arXiv subjects

Pierrick Siest

Publications and source records attributed to Pierrick Siest.

3 recordsLinked to original sources

arXiv2025

Richardson's model and the contact process with stirring: long time behavior

We study two famous interacting particle systems, the so-called Richardson's model and the contact process, when we add a stirring dynamics to them. We prove that they both satisfy an asymptotic shape theorem, as their analogues without stirring, but only for high enough infection rates, using couplings and restart techniques. We also show that for Richardson's model with stirring, for high enough infection rates, each site is forever infected after a certain time almost surely. Finally, we study weak and strong survival for both models on a homogeneous infinite tree, and show that there are two phase transitions for certain values of the parameters and the dimension, which is a result similar to what is proved for the contact process.

math.PR↗

arXiv2022

Corner percolation with preferential directions

Corner percolation is a dependent bond percolation model on Z^2 introduced by Bálint Tóth, in which each vertex has exactly two incident edges, perpendicular to each other. Gábor Pete has proven in 2008 that under the maximal entropy probability measure, all connected components are finite cycles almost surely. We consider here a regime where West and North directions are preferred with probability p and q respectively, with (p,q) different from (1/2,1/2). We prove that there exists almost surely an infinite number of infinite connected components, which are in fact infinite paths. Furthermore, they all have the same asymptotic slope (2q-1)/(1-2p).

math.PR↗

arXiv2015

An asymptotic shape theorem for additive random linear growth models

In this paper, we define a class of additive random growth models whose growth is at least and at most linear and prove an asymptotic shape theorem for these models. This proof generalizes already known proofs for the classical contact process or some of its variants and allows us to obtain conjectured asymptotic shape theorems for Richardson's model with stirring and the contact process with stirring.

math.PR↗