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 ${\mathcal{S}}_{\alpha }={\sum }_{x=-\infty }^{\infty }{\left(n_x\right)}^{\alpha }$, with $n_x$ being the number of visits to site $x$. Here $0<\alpha =T/T_g<1$ 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 ${\mathcal{S}}_{\alpha }$ is changed to laboratory time $t$ with a Lévy time transformation. Investigation of Brownian motion stopped at time ${\mathcal{S}}_{\alpha }$ yields the diffusion front of the quenched trap model, which is favorably compared with numerical simulations. In the zero-temperature limit of $\alpha \to 0$ 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 $\alpha \to 1$. Above the critical dimension 2, mapping the problem to a continuous time random walk becomes feasible, though still not trivial.

• Received 29 May 2012

DOI:https://doi.org/10.1103/PhysRevE.86.041137