Tail distributions of cover times of once-reinforced random walks

TL;DR

This paper studies the tail distribution of the edge cover time for a specific non-Markov process called once-reinforced random walk on finite graphs, revealing exponential decay and phase transition phenomena.

Contribution

It provides the first analysis of the tail behavior and phase transition of the edge cover time for once-reinforced random walks, including a variational characterization of the critical exponent.

Findings

Tail distribution decays exponentially

Identifies phase transition in exponential integrability

Provides variational representation of critical exponent

Abstract

We consider the tail distribution of the edge cover time of a specific non-Markov process, $\delta$ once-reinforced random walk, on finite connected graphs, whose transition probability is proportional to weights of edges. Here the weights are $1$ on edges not traversed and $\delta\in(0,\infty)$ otherwise. In detail, we show that its tail distribution decays exponentially, and obtain a phase transition of the exponential integrability of the edge cover time with critical exponent $\alpha_c^1(\delta)$, which has a variational representation and some interesting analytic properties including $\alpha_c^1(0+)$ reflecting the graph structures.