Course 8 Dynamics of trap models

Abstract

The chapter discusses the dynamics of trap models; the Bouchaud trap model and the one-dimensional trap model. The definition of the Bouchaud trap model for a general graph and a general “depth” landscape is given. The very specific one-dimensional case (the graph here is Z), is studied in much detail. The results and the methods are different from all other situations studied and depend on the scaling limit introduced by Fontes, Isopi, and Newman. This scaling limit is an interesting self-similar singular diffusion, which quite easily gives results about aging, subaging and the “environment seen from the particle”. It considers BTM (Z, τ, a) on the one-dimensional lattice, Z, with nearest-neighbor edges. At present, the one-dimensional BTM is the only case where the asymmetric variant (a > 0) is rigorously studied. The most useful feature of the one-dimensional BTM is that its scaling limit, identified as an interesting one-dimensional singular diffusion in random environment introduced by Fontes, Isopi, and Newman. The Random Energy Model is the simplest mean-field model for spin-glasses and its static behaviour is well understood. The studies of dynamics are much sparser.