Stochastic Ordering of Infinite Binomial Galton-Watson Trees

We consider Galton-Watson trees with ${\rm Bin}(d,p)$ offspring distribution. We let $T_{\infty}(p)$ denote such a tree conditioned on being infinite. For $d=2,3$ and any $1/d\leq p_1 <p_2 \leq 1$, we show that there exists a coupling between $T_{\infty}(p_1)$ and $T_{\infty}(p_2)$ such that ${\mathbb P}(T_{\infty}(p_1) \subseteq T_{\infty}(p_2))=1.$Comment: 19 page

One-dependent trigonometric determinantal processes are two-block-factors

Given a trigonometric polynomial f:[0,1]\to[0,1] of degree m, one can define a corresponding stationary process {X_i}_{i\in Z} via determinants of the Toeplitz matrix for f. We show that for m=1 this process, which is trivially one-dependent, is a two-block-factor.Comment: Published at http://dx.doi.org/10.1214/009117904000000595 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Connectedness of Poisson cylinders in Euclidean space

We consider the Poisson cylinder model in ${\mathbb R}^d$, $d\ge 3$. We show that given any two cylinders ${\mathfrak c}_1$ and ${\mathfrak c}_2$ in the process, there is a sequence of at most $d-2$ other cylinders creating a connection between ${\mathfrak c}_1$ and ${\mathfrak c}_2$. In particular, this shows that the union of the cylinders is a connected set, answering a question appearing in a previous paper. We also show that there are cylinders in the process that are not connected by a sequence of at most $d-3$ other cylinders. Thus, the diameter of the cluster of cylinders equals $d-2$.Comment: 30 page

Phase transition and uniqueness of levelset percolation

The main purpose of this paper is to introduce and establish basic results of a natural extension of the classical Boolean percolation model (also known as the Gilbert disc model). We replace the balls of that model by a positive non-increasing attenuation function $l:(0,\infty) \to (0,\infty)$ to create the random field $\Psi(y)=\sum_{x\in \eta}l(|x-y|),$ where $\eta$ is a homogeneous Poisson process in ${\mathbb R}^d.$ The field $\Psi$ is then a random potential field with infinite range dependencies whenever the support of the function $l$ is unbounded. In particular, we study the level sets $\Psi_{\geq h}(y)$ containing the points $y\in {\mathbb R}^d$ such that $\Psi(y)\geq h.$ In the case where $l$ has unbounded support, we give, for any $d\geq 2,$ exact conditions on $l$ for $\Psi_{\geq h}(y)$ to have a percolative phase transition as a function of $h.$ We also prove that when $l$ is continuous then so is $\Psi$ almost surely. Moreover, in this case and for $d=2,$ we prove uniqueness of the infinite component of $\Psi_{\geq h}$ when such exists, and we also show that the so-called percolation function is continuous below the critical value $h_c$.Comment: 25 page

Universal Behavior of Connectivity Properties in Fractal Percolation Models

Partially motivated by the desire to better understand the connectivity phase transition in fractal percolation, we introduce and study a class of continuum fractal percolation models in dimension d greater than or equal to 2. These include a scale invariant version of the classical (Poisson) Boolean model of stochastic geometry and (for d=2) the Brownian loop soup introduced by Lawler and Werner. The models lead to random fractal sets whose connectivity properties depend on a parameter lambda. In this paper we mainly study the transition between a phase where the random fractal sets are totally disconnected and a phase where they contain connected components larger than one point. In particular, we show that there are connected components larger than one point at the unique value of lambda that separates the two phases (called the critical point). We prove that such a behavior occurs also in Mandelbrot's fractal percolation in all dimensions d greater than or equal to 2. Our results show that it is a generic feature, independent of the dimension or the precise definition of the model, and is essentially a consequence of scale invariance alone. Furthermore, for d=2 we prove that the presence of connected components larger than one point implies the presence of a unique, unbounded, connected component.Comment: 29 pages, 4 figure

Random cover times using the Poisson cylinder process

In this paper we deal with the classical problem of random cover times. We investigate the distribution of the time it takes for a Poisson process of cylinders to cover a set $A \subset \mathbb{R}^d.$ This Poisson process of cylinders is invariant under rotations, reflections and translations, and in addition we add a time component so that cylinders are "raining from the sky" at unit rate. Our main results concerns the asymptotic of this cover time as the set $A$ grows. If the set $A$ is discrete and well separated, we show convergence of the cover time to a Gumbel distribution. If instead $A$ has positive box dimension (and satisfies a weak additional assumption), we find the correct rate of convergence

Stochastic domination for a hidden Markov chain with applications to the contact process in a randomly evolving environment

The ordinary contact process is used to model the spread of a disease in a population. In this model, each infected individual waits an exponentially distributed time with parameter 1 before becoming healthy. In this paper, we introduce and study the contact process in a randomly evolving environment. Here we associate to every individual an independent two-state, $\{0,1\},$ background process. Given $\delta_0<\delta_1,$ if the background process is in state $0,$ the individual (if infected) becomes healthy at rate $\delta_0,$ while if the background process is in state $1,$ it becomes healthy at rate $\delta_1.$ By stochastically comparing the contact process in a randomly evolving environment to the ordinary contact process, we will investigate matters of extinction and that of weak and strong survival. A key step in our analysis is to obtain stochastic domination results between certain point processes. We do this by starting out in a discrete setting and then taking continuous time limits.Comment: Published in at http://dx.doi.org/10.1214/0091179606000001187 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org