Abstract
We consider the geometry of random interlacements on the $d$-dimensional lattice. We use ideas from stochastic dimension theory developed in [1] to prove the following: Given that two vertices $x,y$ belong to the interlacement set, it is possible to find a path between $x$ and $y$ contained in the trace left by at most $\lceil d/2 \rceil$ trajectories from the underlying Poisson point process. Moreover, this result is sharp in the sense that there are pairs of points in the interlacement set which cannot be connected by a path using the traces of at most $\lceil d/2 \rceil-1$ trajectories.
Highlights
The model of random interlacements was introduced by Sznitman in [Szn10], on the graph Zd, d ≥ 3
We consider the geometry of random interlacements on the d-dimensional lattice
We use ideas from stochastic dimension theory developed in [BKPS04] to prove the following: Given that two vertices x, y belong to the interlacement set, it is possible to find a path between x and y contained in the trace left by at most d/2 trajectories from the underlying Poisson point process
Summary
The model of random interlacements was introduced by Sznitman in [Szn10], on the graph Zd , d ≥ 3. In this paper we prove a stronger statement (for precise formulation, see Theorem 2.2): Given that two vertices x, y ∈ Zd belong to the interlacement set, it is a.s. possible to find a path between x and y contained in the trace left by at most d/2 trajectories from the underlying Poisson. Our method is based on the concept of stochastic dimension (see Section 2.2 below) introduced by Benjamini, Kesten, Peres and Schramm, [BKPS04]. They studied the geometry of the so called uniform spanning forest, and here we show how their techniques can be adapted to the study of the geometry of the random interlacements. Their proof is significantly different from the proof we present in this paper
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