Localization of a vertex reinforced random walks on $\Z$ with sub-linear weights

TL;DR

This paper studies vertex reinforced random walks on the integer lattice with sub-linear weights, characterizing conditions for localization on finite intervals and estimating the size of the localization set.

Contribution

It provides a characterization of localization behavior for sub-linearly reinforced walks and constructs examples with specific localization sizes.

Findings

Localization occurs on finite intervals under certain weight regularity conditions.

For any odd number N ≥ 5, there exists a walk localizing on exactly N sites with positive probability.

The size of the localization set can be precisely estimated based on the weight function.

Abstract

We consider a vertex reinforced random walk on the integer lattice with sub-linear reinforcement. Under some assumptions on the regular variation of the weight function, we characterize whether the walk gets stuck on a finite interval. When this happens, we estimate the size of the localization set. In particular, we show that, for any odd number $N$ larger than or equal to 5, there exists a vertex reinforced random walk which localizes with positive probability on exactly $N$ consecutive sites.