Weak subordination breaking for the quenched trap model

Abstract

We map the problem of diffusion in the quenched trap model onto a different stochastic process: Brownian motion that is terminated at the coverage time S(α)=∑(x=-∞)(∞)(n(x))(α), with n(x) being the number of visits to site x. Here 0<α=T/T(g)<1 is a measure of the disorder in the original model. This mapping allows us to treat the intricate correlations in the underlying random walk in the random environment. The operational time S(α) is changed to laboratory time t with a Lévy time transformation. Investigation of Brownian motion stopped at time S(α) yields the diffusion front of the quenched trap model, which is favorably compared with numerical simulations. In the zero-temperature limit of α→0 we recover the renormalization group solution obtained by Monthus [Phys. Rev. E 68, 036114 (2003)]. Our theory surmounts the critical slowing down that is found when α→1. Above the critical dimension 2, mapping the problem to a continuous time random walk becomes feasible, though still not trivial.