Lilypad Percolation

Let r be a nonnegative real number. Attach a disc of radius r to infinitely many random points (including the origin). Lilypad percolation asks whether we can reach infinity from the origin by walking through ‘lilypads’, that is, moving from one disc to another only if the discs overlap. In this paper, we explain what we mean by infinitely many random points, giving a definition of a Poisson Random Process. We also prove a theorem showing some lower bounds for r where we percolate and some upper bounds for r where we are stuck in finite land. We end with some remarks on other developments and numerical results, connecting lattice percolation models with continuum percolation models.