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

• Autores: Erhard, Dirk; Martínez Linares, Julián Facundo; Poisat, Julien
• Año de publicación: 2016
• Idioma: inglés
• Tipo de recurso: artículo
• Materia: Continuum Percolation; Brownian Motion
• Institución: Consejo Nacional de Investigaciones Científicas y Técnicas

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.
Fil: Erhard, Dirk. University Of Warwick; Reino Unido
Fil: Martínez Linares, Julián Facundo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina
Fil: Poisat, Julien. Université Paris-Dauphine; Francia