Glauber dynamics of spin glasses at low and high temperature
The paper deals with Glauber dynamics for the Edwards-Anderson spin-glass model in two dimensions. The corresponding dynamics is a single site dynamics. The model is a short ranged Ising model spin-glass with interactions only between nearest neighbors. All interactions are taken with equal absolute value. The Hamiltonian is calculated from the pair interactions of spins within a finite region. The interaction parameters are ndependently identically distributed random variables with a Bernoulli distribution of parameter. It is found that there exists a critical temperature, separating two regions with qualitatively different relaxation speed to equilibrium. For low temperatures there is a slow relaxation to equilibrium, while for high temperatures there is a fast relaxation. More specifically the main steps are: Starting from the Hamiltonian the Gibbs measure is defined, and local irreducible Markov chains are considered, having the Gibbs measure as stationary measure. For two measures defined on the regular lattice a distance is defined. It is shown that there are spins, for which the autocorrelation time decays slowly to zero. The topology induced by the distance is used to study convergence of the measure to the stationary measure. For high temperature this distance decreases at least exponentially in time. For low temperatures a lower bound for the distance is given. The idea behind this is the observation that whenever there are some ferromagnetic zones in the spin system independent of the rest of the system, and if these regions are arbitrarily large, then in these regions the results on ferromagnetic systems can be used to prove a slow convergence to the equilibrium measure.
scientific article; zbMATH DE number 1879931