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

2020/09/24 by Mikami, Tatsuya
#82B43 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2009.11679

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 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.

Related

• First passage percolation in Euclidean space and on random tessellations
• Weak shape theorem in first passage percolation with infinite passage times
• Classification of the limit shape for 1+1-dimensional FPP
• Brochette first-passage percolation
• Monotonicity in first-passage percolation