We consider a statistically semi-scale-invariant collection of bi-infinite cylinders in \R^d, chosen according to a Poisson line process of intensity \lambda. The complement of the union of these cylinders is a random fractal which we denote by V. This fractal exhibits long-range dependence, complicating its analysis. Nevertheless, we show that this random fractal undergoes two different phase transitions. First and foremost we determine the critical value of \lambda for which V is non-empty.

We additionally show that for dm\geq 4 this random fractal exhibits a connectivity phase transition in the sense that the random fractal is not totally disconnected for \lambda small enough but positive. For d=3 we obtain a partial result showing that the fractal restricted to a fixed plane is always totally disconnected.

An important tool in understanding the connectivity phase transition is the study of a continuum percolation model which we call the fractal random ellipsoid model. This model is obtained as the intersection between the semi-scale-invariant Poisson cylinder model and a k-dimensional linear subspace of R^d. Moreover, this model can be understood as a Poisson point process in its own right with intensity measure \Leb_k \times \xi_{k,d}, where \Leb_k denotes the Lebesgue measure on \R^k and \xi_{k,d} is the shape measure describing the random ellipsoid.

Joint with Erik Broman, Filipe Mussini, Johan Tykesson.