Volume 20, 2016
|Page(s)||261 - 292|
|Published online||18 July 2016|
Moderate deviations for shortest-path lengths on random segment processes
1 Weierstrass Institute for Applied
Analysis and Stochastics, 10117
2 Institute of Stochastics, Ulm University, 89069 Ulm, Germany.
Accepted: 4 May 2016
We consider first-passage percolation on segment processes and provide concentration results concerning moderate deviations of shortest-path lengths from a linear function in the distance of their endpoints. The proofs are based on a martingale technique developed by [H. Kesten, Ann. Appl. Probab. 3 (1993) 296–338.] for an analogous problem on the lattice. Our results are applicable to graph models from stochastic geometry. For example, they imply that the time constant in Poisson−Voronoi and Poisson−Delaunay tessellations is strictly greater than 1. Furthermore, applying the framework of Howard and Newman, our results can be used to study the geometry of geodesics in planar shortest-path trees.
Mathematics Subject Classification: 60D05 / 05C80 / 82B43
Key words: Random segment process / first-passage percolation / moderate deviation / shortest-path
© EDP Sciences, SMAI, 2016
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