Percolative Properties of Brownian Interlacements and Its Vacant Set
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Percolative Properties of Brownian Interlacements and Its Vacant Set Xinyi Li1 Received: 27 July 2018 / Revised: 23 August 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract In this article, we investigate the percolative properties of Brownian interlacements, a model introduced by Sznitman (Bull Braz Math Soc New Ser 44(4):555–592, 2013), and show that: the interlacement set is “well-connected”, i.e., any two “sausages” in d-dimensional Brownian interlacements, d ≥ 3, can be connected via no more than (d − 4)/2 intermediate sausages almost surely; while the vacant set undergoes a non-trivial percolation phase transition when the level parameter varies. Keywords Brownian interlacements · Random interlacements · Wiener sausage · Percolation Mathematics Subject Classification (2010) 60J65 · 60K35 · 82B43
1 Introduction In this article, we investigate various aspects of the percolative properties of Brownian interlacements and show that the interlacements are well-connected and that the vacant set undergoes a non-trivial phase transition. The model of Brownian interlacements, recently introduced by Sznitman in [29], is the continuous counterpart of random interlacements, a model that has already attracted a lot of attention and has been relatively thoroughly studied (see [27] for the seminal paper on this model and see [4,8] for a comprehensive introduction). Roughly speaking, Brownian interlacements can be described as a certain Poissonian cloud of doubly infinite continuous trajectories in the d-dimensional Euclidean space, d ≥ 3, with the intensity measure governed by a parameter α > 0. We are interested in both the interlacement set, which is an r -enlargement (sometimes colloquially referred to as
B 1
Xinyi Li [email protected] Beijing International Center for Mathematical Research, Peking University, Beijing 100871, China
123
Journal of Theoretical Probability
“the sausages”) of the union of the trace in the aforementioned cloud (of trajectories), for some r > 0, and the vacant set, which is the complement of the interlacement set. Brownian interlacements bear similar properties, for instance long-range dependence, to random interlacements, due to similarities in the construction. Moreover, this model plays a crucial role in both the study of the limiting behaviors of various aspects of random interlacements (see for example [13,29]), and the interconnection of random interlacements, loop soups, and Gaussian free fields. Brownian interlacements, as a model of continuous percolation, could also shed light on the study of other models. For example, the visibility in the vacant set of Brownian interlacements is studied and compared with that of the Brownian excursion process in [9]. We now describe the model and our results in a more precise fashion. Readers are referred to Sect. 2 for notations and definitions. We consider Brownian interlacements on Rd , d ≥ 3. We denote by P the canonical law of Brownian interlacements and by Irα (resp. Vrα ) the corresponding inte
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