Connectivity properties of random interlacement and intersection of random walks

Latin American journal of probability and mathematical statistics (Impact Factor: 0.52). 12/2010; 9(1).
Source: arXiv


We consider the interlacement Poisson point process on the space of
doubly-infinite Z^d-valued trajectories modulo time-shift, tending to infinity
at positive and negative infinite times. The set of vertices and edges visited
by at least one of these trajectories is the random interlacement at level u of
Sznitman arXiv:0704.2560 . We prove that for any u>0, almost surely, (1) any
two vertices in the random interlacement at level u are connected via at most
ceiling(d/2) trajectories of the point process, and (2) there are vertices in
the random interlacement at level u which can only be connected via at least
ceiling(d/2) trajectories of the point process. In particular, this implies the
already known result of Sznitman arXiv:0704.2560 that the random interlacement
at level u is connected.

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    • "The same authors in [13], as well as Procaccia and Tykesson [11] have shown by essentially different methods (using ideas from the field of potential theory on the one hand, and stochastic dimension on the other hand) that any two points of the set I u can be connected by using at most ⌈d/2⌉ trajectories from the constructive definition described above. Recently, using in parts extensions of the techniques in [13], this result has been generalized to an arbitrary number of points by Lacoin and Tykesson [6]. Another step in showing that the geometry of random interlacements resembles that of Z d has been undertaken byČern´y and Popov [3], where the authors prove that the chemical distance (also called graph distance or internal distance) in the set I u is comparable to that of Z d . "
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    ABSTRACT: The model of random interlacements is a one-parameter family $\mathcal I^u,$ $u \ge 0,$ of random subsets of $\mathbb{Z}^d,$ which locally describes the trace of simple random walk on a $d$-dimensional torus run up to time $u$ times its volume. Its complement, the so-called vacant set $\mathcal V^u$, has been shown to undergo a non-trivial percolation phase-transition in $u;$ i.e., there exists $u_*(d) \in (0, \infty)$ such that for $u \in [0, u_*(d))$ the vacant set $\mathcal V^u$ contains a unique infinite connected component $\mathcal V_\infty^u,$ while for $u > u_*(d)$ it consists of finite connected components. Sznitman \cite{SZ11,SZ11B} showed that $u_*(d) \sim \log d,$ and in this article we show the existence of $u(d) > 0$ with $\frac{u(d)}{u_*(d)} \to 1$ as $d \to \infty$ such that $\mathcal V_\infty^{u}$ is transient for all $u \in [0, u(d)).$
    Preview · Article · Dec 2013 · Annales de l Institut Henri Poincaré Probabilités et Statistiques
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    • "In particular, the fact that the conclusions of Theorems 1.3 and 1.5 hold for the random interlacements I u at level u > 0 has been recently established in [8, Theorems 1.3 and 1.1]. The key idea in the proofs of [8] is a certain refinement of the strategy developed in [20], which crucially relies on the underlying random walk structure of random interlacements (see [26, (1.53)]). The goal of this section is to observe (based on earlier results about random interlacements) that the distributions of random interlacements at level u satisfy conditions P1 – P3 and S1 – S2. "
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    ABSTRACT: In this paper we provide general conditions on a one parameter family of random infinite subsets of Z^d to contain a unique infinite connected component for which the chemical distances are comparable to the Euclidean distances, focusing primarily on models with long-range correlations. Our results are in the spirit of those by Antal and Pisztora proved for Bernoulli percolation. We also prove a shape theorem for balls in the chemical distance under such conditions. Our general statements give novel results about the structure of the infinite connected component of the vacant set of random interlacements and the level sets of the Gaussian free field. We also obtain alternative proofs to the main results in arXiv:1111.3979. Finally, as a corollary, we obtain new results about the (chemical) diameter of the largest connected component in the complement of the trace of the random walk on the torus.
    Preview · Article · Dec 2012 · Journal of Mathematical Physics
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    • "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]. Random interlacements have further been helpful in some questions of cover times, see [1], [2]. "
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    ABSTRACT: We consider continuous time interlacements on Z^d, with d bigger or equal to 3, and investigate the scaling limit of their occupation times. In a suitable regime, referred to as the constant intensity regime, this brings Brownian interlacements on R^d into play, whereas in the high intensity regime the Gaussian free field shows up instead. We also investigate the scaling limit of the isomorphism theorem of arXiv:1111.4818. As a by-product, when d=3, we obtain an isomorphism theorem for Brownian interlacements.
    Preview · Article · Sep 2012 · Bulletin Brazilian Mathematical Society
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