First passage percolation in Euclidean space and on random tessellations

Abstract

There are various models of first passage percolation (FPP) in $\mathbb R^d$. We want to start a very general study of this topic. To this end we generalize the first passage percolation model on the lattice $\mathbb Z^d$ to $\mathbb R^d$ and adapt the results of \cite{boivin1990first} to prove a shape theorem for ergodic random pseudometrics on $\mathbb R^d$. A natural application of this result will be the study of FPP on random tessellations where a fluid starts in the zero cell and takes a random time to pass through the boundary of a cell into a neighbouring cell. We find that a tame random tessellation, as introduced in the companion paper \cite{ziesche2016bernoulli}, has a positive time constant. This is used to derive a spatial ergodic theorem for the graph induced by the tessellation. Finally we take a look at the Poisson hyperplane tessellation, give an explicit formula to calculate it's FPP limit shape and bound the speed of convergence in the corresponding shape theorem.