Enhanced Diffusivity in Perturbed Senile Reinforced Random Walk Models

TL;DR

This paper investigates how small perturbations to senile reinforced random walks enhance their diffusivity, providing quantitative bounds and extending results to higher dimensions where the unperturbed models are already diffusive.

Contribution

It introduces and analyzes perturbed senile reinforced random walk models, demonstrating enhanced diffusivity and deriving bounds in one and higher dimensions.

Findings

Perturbed SeRW models become diffusive with enhanced diffusivity for any positive perturbation.

Enhanced diffusivity scales with the perturbation size, e.g., \\gg O(\\delta^2) in small \\delta regime.

In higher dimensions, the diffusivity can be as large as O(\\log^{-2} \\delta).

Abstract

We consider diffusivity of random walks with transition probabilities depending on the number of consecutive traversals of the last traversed edge, the so called senile reinforced random walk (SeRW). In one dimension, the walk is known to be sub-diffusive with identity reinforcement function. We perturb the model by introducing a small probability $\delta$ of escaping the last traversed edge at each step. The perturbed SeRW model is diffusive for any $\delta >0 $, with enhanced diffusivity ($\gg O(\delta^2)$) in the small $\delta$ regime. We further study stochastically perturbed SeRW models by having the last edge escape probability of the form $\delta\, \xi_n$ with $\xi_n$'s being independent random variables. Enhanced diffusivity in such models are logarithmically close to the so called residual diffusivity (positive in the zero $\delta$ limit), with diffusivity between $O\left(\frac{1}{|\log\delta |}\right)$ and $O\left(\frac{1}{\log|\log\delta|}\right)$. Finally, we generalize our results to higher dimensions where the unperturbed model is already diffusive. The enhanced diffusivity can be as much as $O(\log^{-2}\delta)$.