Percolation for the vacant set of random interlacements

Abstract

We investigate random interlacements on Z d , d 3. This model, recently introduced in [8], corresponds to a Poisson cloud on the space of doubly infinite trajectories modulo time shift tending to infinity at positive and negative infinite times. A nonnegative parameter u measures how many trajectories enter the picture. Our main interest lies in the percolative properties of the vacant set left by random interlacements at level u. We show that for all d 3 the vacant set at level u percolates when u is small. This solves an open problem of [8], where this fact has only been established when d 7. It also completes the proof of the nondegeneracy in all dimensions d 3 of the critical parameter u of [8].

Vacant set of random interlacements and percolation

“…(2.6), shows that when d 7, for small u > 0, ᐂ u percolates in ‫ޚ‬ 2 ‫ޚ‬ d . This feature remains true for all d 3, see Theorem 3.4 of [15].…”

“…positive and finite), when d 7, and finite for all d 3. The results presented here have triggered some progress on the questions mentioned above; see in particular [15], [20], [21], [22], and [25].…”

“…Similarly (0.15) offers evidence that the lower bound on T N in [7], which shows the tightness of N 2d =T N when d 17, should hold as soon as d 6 (and in fact for all d 2, in view of the recent work [15]). Analogously in the context of [2], we see that (0.14), (0.15), and [15] give support for the typical absence for large N of a giant component in the vacant set left by simple random walk on .‫=ޚ‬N ‫/ޚ‬ d , d 3, run up to time uN d , if u is large, and for its typical presence when u is chosen small.…”

“…Is u > 0, when 3 Ä d Ä 6, as suggested by simulations? This is indeed the case; see [15]. However it is presently unknown whether the vacant set percolates at criticality, i.e.…”

On scaling limits and Brownian interlacements

“…Their vacant set has non-trivial percolative properties, see for instance [9], [31], [38], [43], which are useful in the study of certain disconnection and fragmentation problems, see [6], [7], [37], [44]. Their connectivity properties have been actively investigated, see [5], [20], [28], [29], [30].…”