Derivation of a stretched-exponential time relaxation

Abstract

On the basis of a simple model of time relaxation in disordered systems (such as glasses or spin glasses), the authors study the diffusion on clusters built on high-dimensional hypercubes. They show there exist two particular concentrations of available sites on the hypercube. The first one defines two distinct behaviours of relaxation: above p=1/2, the relaxation is purely exponential; below p=1/2, the relaxation is no longer exponential. The other case is of more interest: it is the percolation concentration. Using the unproven but argued assumption that, because of the high dimensionality of the considered space, the calculation may be mapped onto the problem of diffusion on a standard mean-field percolation cluster, they show that the relaxation should follow a stretched exponential law there with exponent 1/3. They briefly discuss the physical implications of this approach.