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

TL;DR

This paper extends first passage percolation to crystal lattices, demonstrating that the limit shapes are monotonic under covering maps, which enhances understanding of the shape behavior in lattice models.

Contribution

It introduces a general FPP model on crystal lattices and proves the monotonicity of limit shapes under covering maps, linking lattice structures to shape properties.

Findings

Limit shapes are monotonic under covering maps.

The results apply to FPP on crystal lattices.

Provides insight into the shape behavior of cubic FPP.

Abstract

This paper studies the first passage percolation (FPP) model: each edge in the cubic 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 $t > 0$. Cox and Durrett showed the shape theorem for the percolation region, saying that the normalized region $B(t)/t$ converges to some limit shape $\mathcal{B}$. This paper introduces a general FPP model defined on crystal lattices, and shows the monotonicity of the limit shapes under covering maps, thereby providing insight into the limit shape of the cubic FPP model.