The number of unbounded components in the Poisson Boolean model of continuum percolation in hyperbolic space

Journal article, 2007

We show that for the Poisson Boolean model in hyperbolic space, there are intensities for the underlying Poisson process that produce infinitely many components in the covered region, if the radius of the balls in the model are big enough. In 2 dimensions, this is shown for any radius, and moreover we also consider unbounded components in the vacant region.