Brownian Paths Homogeneously Distributed in Space: Percolation Phase Transition and Uniqueness of the Unbounded Cluster

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 re...

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Detalles Bibliográficos
Publicado: 2017
Materias:
Boolean percolation
Brownian motion
Continuum percolation
Phase transition
Poisson point process
Acceso en línea:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_08949840_v30_n3_p784_Erhard
http://hdl.handle.net/20.500.12110/paper_08949840_v30_n3_p784_Erhard
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
id paper:paper_08949840_v30_n3_p784_Erhard
record_format dspace
spelling paper:paper_08949840_v30_n3_p784_Erhard2023-06-08T15:47:59Z Brownian Paths Homogeneously Distributed in Space: Percolation Phase Transition and Uniqueness of the Unbounded Cluster Boolean percolation Brownian motion Continuum percolation Phase transition Poisson point process 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. 2017 https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_08949840_v30_n3_p784_Erhard http://hdl.handle.net/20.500.12110/paper_08949840_v30_n3_p784_Erhard
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Boolean percolation
Brownian motion
Continuum percolation
Phase transition
Poisson point process
spellingShingle Boolean percolation
Brownian motion
Continuum percolation
Phase transition
Poisson point process
Brownian Paths Homogeneously Distributed in Space: Percolation Phase Transition and Uniqueness of the Unbounded Cluster
topic_facet Boolean percolation
Brownian motion
Continuum percolation
Phase transition
Poisson point process
description 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.
title Brownian Paths Homogeneously Distributed in Space: Percolation Phase Transition and Uniqueness of the Unbounded Cluster
title_short Brownian Paths Homogeneously Distributed in Space: Percolation Phase Transition and Uniqueness of the Unbounded Cluster
title_full Brownian Paths Homogeneously Distributed in Space: Percolation Phase Transition and Uniqueness of the Unbounded Cluster
title_fullStr Brownian Paths Homogeneously Distributed in Space: Percolation Phase Transition and Uniqueness of the Unbounded Cluster
title_full_unstemmed Brownian Paths Homogeneously Distributed in Space: Percolation Phase Transition and Uniqueness of the Unbounded Cluster
title_sort brownian paths homogeneously distributed in space: percolation phase transition and uniqueness of the unbounded cluster
publishDate 2017
url https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_08949840_v30_n3_p784_Erhard
http://hdl.handle.net/20.500.12110/paper_08949840_v30_n3_p784_Erhard
_version_ 1768545056547930112

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