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

Brownian Paths Homogeneously Distributed in Space: Percolation Phase Transition and Uniqueness of... We consider a continuum percolation model on $$\mathbb {R}^d$$ R d , $$d\ge 1$$ d ≥ 1 . For $$t,\lambda \in (0,\infty )$$ t , λ ∈ ( 0 , ∞ ) and $$d\in \{1,2,3\}$$ 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 $$\lambda >0$$ λ > 0 . When $$d\ge 4$$ d ≥ 4 , the Brownian paths are replaced by Wiener sausages with radius $$r>0$$ r > 0 . We establish that, for $$d=1$$ d = 1 and all choices of t, no percolation occurs, whereas for $$d\ge 2$$ d ≥ 2 , there is a non-trivial percolation transition in t, provided $$\lambda $$ λ and r are chosen properly. The last statement means that $$\lambda $$ 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 $$d\in \{2,3\}$$ d ∈ { 2 , 3 } , but finite and dependent on r when $$d\ge 4$$ d ≥ 4 ). We further show that for all $$d\ge 2$$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Theoretical Probability Springer Journals

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

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Publisher
Springer US
Copyright
Copyright © 2016 by Springer Science+Business Media New York
Subject
Mathematics; Probability Theory and Stochastic Processes; Statistics, general
ISSN
0894-9840
eISSN
1572-9230
D.O.I.
10.1007/s10959-015-0661-5
Publisher site
See Article on Publisher Site

Abstract

We consider a continuum percolation model on $$\mathbb {R}^d$$ R d , $$d\ge 1$$ d ≥ 1 . For $$t,\lambda \in (0,\infty )$$ t , λ ∈ ( 0 , ∞ ) and $$d\in \{1,2,3\}$$ 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 $$\lambda >0$$ λ > 0 . When $$d\ge 4$$ d ≥ 4 , the Brownian paths are replaced by Wiener sausages with radius $$r>0$$ r > 0 . We establish that, for $$d=1$$ d = 1 and all choices of t, no percolation occurs, whereas for $$d\ge 2$$ d ≥ 2 , there is a non-trivial percolation transition in t, provided $$\lambda $$ λ and r are chosen properly. The last statement means that $$\lambda $$ 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 $$d\in \{2,3\}$$ d ∈ { 2 , 3 } , but finite and dependent on r when $$d\ge 4$$ d ≥ 4 ). We further show that for all $$d\ge 2$$ 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.

Journal

Journal of Theoretical ProbabilitySpringer Journals

Published: Jan 14, 2016

References

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