Coupling and an application to level-set percolation of the Gaussian free field

We consider a general enough set-up and obtain a refinement of the coupling between the Gaussian free field and random interlacements recently constructed by Titus Lupu in arXiv:1402.0298. We apply our results to level-set percolation of the Gaussian free field on a $(d+1)$-regular tree, when $d \ge 2$, and derive bounds on the critical value $h_*$. In particular, we show that $0 < h_* < \sqrt{2u_*}$, where $u_*$ denotes the critical level for the percolation of the vacant set of random interlacements on a $(d+1)$-regular tree.Comment: 28 pages, appeared in the Electronic Journal of Probabilit

On scaling limits and Brownian interlacements

We consider continuous time interlacements on Z^d, with d bigger or equal to 3, and investigate the scaling limit of their occupation times. In a suitable regime, referred to as the constant intensity regime, this brings Brownian interlacements on R^d into play, whereas in the high intensity regime the Gaussian free field shows up instead. We also investigate the scaling limit of the isomorphism theorem of arXiv:1111.4818. As a by-product, when d=3, we obtain an isomorphism theorem for Brownian interlacements.Comment: 28 pages, typos corrected, appeared in the special issue of the Bulletin of the Brazilian Mathematical Society-IMPA 60 year

On the critical parameter of interlacement percolation in high dimension

The vacant set of random interlacements on ${\mathbb{Z}}^d$, $d\ge3$, has nontrivial percolative properties. It is known from Sznitman [Ann. Math. 171 (2010) 2039--2087], Sidoravicius and Sznitman [Comm. Pure Appl. Math. 62 (2009) 831--858] that there is a nondegenerate critical value $u_*$ such that the vacant set at level $u$ percolates when $u<u_*$ and does not percolate when $u>u_*$. We derive here an asymptotic upper bound on $u_*$, as $d$ goes to infinity, which complements the lower bound from Sznitman [Probab. Theory Related Fields, to appear]. Our main result shows that $u_*$ is equivalent to $\log d$ for large $d$ and thus has the same principal asymptotic behavior as the critical parameter attached to random interlacements on $2d$-regular trees, which has been explicitly computed in Teixeira [Electron. J. Probab. 14 (2009) 1604--1627].Comment: Published in at http://dx.doi.org/10.1214/10-AOP545 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

A lower bound on the critical parameter of interlacement percolation in high dimension

We investigate the percolative properties of the vacant set left by random interlacements on Z^d, when d is large. A non-negative parameter u controls the density of random interlacements on Z^d. It is known from arXiv:0704.2560, and arXiv:0808.3344, that there is a non-degenerate critical value u_*, such that the vacant set at level u percolates when u < u_*, and does not percolate when u > u_*. Little is known about u_*, however for large d, random interlacements on Z^d, ought to exhibit similarities to random interlacements on a (2d)-regular tree, for which the corresponding critical parameter can be explicitly computed, see arXiv:0907.0316. We prove in this article a lower bound on u_*, which is equivalent to log(d) as d goes to infinity. This lower bound is in agreement with the above mentioned heuristics.Comment: 31 pages, 1 figure, accepted for publication in Probability Theory and Related Field

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 discrete cylinder, which as a by-product shows the tightness of the laws of the ratio of the square of the number of sites in the base to the disconnection time. This fact had previously only been established when d is at least 17, in arXiv: math/0701414.Comment: 33 pages, accepted for publication in the Electronic Journal of Probabilit