A sluggish random walk with subdiffusive spread

TL;DR

This paper analyzes a one-dimensional sluggish random walk with space-dependent transition probabilities, revealing subdiffusive behavior, a nontrivial position distribution, and survival probabilities, with a generalization for different decay rates.

Contribution

It introduces a model with logarithmically increasing trap depths, showing subdiffusive scaling and nontrivial distribution features, extending understanding of trap-influenced random walks.

Findings

Typical position scales as t^{1/3}

Position distribution has a cusp at the origin

Survival probability decays as t^{-1/3}

Abstract

We study a one-dimensional sluggish random walk with space-dependent transition probabilities between nearest-neighbour lattice sites. Motivated by trap models of slow dynamics, we consider a model in which the trap depth increases logarithmically with distance from the origin. This leads to a random walk which has symmetric transition probabilities that decrease with distance $|k|$ from the origin as $1/|k|$ for large $|k|$. We show that the typical position after time $t$ scales as $t^{1/3}$ with a nontrivial scaling function for the position distribution which has a trough (a cusp singularity) at the origin. Therefore an effective central bias away from the origin emerges even though the transition probabilities are symmetric. We also compute the survival probability of the walker in the presence of a sink at the origin and show that it decays as $t^{-1/3}$ at late times. Furthermore we compute the distribution of the maximum position, $M(t)$, to the right of the origin up to time $t$, and show that it has a nontrivial scaling function. Finally we provide a generalisation of this model where the transition probabilities decay as $1/|k|^\alpha$ with $\alpha >0$.