Almost sure behavior of linearly edge-reinforced random walks on the half-line
| Autor: | Masato Takei |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Statistics and Probability
Phase transition Probability (math.PR) Law of the iterated logarithm Edge (geometry) Simple random sample Random walk Combinatorics Mathematics::Probability FOS: Mathematics Half line Limit (mathematics) Statistics Probability and Uncertainty Mathematics - Probability Mathematics |
| Zdroj: | Electronic Journal of Probability. 26 |
| ISSN: | 1083-6489 |
| Popis: | We study linearly edge-reinforced random walks on $\mathbb{Z}_+$, where each edge $\{x,x+1\}$ has the initial weight $x^{\alpha} \vee 1$, and each time an edge is traversed, its weight is increased by $\Delta$. It is known that the walk is recurrent if and only if $\alpha \leq 1$. The aim of this paper is to study the almost sure behavior of the walk in the recurrent regime. For $\alpha0$, 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 $\Delta>0$ is much slower than $\Delta=0$. In the critical case $\alpha=1$, our (almost sure) bounds for the trajectory of the walk shows that there is a phase transition of the speed at $\Delta=2$. Comment: 18 pages, with minor updates |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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