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

Dirk Erhard (), Julián Martínez () and Julien Poisat ()
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Dirk Erhard: Warwick University
Julián Martínez: Conicet
Julien Poisat: Université Paris-Dauphine, PSL Research University

Journal of Theoretical Probability, 2017, vol. 30, issue 3, 784-812

Abstract: 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.

Keywords: Continuum percolation; Brownian motion; Poisson point process; Phase transition; Boolean percolation; Primary 60K35; 60J65; 60G55; Secondary 82B26 (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s10959-015-0661-5

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