# Anderson-like localization transition of random walks with resetting

## TL;DR

This paper investigates a phase transition in lattice random walks with stochastic resetting, revealing a localization transition akin to Anderson localization, with implications for understanding spatial learning in organisms.

## Contribution

It introduces analytically tractable models of localization transitions in path-dependent stochastic processes with resetting and impurities.

## Key findings

- Localization occurs above a critical resetting rate.
- Critical exponents match those of Anderson localization.
- A single impurity site induces spontaneous localization.

## Abstract

We study several lattice random walk models with stochastic resetting to previously visited sites which exhibit a phase transition between an anomalous diffusive regime and a localization regime where diffusion is suppressed. The localized phase settles above a critical resetting rate, or rate of memory use, and the probability density asymptotically adopts in this regime a non-equilibrium steady state similar to that of the well known problem of diffusion with resetting to the origin. The transition occurs because of the presence of a single impurity site where the resetting rate is lower than on other sites, and around which the walker spontaneously localizes. Near criticality, the localization length diverges with a critical exponent that falls in the same class as the self-consistent theory of Anderson localization of waves in random media. The critical dimensions are also the same in both problems. Our study provides analytically tractable examples of localization transitions in path-dependent, reinforced stochastic processes, which can be also useful for understanding spatial learning by living organisms.