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Abstract:

We consider a continuum percolation model on Rd, d≥ 1. For t, λ∈ (0 , ∞) and d∈ { 1 , 2 , 3 } , the occupied set is given by the union of independent Brownian paths running up to time t whose initial points form a Poisson point process with intensity λ> 0. When d≥ 4 ,the Brownian paths are replaced by Wiener sausages with radius r> 0. We establish that, for d= 1 and all choices of t, no percolation occurs, whereas for d≥ 2 , there is a non-trivial percolation transition in t, provided λ and r are chosen properly. The last statement means that λ has to be chosen to be strictly smaller than the critical percolation parameter for the occupied set at time zero (which is infinite when d∈ {2,3} , but finite and dependent on r when d≥ 4). We further show that for all d≥ 2 , the unbounded cluster in the supercritical phase is unique. Along the way a finite box criterion for non-percolation in the Boolean model is extended to radius distributions with an exponential tail. This may be of independent interest. The present paper settles the basic properties of the model and should be viewed as a springboard for finer results. © 2016, Springer Science+Business Media New York.

Registro:

Documento: Artículo
Título:Brownian Paths Homogeneously Distributed in Space: Percolation Phase Transition and Uniqueness of the Unbounded Cluster
Autor:Erhard, D.; Martínez, J.; Poisat, J.
Filiación:Mathematics Institute, Warwick University, Coventry, CV4 7AL, United Kingdom
Instituto de Investigaciones Matemáticas Luis A. Santaló, Conicet, Buenos Aires, C1428EGA, Argentina
CEREMADE, UMR CNRS 7534, Université Paris-Dauphine, PSL Research University, Place du Maréchal de Lattre de Tassigny, Paris Cedex-16, 75775, France
Palabras clave:Boolean percolation; Brownian motion; Continuum percolation; Phase transition; Poisson point process
Año:2017
Volumen:30
Número:3
Página de inicio:784
Página de fin:812
DOI: http://dx.doi.org/10.1007/s10959-015-0661-5
Título revista:Journal of Theoretical Probability
Título revista abreviado:J. Theor. Probab.
ISSN:08949840
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_08949840_v30_n3_p784_Erhard

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Citas:

---------- APA ----------
Erhard, D., Martínez, J. & Poisat, J. (2017) . Brownian Paths Homogeneously Distributed in Space: Percolation Phase Transition and Uniqueness of the Unbounded Cluster. Journal of Theoretical Probability, 30(3), 784-812.
http://dx.doi.org/10.1007/s10959-015-0661-5
---------- CHICAGO ----------
Erhard, D., Martínez, J., Poisat, J. "Brownian Paths Homogeneously Distributed in Space: Percolation Phase Transition and Uniqueness of the Unbounded Cluster" . Journal of Theoretical Probability 30, no. 3 (2017) : 784-812.
http://dx.doi.org/10.1007/s10959-015-0661-5
---------- MLA ----------
Erhard, D., Martínez, J., Poisat, J. "Brownian Paths Homogeneously Distributed in Space: Percolation Phase Transition and Uniqueness of the Unbounded Cluster" . Journal of Theoretical Probability, vol. 30, no. 3, 2017, pp. 784-812.
http://dx.doi.org/10.1007/s10959-015-0661-5
---------- VANCOUVER ----------
Erhard, D., Martínez, J., Poisat, J. Brownian Paths Homogeneously Distributed in Space: Percolation Phase Transition and Uniqueness of the Unbounded Cluster. J. Theor. Probab. 2017;30(3):784-812.
http://dx.doi.org/10.1007/s10959-015-0661-5