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- W1977429543 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." @default.
- W1977429543 created "2016-06-24" @default.
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- W1977429543 date "2014-02-18" @default.
- W1977429543 modified "2023-10-17" @default.
- W1977429543 title "Geodesics of Random Riemannian Metrics" @default.
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- W1977429543 doi "https://doi.org/10.1007/s00220-014-1901-8" @default.
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