A sluggish random walk with subdiffusive spread

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