Localization on 4 sites for Vertex-reinforced random walks on $\mathbb Z$

TL;DR

This paper characterizes the conditions under which a vertex reinforced random walk on the integer line localizes on exactly 4 sites, revealing a phase transition based on the growth rate of the weight function.

Contribution

It provides a precise characterization of weight functions leading to localization on 4 sites and identifies a phase transition at the critical growth rate.

Findings

For weights growing faster than n log log n, VRRW localizes on at most 4 sites.

For weights growing slower than n log log n, VRRW cannot localize on fewer than 5 sites.

At the critical growth rate, localization occurs on either 4 or 5 sites with positive probability.

Abstract

We characterize non-decreasing weight functions for which the associated one-dimensional vertex reinforced random walk (VRRW) localizes on 4 sites. A phase transition appears for weights of order $n\log \log n$: for weights growing faster than this rate, the VRRW localizes almost surely on at most 4 sites whereas for weights growing slower, the VRRW cannot localize on less than 5 sites. When $w$ is of order $n\log \log n$, the VRRW localizes almost surely on either 4 or 5 sites, both events happening with positive probability.