Title :

Existence and percolation results for stopped germ-grain models with unbounded velocities

Author(s) :

Coupier, David [Auteur]
Laboratoire de Mathématiques et leurs Applications de Valenciennes - EA 4015 [LAMAV]
Dereudre, David [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Le Stum, Simon [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]

Journal title :

Stochastic Processes and their Applications

Pages :

549-579

Publisher :

Elsevier

Publication date :

2021-12

ISSN :

0304-4149

English keyword(s) :

continuum percolation geometric random graph lilypond model Brownian dynamic
continuum percolation
geometric random graph
lilypond model
Brownian dynamic

HAL domain(s) :

Mathématiques [math]

English abstract : [en]

We investigate the existence and first percolation properties of general stopped germ-grain models. They are defined via a random set of germs generated by a homogeneous planar Poisson point process in R2. From each germ, ...
Show more >We investigate the existence and first percolation properties of general stopped germ-grain models. They are defined via a random set of germs generated by a homogeneous planar Poisson point process in R2. From each germ, a grain, composed of a random number of branches, grows. This grain stops growing whenever one of its branches hits another grain. The classical and historical example is the line segment model for which the grains are segments growing in a random direction in [0,2π) with random velocity. In the bilateral line segment model the segments grow in both directions. Other examples are considered here such as the Brownian model where the branches are simply given by independent Brownian motions in R2. The existence of such dynamics for an infinite number of germs is not obvious and our first result ensures it in a very general setting. In particular the existence of the line segment model is proved as soon as the random velocity admits a moment of order 4 which extends the result by Daley et al. (Theorem 4.3 in Daley et al. (2014)) for bounded velocity. Our result covers also the Brownian dynamic model. In the second part of the paper, we show that the line segment model with random velocity admitting a super exponential moment does not percolate. This improves a recent result (Theorem 3.2 in Coupier et al .(2020)) in the case of bounded velocity.Show less >

Language :

Anglais

Peer reviewed article :

Oui

Audience :

Internationale

Popular science :

Non

ANR Project :

Centre Européen pour les Mathématiques, la Physique et leurs Interactions

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL