Random Walk, Green Function, Equilibrium Measure.- Random Interlacements: First Definition and Basic Properties.- Random Walk on the Torus and Random Interlacements.- Poisson Point Processes.- Random Interlacements Point Process.- Percolation of the Vacant Set.- Source of Correlations and Decorrelation via Coupling.- Decoupling Inequalities.- Phase Transition of Vu.- Coupling of Point Measures of Excursions.

/pdf/an-introduction-to-random-interlacements-3jofrs9mcr.pdf

An Introduction to Random Interlacements

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 distance. In addition, we show that these conditions also imply a shape theorem for the corresponding infinite connected component. By verifying these conditions for specific models, we obtain 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. As a byproduct, we obtain alternative proofs to the corresponding results for random interlacements in the work of Cerný and Popov [“On the internal distance in the interlacement set,” Electron. J. Probab.17(29), 1–25 (2012)], and while our main interest is in percolation models with long-range correlations, we also recover results in the spirit of the work of Antal and Pisztora [“On the chemical distance for supercritical Bernoulli percolation,” Ann Probab.24(2), 1036–1048 (1996)] for Bernoulli percolation. Finally, as a corollary, we derive new results about the (chemical) diameter of the largest connected component in the complement of the trace of the random walk on the torus.

On chemical distances and shape theorems in percolation models with long-range correlations

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.

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.

On the internal distance in the interlacement set

We consider the interlacement Poisson point process on the space of doubly-infinite Z d -valued trajectories modulo time shift, tending to infinity at pos- itive 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 (2010). 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 dd/2e 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 dd/2e trajectories of the point process. In par- ticular, this implies the already known result of Sznitman (2010) that the random interlacement at level u is connected.

/pdf/connectivity-properties-of-random-interlacement-and-4o6eto65cu.pdf

Connectivity properties of random interlacement and intersection of random walks

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.

/pdf/geometry-of-the-random-interlacement-4hdkzlpt93.pdf

Geometry of the random interlacement

We consider a Poisson point process on the space of lines in R^d, where a multiplicative factor u>0 of the intensity measure determines the density of lines Each line in the process is taken as the axis of a bi-infinite cylinder of radius 1 We investigate percolative properties of the vacant set, defined as the subset of R^d that is not covered by any such cylinder We show that in dimensions d >= 4, there is a critical value u_*(d) \in (0,\infty), such that with probability 1, the vacant set has an unbounded component if u u_*(d) For d=3, we prove that the vacant set does not percolate for large u and that the vacant set intersected with a two-dimensional subspace of R^d does not even percolate for small u>0

Percolation in the vacant set of Poisson cylinders

We consider the Poisson Boolean model of continuum percolation with balls of fixed radius $R$ in $n$-dimensional hyperbolic space $H^n$. Let $\lambda$ be the intensity of the underlying Poisson process, and let $N_C$ denote the number of unbounded components in the covered region. For the model in any dimension we show that there are intensities such that $N_C=\infty$ a.s. if $R$ is big enough. In $H^2$ we show a stronger result: for any $R$ there are two intensities $\lambda_c$ and $\lambda_u$ where $0< \lambda_c < \lambda _u < \infty$, such that$N_C=0$ for $\lambda \in [0,\lambda_c]$, $N_C=\infty$ for $\lambda \in (\lambda_c,\lambda_u)$ and $N_C=1$ for $\lambda \in [\lambda_u, \infty)$.

/pdf/the-number-of-unbounded-components-in-the-poisson-boolean-1hh69479i0.pdf

The number of unbounded components in the Poisson Boolean model of continuum percolation in hyperbolic space

We consider the random interlacements process with intensity $u$ on ${\mathbb Z}^d$, $d\ge 5$ (call it $I^u$), built from a Poisson point process on the space of doubly infinite nearest neighbor trajectories on ${\mathbb Z}^d$. For $k\ge 3$ we want to determine the minimal number of trajectories from the point process that is needed to link together $k$ points in $\mathcal I^u$. Let $$n(k,d):=\lceil \frac d 2 (k-1) \rceil - (k-2).$$ We prove that almost surely given any $k$ points $x_1,...,x_k\in \mathcal I^u$, there is a sequence ofof $n(k,d)$ trajectories $\gamma^1,...,\gamma^{n(k,d)}$ from the underlying Poisson point process such that the union of their traces $\bigcup_{i=1}^{n(k,d)}\tr(\gamma^{i})$ is a connected set containing $x_1,...,x_k$. Moreover we show that this result is sharp, i.e. that a.s. one can find $x_1,...,x_k in I^u$ that cannot be linked together by $n(k,d)-1$ trajectories.

/pdf/on-the-easiest-way-to-connect-k-points-in-the-random-23dwc8ngdh.pdf

On the easiest way to connect k points in the Random Interlacements process

In this paper, we initiate the study of "Generalized Divide and Color Models". A very interesting special case of this is the "Divide and Color Model" (which motivates the name we use) introduced and studied by Olle Haggstrom. In this generalized model, one starts with a finite or countable set V, a random partition of V and a parameter p ∈ [0; 1]. The corresponding Generalized Divide and Color Model is the [0; 1]-valued process indexed by V obtained by independently, for each partition element in the random partition chosen, with probability p, assigning all the elements of the partition element the value 1, and with probability 1 - p, assigning all the elements of the partition element the value 0. Some of the questions which we study here are the following. Under what situations can different random partitions give rise to the same color process? What can one say concerning exchangeable random partitions? What is the set of product measures that a color process stochastically dominates? For random partitions which are translation invariant, what ergodic properties do the resulting color processes have? The motivation for studying these processes is twofold; on the one hand, we believe that this is a very natural and interesting class of processes that deserves investigation and on the other hand, a number of quite varied well-studied processes actually fall into this class such as (1) the Ising model, (2) the fuzzy Potts model, (3) the stationary distributions for the Voter Model, (4) random walk in random scenery and of course (5) the original Divide and Color Model.

/pdf/generalized-divide-and-color-models-3sj110arqx.pdf

Johan Tykesson

Papers

Geometry of the random interlacement

Percolation in the vacant set of Poisson cylinders

The number of unbounded components in the Poisson Boolean model of continuum percolation in hyperbolic space

On the easiest way to connect k points in the Random Interlacements process

Generalized Divide and Color models