Localization of random walks to competing manifolds of distinct dimensions

TL;DR

This paper investigates how random walks localize near multiple manifolds of different dimensions, revealing complex phase behavior and confirming theoretical predictions through numerical simulations with surprising reentrant localization phenomena.

Contribution

It introduces a detailed analysis of random walk localization on competing manifolds of different dimensions, identifying multiple phases and a multi-critical point, supported by extensive numerical simulations.

Findings

Four distinct localization phases identified.

Numerical results confirm continuum theory predictions.

Unexpected reentrant localization to corners observed.

Abstract

We consider localization of a random walk (RW) when attracted or repelled by multiple extended manifolds of different dimensionalities. In particular, we focus on $(d-1)$- and $(d-2)$-dimensional manifolds in $d$-dimensional space, where attractive interactions are (fully or marginally) relevant. The RW can then be in one of four phases where it is localized to neither, one, or both manifolds. The four phases merge at a special multi-critical point where (away from the manifolds) the RW spreads diffusively. Extensive numerical analyses on two dimensional RWs confined inside or outside a rectangular wedge confirm general features expected from a continuum theory, but also exhibit unexpected attributes, such as a reentrant localization to the corner while repelled by it.