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

TL;DR

This work studies the limit behavior of linearly edge-reinforced random walks on the half-line with reinforcement $\delta$ and initial edge weights $w_0(x)=1$ for $x=0,1$ and $w_0(x)=x^{\alpha}\ln^{\beta}x$ for $x\ge 2$. Building on Takei's recurrence/transience dichotomy, the authors establish precise almost-sure limsup growth rates in the recurrent regime, including a logarithmic scaling for $\alpha<1$ and nuanced borderline behavior when $\alpha=1$ depending on $\beta$. The analysis combines a random-environment representation for the reinforced process, detailed asymptotics of edge weights via $S_x=\ln\gamma_x$, and Stolz–Cesàro arguments to translate weight-driven quantities into pathwise growth of $X_n$. The results refine the understanding of reinforcement effects on long-term behavior and extend Takei's results to a broader class of slowly varying initial weights with potential implications for related random walks in random environments.

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 $δ>0$, and each edge $\{x,x+1\}$ has the initial weight $x^α\ln^βx$ 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.

Theorems & Definitions (18)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Remark 2.7
• Theorem 2.8
• Lemma 3.1
• Lemma 3.2