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

Abstract

We consider a continuum percolation model on R d , 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 jumpboard for finer results.MSC 2010. Primary 60K35, 60J65, 60G55; Secondary 82B26.

Asymptotics of the critical time in Wiener sausage percolation with a small radius

“…(1) For completeness, we state that r → t c (r) stays bounded as r → 0 when d ∈ {2, 3}, since, by monotonicity, lim sup r→0 t c (r) ≤ t c (0) < ∞. This follows from [6,Theorem 2]. Continuity at r = 0 is not immediate, but we expect that this follows from a finite-box criterion of percolation.…”

“…The rigorous construction found in [6] yields ergodicity of O t,r with respect to shifts in space. For d ≥ 4,Cerný, Funken and Spodarev [3] used this model to describe the target detection area of a network of mobile sensors initially distributed at random and moving according to Brownian motions.…”

“…Let E be a Poisson point process with intensity λ Leb d , where λ > 0. Conditionally on E, we define a collection of independent Brownian motions {(B x t ) t≥0 , x ∈ E} such that for each x ∈ E, B x 0 = x and (B x t − x) t≥0 is independent of E. We refer the reader to Section 1.4 in [6] for a rigorous construction. Let P and E be the probability measure and expectation of Brownian motion, respectively.…”

“…A set is said to percolate if it contains an unbounded connected component. In [6] it was shown that O t,r undergoes a non-trivial percolation phase transition for all d ≥ 2. More precisely it was shown that if d ∈ {2, 3}, then for all λ > 0 there exists t c (λ) ∈ (0, ∞) such that for all t < t c (λ) the set O t only contains bounded connected components, whereas for t > t c (λ), the set O t percolates with a unique unbounded cluster.…”

“…In this case, denote by λ c (r) the critical value such that the set O 0,r a.s. percolates for all λ > λ c (r), and a.s. does not for λ < λ c (r), see Section 3.3 in Meester and Roy [15]. Theorem 1.3 in [6] states that when r > 0 and λ < λ c (r), then there is a critical time t c (λ, r) ∈ (0, ∞) which separates a percolation regime (t > t c (λ, r)) from a non-percolation regime (t < t c (λ, r)). Equivalently, a phase transition occurs when λ is fixed and the radius is chosen smaller than a critical radius r c (λ).…”