论文标题
平衡兴奋的随机步行的概述
An overview of the balanced excited random walk
论文作者
论文摘要
Benjamini，Kozma和Schapira以$ 2011 $引入的平衡兴奋的随机步行被定义为$ \ Mathbb Z^D $中的一个离散时间随机过程，具体取决于两个整数参数$ 1 \ le d_1，d_2 \ le d $，nest $ x $ x $ x d $ e_i$ with uniform probability, where $e_1,\ldots,e_d$ are the canonical vectors, for $1\le i\le d_1$, if the site $x$ was visited for the first time at time $n$, while it jumps to $x\pm e_i$ with uniform probability, for $1+d-d_2\le i\le d$, if the site $x$ was already visited before时间$ n $。在这里，当$ d_1+d_2 = d $时，我们概述了此模型，并在$ d_1+d_2> d $中介绍和研究案例。特别是，我们证明，在所有情况下，$ d \ ge 5 $和大多数情况$ d = 4 $，平衡的兴奋随机步行是短暂的。
The balanced excited random walk, introduced by Benjamini, Kozma and Schapira in $2011$, is defined as a discrete time stochastic process in $\mathbb Z^d$, depending on two integer parameters $1\le d_1,d_2\le d$, which whenever it is at a site $x\in\mathbb Z^d$ at time $n$, it jumps to $x\pm e_i$ with uniform probability, where $e_1,\ldots,e_d$ are the canonical vectors, for $1\le i\le d_1$, if the site $x$ was visited for the first time at time $n$, while it jumps to $x\pm e_i$ with uniform probability, for $1+d-d_2\le i\le d$, if the site $x$ was already visited before time $n$. Here we give an overview of this model when $d_1+d_2=d$ and introduce and study the cases when $d_1+d_2>d$. In particular, we prove that for all the cases $d\ge 5$ and most cases $d=4$, the balanced excited random walk is transient.
