Dynamics of an Ising Spin Glass on the Bethe Lattice

Abstract

We study the dynamical low temperature behaviour of the Ising spin glass on the Bethe lattice. Starting from Glauber dynamics we propose a cavity like Ansatz that allows for the treatment of the slow (low temperature) part of dynamics. Assuming a continuous phase transitions and ultrametricity with respect to long time scales we approach the problem perturbatively near the critical temperature. The theory is formulated in terms of correlation-response-functions of arbitrary order. They can, however, be broken down completely to products of pair functions depending on two time arguments only. For binary couplings \(J=\pm I\) a spin glass solution is found which approaches the corresponding solution for the SK-model in the limit of high connectivity. For more general distributions \(P(J)\) no stable or marginal solution of this type appears to exist. The nature of the low temperature phase in this more general case is unclear.