Limit behavior of linearly edge-reinforced random walks on the half-line

TL;DR

This paper investigates the long-term behavior of linearly edge-reinforced random walks on the half-line, focusing on the recurrent regime and extending previous results with specific initial edge weights.

Contribution

It extends prior work by analyzing the almost sure limit behavior of these walks with new initial weight configurations on the half-line.

Findings

Characterizes the limit behavior in the recurrent regime

Identifies conditions for recurrence and transience

Provides new insights into reinforced random walks on the half-line

Abstract

Motivated by the article [M. Takei, Electron. J. Probab. 26 (2021), article no. 104], we study the limit behavior of linearly edge-reinforced random walks on the half-line $\mathbb{Z}_+$ with reinforcement parameter $\delta>0$, and each edge $\{x,x+1\}$ has the initial weight $x^{\alpha}\ln^{\beta}x$ for $x > 1$ and $1$ for $x = 0, 1$. The aim of this paper is to study the almost sure limit behavior of the walk in the recurrent regime, and extend the results of Takei mentioned above.