关于离散圆柱的有偏随机游动的不连续性
On the disconnection of a discrete cylinder by a biased random walk
摘要：
我们考虑一个随机游动上的离散圆柱$({\mathbb{Z}}/N{\mathbb{Z}})^d\times{\mathbb{Z}}$，$d\geq3$与漂移$N^{-dα}$在$\mathbb{Z}$ -方向和调查的大$N$ -行为的断开时间$T^{\mathrm{disc}}_N$，定义为第一次时的随机游动的轨迹断开成两个无限组件的圆柱。我们证明，只要漂移指数$α$严格大于1，渐近行为的$T^{\mathrm{disc}}_N$仍然$N^{2d+o(1)}$，在无偏的情况下考虑Dembo和Sznitman，而对于$α<1$，渐近行为的$T^{\mathrm{disc}}_N$成为指数在$N$。
英文摘要：
We consider a random walk on the discrete cylinder $({\mathbb{Z}}/N{\mathbb{Z}})^d\times{\mathbb{Z}}$, $d\geq3$ with drift $N^{-dα}$ in the $\mathbb{Z}$-direction and investigate the large $N$-behavior of the disconnection time $T^{\mathrm{disc}}_N$, defined as the first time when the trajectory of the random walk disconnects the cylinder into two infinite components. We prove that, as long as the drift exponent $α$ is strictly greater than 1, the asymptotic behavior of $T^{\mathrm{disc}}_N$ remains $N^{2d+o(1)}$, as in the unbiased case considered by Dembo and Sznitman, whereas for $α<1$, the asymptotic behavior of $T^{\mathrm{disc}}_N$ becomes exponential in $N$.
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