Covering monotonicity of the limit shapes of first passage percolation on crystal lattices

Abstract

This paper studies the first passage percolation model on crystal lattices, which is a generalization of that on the cubic lattice. Here, each edge of the graph induced by a crystal lattice is assigned a random passage time, and consideration is given to the behavior of the percolation region 𝐵(𝑡), which consists of those vertices that can be reached from the origin within a time 𝑡 > 0. Our first result is the shape theorem, stating that the normalized region 𝐵(𝑡)/𝑡 converges to some deterministic one, called the limit shape. The second result is the monotonicity of the limit shapes under covering maps. In particular, this provides insight into the limit shape of the cubic first passage percolation model.