The Replica Mean Field Theory For The Glass Matrix Model

We study the dynamical phase transition of the glass matrix model by using the replica method for deterministic models. In the glass matrix model, N dynamical variables are defined by P component vectors which make states of a particle. Each component of a vector is assumed to take +/- 1. To perform replica calculation, auxiliary variables are introduced to control the order of the sum of replicated partition function. These variables work like quenched random variables of the usual replica method. Using the approximation for small P/N and assuming the homogeneity of the component space, we find that the replica theory is similar to that of the previously studied glassy spin models. We study the dynamical solution, which is defined by imposing the marginally stable condition for a one-step replica symmetry breaking ansatz. To find the marginally stable condition, we study the fluctuation modes in replica spaces and component space. The results of simulated annealing is presented to compare with the analytic result. The transition temperatures suggested by the two approaches are consistent. However, the agreement is modest in the sense that simulated annealing shows richer behavior than the replica results, mainly due to the lower energy states, which appear by the slow annealing schedule.