Asymptotic Shapes for Stationary First Passage Percolation

Abstract

This paper deals with first passage percolation where the usual i.i.d. condition is weakened to stationarity (and ergodicity). The well known asymptotic shape result is known to extend to this case. It is easy to give necessary conditions for a compact set $B \subset \mathbb{R}^d$ to arise as the asymptotic shape for some stationary measure on the passage times. Our main result says that these conditions are also sufficient.