Abstract. We prove that the Poisson Boolean model, also known as the Gilbert disc model, is noise sensitive at criticality. This is the first such result for a Continuum Percolation model, and the first which involves a percolation model with critical probability p c = 1/2. Our proof uses a version of the Benjamini-Kalai-Schramm Theorem for biased product measures. A quantitative version of this result was recently proved by Keller and Kindler. We give a simple deduction of the non-quantitative result from the unbiased version. We also develop a quite general method of approximating Continuum Percolation models by discrete models with p c bounded away from zero; this method is based on an extremal result on non-uniform hypergraphs.

Partially motivated by the desire to better understand the connectivity phase transition in fractal percolation, we introduce and study a class of continuum fractal percolation models in dimension d ≥ 2. These include a scale invariant version of the classical (Poisson) Boolean model of stochastic geometry and (for d = 2) the Brownian loop soup introduced by Lawler and Werner.The models lead to random fractal sets whose connectivity properties depend on a parameter λ. In this paper we mainly study the transition between a phase where the random fractal sets are totally disconnected and a phase where they contain connected components larger than one point. In particular, we show that there are connected components larger than one point at the unique value of λ that separates the two phases (called the critical point). We prove that such a behavior occurs also in Mandelbrot's fractal percolation in all dimensions d ≥ 2. Our results show that it is a generic feature, independent of the dimension or the precise definition of the model, and is essentially a consequence of scale invariance alone.Furthermore, for d = 2 we prove that the presence of connected components larger than one point implies the presence of a unique, unbounded, connected component.

We study Mandelbrot's percolation process in dimension d ≥ 2. The process generates random fractal sets by an iterative procedure which starts by dividing the unit cube [0, 1] d in N d subcubes, and independently retaining or discarding each subcube with probability p or 1 − p respectively. This step is then repeated within the retained subcubes at all scales. As p is varied, there is a percolation phase transition in terms of paths for all d ≥ 2, and in terms of (d − 1)dimensional "sheets" for all d ≥ 3.For any d ≥ 2, we consider the random fractal set produced at the path-percolation critical value p c (N, d), and show that the probability that it contains a path connecting two opposite faces of the cube [0, 1] d tends to one as N → ∞. As an immediate consequence, we obtain that the above probability has a discontinuity, as a function of p, at p c (N, d) for all N sufficiently large. This had previously been proved only for d = 2 (for any N ≥ 2). For d ≥ 3, we prove analogous results for sheet-percolation.

We consider the Poisson cylinder model in R d , d ≥ 3. We show that given any two cylinders c 1 and c 2 in the process, there is a sequence of at most d − 2 other cylinders creating a connection between c 1 and c 2 . In particular, this shows that the union of the cylinders is a connected set, answering a question appearing in [13]. We also show that there are cylinders in the process that are not connected by a sequence of at most d − 3 other cylinders. Thus, the diameter of the cluster of cylinders equals d − 2.