Strongly vertex-reinforced jump process on a complete graph

TL;DR

This paper investigates vertex-reinforced jump processes with super-linear weights on complete graphs, proving that such processes almost surely concentrate infinitely on a single vertex while remaining bounded on others.

Contribution

It establishes the almost sure localization of super-linear vertex-reinforced jump processes on a single vertex in complete graphs, a novel result in reinforcement process analysis.

Findings

Almost sure localization on a single vertex

Finite total time on other vertices

Behavior depends on super-linear weight growth

Abstract

The aim of our work is to study vertex-reinforced jump processes with super-linear weight function $w(t) = t^\alpha$ , for some $\alpha>1$. On any complete graph $G = (V, E)$, we prove that there is one vertex $v \in V$ such that the total time spent at $v$ almost surely tends to infinity while the total time spent at the remaining vertices is bounded.