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April 2001 Special Invited Paper: Geodesics And Spanning Tees For Euclidean First-Passage Percolaton
C. Douglas Howard, Charles M. Newman
Ann. Probab. 29(2): 577-623 (April 2001). DOI: 10.1214/aop/1008956686

Abstract

The metric $D_{\alpha}(q,q')$ on the set $Q$ of particle locations of a homogeneous Poisson process on $\mathbb{R}^d$ , defined as the infimum of $(\sum_i |q_i - q_{i+1}|^{\alpha})^{1/\alpha}$ over sequences in $Q$ starting with $q$ and ending with $q'$ (where $|·|$ denotes Euclidean distance) has nontrivial geodesics when $\alpha>1$. The cases $1< \alpha<\infty$ are the Euclidean first­passage percolation (FPP) models introduced earlier by the authors, while the geodesics in the case $\alpha = \infty$ are exactly the paths from the Euclidean minimal spanning trees/forests of Aldous and Steele. We compare and contrast results and conjectures for these two situations. New results for $1 < \alpha < \infty$ (and any $d$) include inequalities on the fluctuation exponents for the metric $(\chi \le 1/2)$ and for the geodesics $(\xi \le 3/4)$ in strong enough versions to yield conclusions not yet obtained for lattice FPP: almost surely, every semiinfinite geodesic has an asymptotic direction and every direction has a semiinfinite geodesic (from every $q$). For $d = 2$ and $2 \le \alpha < \infty$, further results follow concerning spanning trees of semiinfinite geodesics and related random surfaces.

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C. Douglas Howard. Charles M. Newman. "Special Invited Paper: Geodesics And Spanning Tees For Euclidean First-Passage Percolaton." Ann. Probab. 29 (2) 577 - 623, April 2001. https://doi.org/10.1214/aop/1008956686

Information

Published: April 2001
First available in Project Euclid: 21 December 2001

MathSciNet: MR1849171
Digital Object Identifier: 10.1214/aop/1008956686

Subjects:
Primary: 60G55 , 60K35
Secondary: 60F10 , 82D30

Keywords: Combinatorial optimization , first­passage percolation , Geodesic , Minimal spanning tree , Poisson process , Random metric , Random surface , shape theorem

Rights: Copyright © 2001 Institute of Mathematical Statistics

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Vol.29 • No. 2 • April 2001
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