First passage percolation in Euclidean space and on random tessellations

TL;DR

This paper extends first passage percolation models to Euclidean space and random tessellations, establishing a shape theorem and analyzing specific tessellations like the Poisson hyperplane tessellation.

Contribution

It generalizes FPP to $\,\mathbb{R}^d$ and random tessellations, proving a shape theorem and providing explicit results for Poisson hyperplane tessellations.

Findings

Shape theorem for ergodic random pseudometrics in $\,\mathbb{R}^d$

Positive time constant for tame random tessellations

Explicit formula and convergence bounds for Poisson hyperplane tessellation

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.