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2021 Almost sure behavior of linearly edge-reinforced random walks on the half-line
Masato Takei
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Electron. J. Probab. 26: 1-18 (2021). DOI: 10.1214/21-EJP674


We study linearly edge-reinforced random walks on Z+, where each edge {x,x+1} has the initial weight xα for x>0 and 1 for x=0, and each time an edge is traversed, its weight is increased by Δ. It is known that the walk is recurrent if and only if α1. The aim of this paper is to study the almost sure behavior of the walk in the recurrent regime. For α<1 and Δ>0, we obtain a limit theorem which is a counterpart of the law of the iterated logarithm for simple random walks. This reveals that the speed of the walk with Δ>0 is much slower than Δ=0. In the critical case α=1, our (almost sure) bounds for the trajectory of the walk shows that there is a phase transition of the speed at Δ=2.

Funding Statement

M.T. is partially supported by JSPS Grant-in-Aid for Young Scientists (B) No. 16K21039, and JSPS Grant-in-Aid for Scientific Research (B) No. 19H01793 and (C) No. 19K03514.


To the memory of late Professor Munemi Miyamoto.


M.T. thanks an anonymous referee for detailed comments.


Download Citation

Masato Takei. "Almost sure behavior of linearly edge-reinforced random walks on the half-line." Electron. J. Probab. 26 1 - 18, 2021.


Received: 25 July 2020; Accepted: 27 June 2021; Published: 2021
First available in Project Euclid: 14 July 2021

Digital Object Identifier: 10.1214/21-EJP674

Primary: 60K35

Keywords: random walks in random environment , reinforced random walks


Vol.26 • 2021
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