Geodesics of Random Riemannian Metrics

TL;DR

This paper studies the properties of geodesics in a randomly perturbed Riemannian metric, proving that typical geodesics are almost surely not length-minimizing due to encounters with positive curvature regions.

Contribution

It introduces a new analysis of geodesics in disordered Riemannian environments, including the law of the environment observed from a moving particle and the non-minimizing nature of typical geodesics.

Findings

Geodesics almost surely are not minimizing in random Riemannian metrics.

The environment observed from a moving particle has an explicit Radon-Nikodym derivative.

Typical geodesics encounter positive curvature regions, destabilizing their minimizing property.

Abstract

We analyze the disordered Riemannian geometry resulting from random perturbations of the Euclidean metric. We focus on geodesics, the paths traced out by a particle traveling in this quenched random environment. By taking the point of the view of the particle, we show that the law of its observed environment is absolutely continuous with respect to the law of the random metric, and we provide an explicit form for its Radon-Nikodym derivative. We use this result to prove a "local Markov property" along an unbounded geodesic, demonstrating that it eventually encounters any type of geometric phenomenon. We also develop in this paper some general results on conditional Gaussian measures. Our Main Theorem states that a geodesic chosen with random initial conditions (chosen independently of the metric) is almost surely not minimizing. To demonstrate this, we show that a minimizing geodesic is guaranteed to eventually pass over a certain "bump surface," which locally has constant positive curvature. By using Jacobi fields, we show that this is sufficient to destabilize the minimizing property.