We study linearly edge-reinforced random walks on , where each edge has the initial weight for and 1 for , 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 . The aim of this paper is to study the almost sure behavior of the walk in the recurrent regime. For and , 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 is much slower than . In the critical case , our (almost sure) bounds for the trajectory of the walk shows that there is a phase transition of the speed at .
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.
"Almost sure behavior of linearly edge-reinforced random walks on the half-line." Electron. J. Probab. 26 1 - 18, 2021. https://doi.org/10.1214/21-EJP674