Time transformation for random walks in the quenched trap model

TL;DR

This paper introduces a new approach called weak subordination breaking to analyze diffusion in the quenched trap model, mapping it onto Brownian motion and generalizing previous RG solutions to all disorder strengths.

Contribution

It presents an alternative method to the RG approach for studying the quenched trap model, applicable across different disorder regimes.

Findings

Derived the operational time ${\ m S}_\alpha$ for the model

Mapped the quenched trap problem onto Brownian motion

Generalized the solution beyond zero temperature

Abstract

Diffusion in the quenched trap model is investigated with an approach we call weak subordination breaking. We map the problem onto Brownian motion and show that the operational time is ${\cal S}_\alpha = \sum_{x=-\infty} ^\infty (n_x)^\alpha$ where $n_x$ is the visitation number at site $x$ . In the limit of zero temperature we recover the renormalization group (RG) solution found by Monthus. Our approach is an alternative to RG capable of dealing with any disorder strength.