Connectivity properties of random interlacement and intersection of random walks

ABSTRACT 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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    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 $u \in [0, u(d)).$
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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.
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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.
    Bulletin Brazilian Mathematical Society 09/2012; · 0.35 Impact Factor


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Artem Sapozhnikov