论文标题
几乎可以确保线性边缘增强的随机步行在半线上
Almost sure behavior of linearly edge-reinforced random walks on the half-line
论文作者
论文摘要
我们在$ \ mathbb {z} _+$上学习线性边缘增强的随机步行，其中每个边缘$ \ {x，x，x+1 \} $具有初始权重$ x^α\ vee 1 $，并且每次边缘穿越时，其重量都会增加$δ$。众所周知，仅当$α\ leq 1 $时，步行是经常出现的。本文的目的是研究经常性制度中步行的几乎确定行为。对于$α<1 $和$δ> 0 $，我们获得了一个限制定理，这是迭代对数法律法律的对应物，用于简单的随机步行。这表明，$δ> 0 $的步行速度比$δ= 0 $慢得多。在关键情况下，$α= 1 $，我们的步行轨迹（几乎可以肯定）界限表明，速度在$δ= 2 $中存在相变。
We study linearly edge-reinforced random walks on $\mathbb{Z}_+$, where each edge $\{x,x+1\}$ has the initial weight $x^α \vee 1$, 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 $α\leq 1$. 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 $Δ=2$.
