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

How universal are asymptotics of disconnection times in discrete cylinders?

We investigate the disconnection time of a simple random walk in a discrete cylinder with a large finite connected base. In a recent article of A. Dembo and the author it was found that for large $N$ the disconnection time of $G_N\times\mathbb{Z}$ has rough order $|G_N|^2$, when $G_N=(\mathbb{Z}/N\mathbb{Z})^d$. In agreement with a conjecture by I. Benjamini, we show here that this behavior has broad generality when the bases of the discrete cylinders are large connected graphs of uniformly bounded degree.

A lower bound for disconnection by simple random walk

We consider simple random walk on Z^d, d bigger or equal to 3. Motivated by the work of A.-S. Sznitman and the author in arXiv:1304.7477 and arXiv:1310.2177, we investigate the asymptotic behaviour of the probability that a large body gets disconnected from infinity by the set of points visited by a simple random walk. We derive asymptotic lower bounds that bring into play random interlacements. Although open at the moment, some of the lower bounds that we obtain possibly match the asymptotic upper bounds obtained in a recent article of A.-S. Sznitman. This potentially yields special significa…

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…

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…

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 Domination of Random Walk on a Discrete Cylinder by Random Interlacements

We consider simple random walk on a discrete cylinder with base a large d-dimensional torus of side-length N, when d is two or more. We develop a stochastic domination control on the local picture left by the random walk in boxes of side-length almost of order N, at certain random times comparable to the square of the number of sites in the base. We show a domination control in terms of the trace left in similar boxes by random interlacements in the infinite (d+1)-dimensional cubic lattice at a suitably adjusted level. As an application we derive a lower bound on the disconnection time of the …

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…

Solidification of porous interfaces and disconnection

In this article we obtain uniform estimates on the absorption of Brownian motion by porous interfaces surrounding a compact set. An important ingredient is the construction of certain resonance sets, which are hard to avoid for Brownian motion starting in the compact set. As an application of our results, we substantially strengthen the results of arXiv:1412.3960, and obtain when $d \ge 3$, large deviation upper bounds on the probability that simple random walk in $Z^d$, or random interlacements in $Z^d$, when their vacant set is in a strongly percolative regime, disconnect the discrete blow-u…

A Lower Bound on the Disconnection Time of a Discrete Cylinder

We study the asymptotic behavior for large N of the disconnection time T_N of simple random walk on a discrete cylinder with base a d-dimensional discrete torus of side-length N. When d is sufficiently large, we are able to substantially improve the lower bounds obtained by the authors in a previous article when d is bigger or equal to 2. We show here that the laws of N^(2d)/T_N are tight.

Vacant Set of Random Interlacements and Percolation

We introduce a model of random interlacements made of a countable collection of doubly infinite paths on Z^d, d bigger or equal to 3. A non-negative parameter u measures how many trajectories enter the picture. This model describes in the large N limit the microscopic structure in the bulk, which arises when considering the disconnection time of a discrete cylinder with base a d-1-dimensional discrete torus of side-length N, or the set of points visited by simple random walk on the d-dimensional discrete torus of side-length N by times of order uN^d. We study the percolative properties of the …

Disconnection by level sets of the discrete Gaussian free field and entropic repulsion

We derive asymptotic upper and lower bounds on the large deviation probability that the level set of the Gaussian free field on $Z^d$, d bigger or equal to three, below a given level disconnects the discrete blow-up of a compact set A from the boundary of the discrete blow-up of a box that contains A, when the level set of the Gaussian free field above this level is in a strongly percolative regime. These bounds substantially strengthen the results of arXiv:1412.3960, where A was a box and the convexity of A played an important role in the proof. We also derive an asymptotic upper bound on the…

柱体的渗流：解耦及其应用

Cylinders' percolation: decoupling and applications

本文为柱体渗流过程建立了强解耦不等式，引入并证明了快速解耦概念，证明了在柱体密度足够小时，空集连通分量上的随机游走在所有维度 $d \geq 3$ 中都是常返的。