Intersection exponents for biased random walks on discrete cylinders

简单与有偏随机游走对大圆柱体断开时间的精确渐近性

Sharp asymptotics of disconnection time of large cylinders by simple and biased random walks

本文研究了简单与有偏随机游走对大离散圆柱体的渐近断开时间，推导出匹配的精确下界并改进了有偏游走的界，同时提出了随机游走迹与随机交织间的强耦合工具。

On the disconnection of a discrete cylinder by a biased random walk

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…

Mixing time for random walk on supercritical dynamical percolation

We consider dynamical percolation on the $d$-dimensional discrete torus of side length $n$, $\mathbb{Z}_n^d$, where each edge refreshes its status at rate $μ=μ_n\le 1/2$ to be open with probability $p$. We study random walk on the torus, where the walker moves at rate $1/(2d)$ along each open edge. In earlier work of two of the authors with A. Stauffer, it was shown that in the subcritical case $p<p_c(\mathbb{Z}^d)$, the (annealed) mixing time of the walk is $Θ(n^2/μ)$, and it was conjectured that in the supercritical case $p>p_c(\mathbb{Z}^d)$, the mixing time is $Θ(n^2+1/μ)$; here the implie…