Convergence towards an asymptotic shape in first-passage percolation on cone-like subgraphs of the integer lattice

TL;DR

This paper extends the Shape Theorem in first-passage percolation to cone-like subgraphs of the integer lattice, characterizing their asymptotic shapes and convergence properties under various conditions.

Contribution

It identifies the asymptotic shapes for cone-like subgraphs and establishes necessary and sufficient conditions for different modes of convergence.

Findings

Asymptotic shapes are restrictions of the lattice's shape.

Conditions for $L^p$- and almost sure convergence are provided.

Analysis of complete convergence and stability in dynamic environments.

Abstract

In first-passage percolation on the integer lattice, the Shape Theorem provides precise conditions for convergence of the set of sites reachable within a given time from the origin, once rescaled, to a compact and convex limiting shape. Here, we address convergence towards an asymptotic shape for cone-like subgraphs of the $\Z^d$ lattice, where $d\ge2$. In particular, we identify the asymptotic shapes associated to these graphs as restrictions of the asymptotic shape of the lattice. Apart from providing necessary and sufficient conditions for $L^p$- and almost sure convergence towards this shape, we investigate also stronger notions such as complete convergence and stability with respect to a dynamically evolving environment.