Loop-weighted Walk

TL;DR

This paper proves that loop-weighted walks, a non-Markovian model related to statistical mechanics, exhibit diffusive behavior in high dimensions for all non-negative parameters, despite their complex loop-dependent weighting.

Contribution

It demonstrates diffusive behavior of loop-weighted walks in high dimensions using lace expansion, overcoming challenges posed by their non-repulsive nature.

Findings

Loop-weighted walk is diffusive in high dimensions for all λ ≥ 0.

Lace expansion technique successfully applied to a non-repulsive model.

The model relates to the loop O(N) model in statistical mechanics.

Abstract

Loop-weighted walk with parameter $\lambda\geq 0$ is a non-Markovian model of random walks that is related to the loop $O(N)$ model of statistical mechanics. A walk receives weight $\lambda^{k}$ if it contains $k$ loops; whether this is a reward or punishment for containing loops depends on the value of $\lambda$. A challenging feature of loop-weighted walk is that it is not purely repulsive, meaning the weight of the future of a walk may either increase or decrease if the past is forgotten. Repulsion is typically an essential property for lace expansion arguments. This article circumvents the lack of repulsion and proves, via the lace expansion, that for any $\lambda\geq 0$ loop-weighted walk is diffusive in high dimensions.