First passage percolation in Euclidean space and on random tessellations

2016/11/07 by Sebastian Ziesche, Ziesche, Sebastian · 1 citation
Mathematics · Biochemistry, Genetics and Molecular Biology · #Stochastic processes and statistical mechanics #Diffusion and Search Dynamics #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.1611.02005

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 \citeboivin1990first 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 \citeziesche2016bernoulli, 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.

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• Random pseudometrics and applications

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