Analytical results for random walks in the presence of disorder and traps

TL;DR

This paper provides exact analytical results for the behavior of a random walk in a disordered one-dimensional lattice with traps, clarifying previous misconceptions about multifractality and deriving asymptotic decay laws for survival probability.

Contribution

It offers exact calculations of escape probabilities and survival decay in disordered traps, correcting prior numerical claims of multifractality.

Findings

Escape probabilities do not exhibit multifractality.

Survival probability decays as exp(-C t^{1/3} log^{2/3}(t)).

Derived a lower bound with similar decay behavior.

Abstract

In this paper, we study the dynamics of a random walker diffusing on a disordered one-dimensional lattice with random trappings. The distribution of escape probabilities is computed exactly for any strength of the disorder. These probabilities do not display any multifractal properties contrary to previous numerical claims. The explanation for this apparent multifractal behavior is given, and our conclusion are supported by numerical calculations. These exact results are exploited to compute the large time asymptotics of the survival probability (or the density) which is found to decay as $\exp [-Ct^{1/3}\log^{2/3}(t)]$. An exact lower bound for the density is found to decay in a similar way.