On the Domination of Random Walk on a Discrete Cylinder by Random Interlacements

Random Walks on Discrete Cylinders and Random Interlacements

We explore some of the connections between the local picture left by the trace of simple random walk on a discrete cylinder with base a d-dimensional torus, d at least 2, of side-length N running for times of order N^{2d} and the model of random interlacements recently introduced in arXiv:0704.2560. In particular we show that when the base becomes large, in the neighborhood of a point of the cylinder with a vertical component of order N^d, the complement of the set of points visited by the walk up to times of order N^{2d}, is close in distribution to the law of the vacant set of random interla…

Upper bound on the disconnection time of discrete cylinders and random interlacements

We study the asymptotic behavior for large $N$ of the disconnection time $T_N$ of a simple random walk on the discrete cylinder $(\mathbb{Z}/N\mathbb{Z})^d\times\mathbb{Z}$, when $d\ge2$. We explore its connection with the model of random interlacements on $\mathbb{Z}^{d+1}$ recently introduced in [Ann. Math., in press], and specifically with the percolative properties of the vacant set left by random interlacements. As an application we show that in the large $N$ limit the tail of $T_N/N^{2d}$ is dominated by the tail of the first time when the supremum over the space variable of the Brownian…

Random walks on discrete cylinders with large bases and random interlacements

Following the recent work of Sznitman (arXiv:0805.4516), we investigate the microscopic picture induced by a random walk trajectory on a cylinder of the form G_N x Z, where G_N is a large finite connected weighted graph, and relate it to the model of random interlacements on infinite transient weighted graphs. Under suitable assumptions, the set of points not visited by the random walk until a time of order |G_N|^2 in a neighborhood of a point with Z-component of order |G_N| converges in distribution to the law of the vacant set of a random interlacement on a certain limit model describing the…

Brownian interlacements、Poisson cylinders与Boolean models中的可见性

Visibility in Brownain interlacements, Poisson cylinders and Boolean models

本文研究了Brownian interlacements等三种模型空置集内的可见性，获得了距离r可见性概率的精确渐近界，发现在$d\geq4$维度下三者标度一致，并改进了已有文献的主要结果。

关于 Brownian interlacements、Poisson cylinders 和 Boolean models 的可见性窗口

On the visibility window for Brownian interlacements, Poisson cylinders and Boolean models

本文研究了 $\mathbb R^d$ 中三种空间相关性缓慢衰减模型的空缺集内部可见性，证明了在给定可见条件下，最大可见球半径与可见性窗口之比弱收敛于显式强度的指数分布。

Cover times in the discrete cylinder

This article proves that, in terms of local times, the rescaled and recentered cover times of finite subsets of the discrete cylinder by simple random walk converge in law to the Gumbel distribution, as the cardinality of the set goes to infinity. As applications we obtain several other results related to covering in the discrete cylinder. Our method is new and involves random interlacements, which were introduced by Sznitman in arXiv:0704.2560. To enable the proof we develop a new stronger coupling of simple random walk in the cylinder and random interlacements, which is also of independent i…