1206.4939 (Tom LaGatta et al.)

Geodesics of Random Riemannian Metrics I: Random Perturbations of
Euclidean Geometry    [PDF]

Tom LaGatta, Jan Wehr

We analyze the disordered geometry resulting from random permutations of Euclidean space. 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 perturbations, and 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. In Part II, we will use this to prove that a geodesic with random initial conditions is almost surely not minimizing. We also develop in this paper some general results on conditional Gaussian measures.

View original: http://arxiv.org/abs/1206.4939