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

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

2009-09-24

arxiv.org/abs/0909.4422v1

Annals of Probability 2009, Vol. 37, No. 5, 1715-1746

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/09-AOP450 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/09-AOP450

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 local times reaches a certain critical value. As a by-product, we prove the tightness of the laws of $T_N/N^{2d}$, when $d\ge2$.

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