Abstract
We prove a shape theorem for the internal (graph) distance on the interlacement set $\mathcal{I}^u$ of the random interlacement model on $\mathbb Z^d$, $d\ge 3$. We provide large deviation estimates for the internal distance of distant points in this set, and use these estimates to study the internal distance on the range of a simple random walk on a discrete torus.
Highlights
Introduction and the resultsWe study properties of the interlacement set Iu of the random interlacement model
We are mainly interested in its connectivity properties, in particular in the internal distance on the interlacement cluster
The random interlacement model was introduced in [12] in order to describe the microscopic structure in the bulk which arises when studying the disconnection time of a discrete cylinder or the vacant set of random walk on a discrete torus. It can be informally described as a dependent site percolation on Zd, d ≥ 3, which is ‘generated’ by a Poisson cloud of independent simple random walks whose intensity is driven by a non-negative multiplicative parameter u
Summary
We study properties of the interlacement set Iu of the random interlacement model. We are mainly interested in its connectivity properties, in particular in the internal distance (sometimes called the chemical distance) on the interlacement cluster. To obtain the finite range of dependence, we should show that connections within a large box of size m can be constructed using less than Θ(md−2) random walk trajectories (which is the typical number of random walks intersecting this box; here and in the sequel we write f (m) = Θ(g(m)) when for positive constants c1, c2 we have c1g(m) ≤ f (m) ≤ c2g(m) for all m). This proposition roughly states that all points in (a possibly thinned version of) the set Iu within box of size n are at internal distance n2, with a very high probability.
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