Brownian Paths Homogeneously Distributed in Space: Percolation Phase Transition and Uniqueness of the Unbounded Cluster
Résumé
We consider a continuum percolation model on $\R^d$, $d\geq 1$. For $t,\lambda\in (0,\infty)$ and $d\in\{1,2,3\}$, 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 $\lambda>0$. When $d\geq 4$, 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 $d\geq 2$, there is a non-trivial percolation transition in $t$, provided $\lambda$ and $r$ are chosen properly. The last statement means that $\lambda$ has to be chosen to be strictly smaller than the critical percolation parameter for the occupied set at time zero (which is infinite when $d\in\{2,3\}$, but finite and dependent on $r$ when $d\geq 4$). We further show that for all $d\geq 2$, the unbounded cluster in the supercritical phase is unique.\\ Along the line a finite box criterion for non-percolation in the Boolean model is extended to radius distributions with an exponential tails. This may be of independent interest.
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