176 research outputs found
Subcritical regimes in the Poisson Boolean model of continuum percolation
We consider the Poisson Boolean model of continuum percolation. We show that
there is a subcritical phase if and only if is finite, where
denotes the radius of the balls around Poisson points and denotes the
dimension. We also give related results concerning the integrability of the
diameter of subcritical clusters.Comment: Published in at http://dx.doi.org/10.1214/07-AOP352 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Visibility to infinity in the hyperbolic plane, despite obstacles
Suppose that is a random closed subset of the hyperbolic plane \H^2,
whose law is invariant under isometries of \H^2. We prove that if the
probability that contains a fixed ball of radius 1 is larger than some
universal constant , then there is positive probability that contains
(bi-infinite) lines.
We then consider a family of random sets in \H^2 that satisfy some
additional natural assumptions. An example of such a set is the covered region
in the Poisson Boolean model. Let be the probability that a line segment
of length is contained in such a set . We show that if decays
fast enough, then there are almost surely no lines in . We also show that if
the decay of is not too fast, then there are almost surely lines in .
In the case of the Poisson Boolean model with balls of fixed radius we
characterize the critical intensity for the almost sure existence of lines in
the covered region by an integral equation.
We also determine when there are lines in the complement of a Poisson process
on the Grassmannian of lines in \H^2
Limit laws for k-coverage of paths by a Markov-Poisson-Boolean model
Let P := {X_i,i >= 1} be a stationary Poisson point process in R^d, {C_i,i >=
1} be a sequence of i.i.d. random sets in R^d, and {Y_i^t; t \geq 0, i >= 1} be
i.i.d. {0,1}-valued continuous time stationary Markov chains. We define the
Markov-Poisson-Boolean model C_t := {Y_i^t(X_i + C_i), i >= 1}. C_t represents
the coverage process at time t. We first obtain limit laws for k-coverage of an
area at an arbitrary instant. We then obtain the limit laws for the k-coverage
seen by a particle as it moves along a one-dimensional path.Comment: 1 figure. 24 Pages. Accepted at Stochastic Models. Theorems 6 and 7
corrected. Theorem 9 and Appendix adde
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