer to physical intuition. This jump process ansatz was introduced in [Som87]. Some rigorous results for asymmetric dynamics were proved in [Gru92]. There is a strong advantage in the use of Glauber dynamics. The Girsanov exponent, used to describe the interacting model, is fairly well behaved, which permits proving large deviation results without restriction on time and temperature. These large deviation results are strong enough to guaranty direct convergence results to limiting dynamics. As an additional present from exponential bounds, cpnvergence holds for almost all realizations of couplings J. The price we have to pay for the use of Glauber dynamics is the loss of Gaussian techniques, which makes it necessary to consider an enlarged state space (we include the “fields” in our picture), which in the end gives a more transparent description of limiting dynamics.