Percolation on a non-homogeneous Poisson blob process

One of the most well known examples of phenomena that introduces and motivates the study of continuum percolation is the process of the ground getting wet during a period of rain. At each point hit by a raindrop, one sees a circular wet patch. Right after the rain begins to fall what one sees is a small wet region inside a large dry region. At some instant, so many raindrops have hit the ground that the situation changes from that to a small dry region inside a large wet region. Typically, the parameter in which there is a phase transition behaviour is the density of the raindrops. Continuum percolation models in which each point of a two-dimensional homogeneous Poisson point process is the centre of a disk of given (or random) radius r, have been extensively studied. In this note we present phase transition results for a sequence of Poisson point process which defines Poisson Boolean models and whose rates depend on the past. In order to prove our results we rely on a multiscale percolation structure. General reference for percolation and continuum percolation are the books of Grimmett [2] and Meester and Roy [3]. A nice example of the use of multi-scale percolation technique can be found is Fontes et al [1].