论文标题
$ \ mathbb z $上的两个排斥随机步行
Two repelling random walks on $\mathbb Z$
论文作者
论文摘要
我们考虑在$ \ mathbb {z} $上进行两次相互作用的随机步行，以使一个方向行走的过渡概率随着另一个方向的过渡次数而呈指数下降。因此，联合过程可以看作是两次随机步行互相排斥。在我们的模型中，通过参数$β\ geq 0 $进一步调节了排斥的强度。当$β= 0 $时，这两个过程都是在$ \ mathbb {z} $上进行独立的对称随机步行，因此经常性。我们表明，如果$β\在（0,1] $中的$β\中，我们还表明，如果$β> 2 $。如果$β> 2 $。$β> 2 $。$β\ in（1,2] $仍然广泛开放。我们的结果是通过考虑动态系统的定位近似方法来获得的。
We consider two interacting random walks on $\mathbb{Z}$ such that the transition probability of one walk in one direction decreases exponentially with the number of transitions of the other walk in that direction. The joint process may thus be seen as two random walks reinforced to repel each other. The strength of the repulsion is further modulated in our model by a parameter $β\geq 0$. When $β= 0$ both processes are independent symmetric random walks on $\mathbb{Z}$, and hence recurrent. We show that both random walks are further recurrent if $β\in (0,1]$. We also show that these processes are transient and diverge in opposite directions if $β> 2$. The case $β\in (1,2]$ remains widely open. Our results are obtained by considering the dynamical system approach to stochastic approximations.
