On the cost of the bubble set for random interlacements

Abstract

The main focus of this article concerns the strongly percolative regime of the vacant set of random interlacements on ${\mathbb{Z}}^{d}$ Z d , $d \ge 3$ d ≥ 3 . We investigate the occurrence in a large box of an excessive fraction of sites that get disconnected by the interlacements from the boundary of a concentric box of double size. The results significantly improve our findings in Sznitman (Probab. Math. Phys. 2–3:563–311, 2021). In particular, if, as expected, the critical levels for percolation and for strong percolation of the vacant set of random interlacements coincide, the asymptotic upper bound that we derive here matches in principal order a previously known lower bound. A challenging difficulty revolves around the possible occurrence of droplets that could get secluded by the random interlacements and thus contribute to the excess of disconnected sites, somewhat in the spirit of the Wulff droplets for Bernoulli percolation or for the Ising model. This feature is reflected in the present context by the so-called bubble set, a possibly quite irregular random set. A pivotal progress in this work has to do with the improved construction of a coarse grained random set accounting for the cost of the bubble set. This construction heavily draws both on the method of enlargement of obstacles originally developed in the mid-nineties in the context of Brownian motion in a Poissonian potential in Sznitman (Ann. Probab. 25(3):1180–1209, 1997; Brownian Motion, Obstacles and Random Media. Springer, Berlin, 1998), and on the resonance sets recently introduced by Nitzschner and Sznitman in (J. Eur. Math. Soc. 22(8):2629–2672, 2020) and further developed in a discrete set-up by Chiarini and Nitzschner in (Commun. Math. Phys. 386(3):1685–1745, 2021).

A characterization of strong percolation via disconnection

“…For completeness, let us mention that various reinforcements of being 'strongly percolative' have been in circulation. The notion we deal with here has been fruitfully exploited to give strong answers to various problems relating to disconnection and the formation of droplet(s) in the supercritical regime [7,24,26,[39][40][41]43]. For other questions, see, for example, [13,28,31], see also [8,14,20,22] in various other contexts, it is of interest to remove the sprinkling, that is, to require 𝑣 ′ = 𝑣 in (1.5), see, for example, [12, (1.3)].…”

“…for instance (1.8)). For this model, a non-trivial strongly percolative regime is so far known to exist on ℤ 𝑑 for all 𝑑 ⩾ 3 by already involved perturbative arguments, see [46] for 𝑑 ⩾ 5 and [12] for all 𝑑 ⩾ 3, and our understanding of various features of vacant clusters in this regime has witnessed considerable progress over the last decade [7,13,24,26,28,31,[39][40][41]43].…”