Variable range random walk

TL;DR

This paper investigates how variable jump rates in random walks affect diffusion behavior, revealing phase transitions between non-diffusive and diffusive regimes depending on spatial dimension and decay rates.

Contribution

It introduces a model of random walks with distance-dependent jump rates and analyzes phase transitions in diffusion properties across different dimensions.

Findings

Transition from non-diffusive to diffusive behavior in 1D with exponential decay

Existence of transition in 3D with super Gaussian decay

Diffusion properties depend on decay rate and spatial dimension

Abstract

Exploiting the coherent medium approximation, random walk among sites distributed randomly in space is investigated when the jump rate depends on the distance between two adjacent sites. In one dimension, it is shown that when the jump rate decays exponentially in the long distance limit, a non-diffusive to diffusive transition occurs as the density of sites is increased. In three dimensions, the transition exists when the jump rate has a super Gaussian decay.