The asymptotic shape theorem for generalized first passage percolation

TL;DR

This paper extends the asymptotic shape theorem in first passage percolation to general semimetrics, providing new structure theorems, inequalities, and convergence rates for a broader class of models.

Contribution

It introduces a generalized framework for first passage percolation using semimetrics, along with new structure theorems and ergodic results.

Findings

Proved a structure theorem for equivariant semimetrics on topological groups.

Extended maximal inequalities for vector-valued cocycles.

Established convergence rates and generalized ergodic theorems for the models.

Abstract

We generalize the asymptotic shape theorem in first passage percolation on $\mathbb{Z}^d$ to cover the case of general semimetrics. We prove a structure theorem for equivariant semimetrics on topological groups and an extended version of the maximal inequality for $\mathbb{Z}^d$-cocycles of Boivin and Derriennic in the vector-valued case. This inequality will imply a very general form of Kingman's subadditive ergodic theorem. For certain classes of generalized first passage percolation, we prove further structure theorems and provide rates of convergence for the asymptotic shape theorem. We also establish a general form of the multiplicative ergodic theorem of Karlsson and Ledrappier for cocycles with values in separable Banach spaces with the Radon--Nikodym property.