A Shape Theorem for Riemannian First-Passage Percolation

TL;DR

This paper proves a shape theorem for Riemannian first-passage percolation, showing large metric balls converge to a deterministic shape, and demonstrates that smooth random Riemannian metrics are almost surely geodesically complete.

Contribution

It establishes a shape theorem for a continuum FPP model and proves geodesic completeness for smooth random Riemannian metrics.

Findings

Large balls under the Riemannian FPP metric converge to a deterministic shape.

Smooth random Riemannian metrics are geodesically complete with probability one.

The shape theorem provides a new understanding of the asymptotic geometry in this model.

Abstract

Riemannian first-passage percolation (FPP) is a continuum model, with a distance function arising from a random Riemannian metric in $\R^d$. Our main result is a shape theorem for this model, which says that large balls under this metric converge to a deterministic shape under rescaling. As a consequence, we show that smooth random Riemannian metrics are geodesically complete with probability one.