Localization of maximal entropy random walk

TL;DR

This paper introduces a maximal entropy random walk process that localizes in defect-free regions of irregular lattices, revealing a classical localization phenomenon linked to Lifshitz states.

Contribution

It defines a new maximal entropy random walk model and demonstrates its localization behavior on irregular lattices with weak dilution, connecting it to Lifshitz states.

Findings

Maximal entropy random walk localizes in defect-free regions.

Localization is explained via Lifshitz states.

Behavior differs from generic random walk on irregular lattices.

Abstract

We define a new class of random walk processes which maximize entropy. This maximal entropy random walk is equivalent to generic random walk if it takes place on a regular lattice, but it is not if the underlying lattice is irregular. In particular, we consider a lattice with weak dilution. We show that the stationary probability of finding a particle performing maximal entropy random walk localizes in the largest nearly spherical region of the lattice which is free of defects. This localization phenomenon, which is purely classical in nature, is explained in terms of the Lifshitz states of a certain random operator.