Geodesics of Random Riemannian Metrics

Tom LaGatta, Jan Wehr

Research output: Contribution to journalArticle

5 Citations (Scopus)

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.

Original languageEnglish (US)
Pages (from-to)181-241
Number of pages61
JournalCommunications in Mathematical Physics
Volume327
Issue number1
DOIs
StatePublished - 2014

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Riemannian Metric
Geodesic
radon
Metric
encounters
theorems
curvature
Jacobi Field
Radon-Nikodym Derivative
perturbation
Markov Property
Riemannian geometry
Gaussian Measure
Positive Curvature
Random Perturbation
Random Environment
Local Properties
Absolutely Continuous
geometry
Euclidean

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Geodesics of Random Riemannian Metrics. / LaGatta, Tom; Wehr, Jan.

In: Communications in Mathematical Physics, Vol. 327, No. 1, 2014, p. 181-241.

Research output: Contribution to journalArticle

LaGatta, Tom ; Wehr, Jan. / Geodesics of Random Riemannian Metrics. In: Communications in Mathematical Physics. 2014 ; Vol. 327, No. 1. pp. 181-241.
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