### 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 language | English (US) |
---|---|

Pages (from-to) | 181-241 |

Number of pages | 61 |

Journal | Communications in Mathematical Physics |

Volume | 327 |

Issue number | 1 |

DOIs | |

State | Published - 2014 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*327*(1), 181-241. https://doi.org/10.1007/s00220-014-1901-8

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

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 327, no. 1, pp. 181-241. https://doi.org/10.1007/s00220-014-1901-8

}

TY - JOUR

T1 - Geodesics of Random Riemannian Metrics

AU - LaGatta, Tom

AU - Wehr, Jan

PY - 2014

Y1 - 2014

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84896396199&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84896396199&partnerID=8YFLogxK

U2 - 10.1007/s00220-014-1901-8

DO - 10.1007/s00220-014-1901-8

M3 - Article

AN - SCOPUS:84896396199

VL - 327

SP - 181

EP - 241

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -