Results on the contact process with dynamic edges or under renewals

We analyze variants of the contact process that are built by modifying the percolative structure given by the graphical construction and develop a robust renormalization argument for proving extinction in such models. With this method, we obtain results on the phase diagram of two models: the Contact Process on Dynamic Edges introduced by Linker and Remenik and a generalization of the Renewal Contact Process introduced by Fontes, Marchetti, Mountford and Vares.

Euclidean and chemical distances in ellipses percolation

The ellipses model is a continuum percolation process in which ellipses with random orientation and eccentricity are placed in the plane according to a Poisson point process. A parameter $alpha$ controls the tail distribution of the major axis distribution and we focus on the regime $alpha in (1,2)$ for which there exists a unique infinite cluster of ellipses and this cluster fulfills the so called highway property. We prove that the distance within this infinite cluster behaves asymptotically like the (unrestricted) Euclidean distance in the plane. We also show that the chemical distance between points $x$ and $y$ behaves roughly as $c loglog |x-y|$.

Renewal Contact Processes: phase transition and survival

We refine some previous results concerning the Renewal Contact Processes. We significantly widen the family of distributions for the interarrival times for which the critical value can be shown to be strictly positive. The result now holds for any spatial dimension $d geq 1$ and requires only a moment condition slightly stronger than finite first moment. We also prove a Complete Convergence Theorem for heavy tailed interarrival times. Finally, for heavy tailed distributions we examine when the contact process, conditioned on survival, can be asymptotically predicted knowing the renewal processes. We close with an example of an interarrival time distribution attracted to a stable law of index 1 for which the critical value vanishes, a tail condition uncovered by previous results.

A note on the phase transition for independent alignment percolation

We study the independent alignment percolation model on $mathbb{Z}^d$ introduced by Beaton, Grimmett and Holmes [arXiv:1908.07203]. It is a model for random intersecting line segments defined as follows. First the sites of $mathbb{Z}^d$ are independently declared occupied with probability $p$ and vacant otherwise. Conditional on the configuration of occupied vertices, consider the set of all line segments that are parallel to the coordinate axis, whose extremes are occupied vertices and that do not traverse any other occupied vertex. Declare independently the segments on this set open with probability $lambda$ and closed otherwise. All the edges that lie on open segments are also declared open giving rise to a bond percolation model in $mathbb{Z}^d$. We show that for any $d geq 2$ and $p in (0,1]$ the critical value for $lambda$ satisfies $lambda_c(p)<1$ completing the proof that the phase transition is non-trivial over the whole interval $(0,1]$. We also show that the critical curve $p mapsto lambda_c(p)$ is continuous at $p=1$, answering a question posed by the authors in [arXiv:1908.07203].

Transience of conditioned walks on the plane: encounters and speed of escape

We consider the two-dimensional simple random walk conditioned on never hitting the origin, which is,formally speaking, the Doobs $h$-transform of the simple random walk with respect to the potential kernel. We then study the behavior of the future minimum distance of the walk to the origin, and also prove that two independent copies of the conditioned walk, although both transient, will nevertheless meet infinitely many times a.s.

Ellipses Percolation

We define a continuum percolation model that provides a collection of random ellipses on the plane and study the behavior of the covered set and the vacant set, the one obtained by removing all ellipses. Our model generalizes a construction that appears implicitly in the Poisson cylinder model of Tykesson and Windisch. The ellipses model has a parameter $alpha > 0$ associated with the tail decay of the major axis distribution; we only consider distributions $rho$ satisfying $rho[r, infty) asymp r^{-alpha}$. We prove that this model presents a double phase transition in $alpha$. For $alpha in (0,1]$ the plane is completely covered by the ellipses, almost surely. For $alpha in (1,2)$ the vacant set is not empty but does not percolate for any positive density of ellipses, while the covered set always percolates. For $alpha in (2, infty)$ the vacant set percolates for small densities of ellipses and the covered set percolates for large densities. Moreover, we prove for the critical parameter $alpha = 2$ that there is a non-degenerate interval of density for which the probability of crossing boxes of a fixed proportion is bounded away from zero and one, a rather unusual phenomenon. In this interval neither the covered set nor the vacant set percolate, a behavior that is similar to critical independent percolation on $mathbb{Z}^2$.