Examples of nonpolygonal limit shapes in i.i.d. first-passage percolation and infinite coexistence in spatial growth models

TL;DR

This paper constructs a specific edge-weight distribution for i.i.d. first-passage percolation on 2 that results in a nonpolygonal limit shape with densely distributed extreme points, enabling infinite coexistence in spatial growth models.

Contribution

It introduces a novel edge-weight distribution leading to nonpolygonal limit shapes and demonstrates infinite coexistence in related spatial growth models.

Findings

Limit shape is not a polygon.

Extreme points are densely distributed.

Infinite coexistence of species is possible.

Abstract

We construct an edge-weight distribution for i.i.d. first-passage percolation on $\mathbb{Z}^2$ whose limit shape is not a polygon and whose extreme points are arbitrarily dense in the boundary. Consequently, the associated Richardson-type growth model can support coexistence of a countably infinite number of distinct species, and the graph of infection has infinitely many ends.