On Anti-Confinement Estimates for Self-Repelling Random Walks

TL;DR

This paper investigates self-repelling random walks in multiple dimensions, establishing lower bounds on diffusion and demonstrating superdiffusive behavior with a focus on the interplay between interaction range and decay rates.

Contribution

It introduces new lower bounds on diffusion constants and characterizes superdiffusive regimes for self-repelling walks using advanced correlation inequalities and multi-scale analysis.

Findings

Lower bounds on diffusion constants for short-range interactions

Superdiffusive behavior for sufficiently long-range interactions

Trade-off between temporal decay and spatial repulsion in superdiffusive regime

Abstract

We study a class of $d$-dimensional random walks, including the two-dimensional simple random walk, reweighted by a self-repelling Gibbsian pair potential. We prove lower bounds on the diffusion constant for short-range interactions, and superdiffusive behavior in case the interaction is sufficiently long-range. Finally, we show that in the superdiffusive regime, faster temporal decay can be compensated by stronger spatial repulsion and vice-versa. Our technique combines GKS-based correlation inequalities on path space with recursive multi-scale estimates.