The random walk penalised by its range in dimensions $d\geq 3$

TL;DR

This paper investigates a self-attractive random walk in dimensions d≥3, showing that its range concentrates around a Euclidean ball with radius proportional to N^{1/(d+2)}, confirming a conjecture for d=2.

Contribution

It extends the understanding of range behavior for self-attractive walks, proving a conjecture for higher dimensions and characterizing the shape of the walk's range.

Findings

Range is close to a Euclidean ball of radius ~ρ_d N^{1/(d+2)}

Confirms Bolthausen's conjecture for d=2

Provides explicit constant ρ_d for the radius approximation

Abstract

We study a self-attractive random walk such that each trajectory of length $N$ is penalised by a factor proportional to $\exp ( - |R_N|)$, where $R_N$ is the set of sites visited by the walk. We show that the range of such a walk is close to a solid Euclidean ball of radius approximately $\rho_d N^{1/(d+2)}$, for some explicit constant $\rho_d >0$. This proves a conjecture of Bolthausen who obtained this result in the case $d=2$.