Dissertation: Geodesics of Random Riemannian Metrics

TL;DR

This paper introduces Riemannian First-Passage Percolation, a new model of random differential geometry, proving a shape theorem and showing that geodesics are almost surely not globally minimizing, revealing novel properties of random Riemannian metrics.

Contribution

It develops the Riemannian FPP model, proves a shape theorem, and demonstrates that geodesics are almost surely not globally minimizing, a novel insight in continuum random geometry.

Findings

Large balls converge to a deterministic shape under rescaling.

Smooth random Riemannian metrics are geodesically complete with probability one.

Geodesics starting from a fixed point in any direction are almost surely not globally minimizing.

Abstract

We introduce Riemannian First-Passage Percolation (Riemannian FPP) as a new model of random differential geometry, by considering a random, smooth Riemannian metric on $\mathbb R^d$. We are motivated in our study by the random geometry of first-passage percolation (FPP), a lattice model which was developed to model fluid flow through porous media. By adapting techniques from standard FPP, we prove a shape theorem for our 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. In differential geometry, geodesics are curves which locally minimize length. They need not do so globally: consider great circles on a sphere. For lattice models of FPP, there are many open questions related to minimizing geodesics; similarly, it is interesting from a geometric perspective when geodesics are globally minimizing. In the present study, we show that for any fixed starting direction $v$, the geodesic starting from the origin in the direction $v$ is not minimizing with probability one. This is a new result which uses the infinitesimal structure of the continuum, and for which there is no equivalent in discrete lattice models of FPP.