The Annals of Applied Probability

Shortest path through random points

Sung Jin Hwang, Steven B. Damelin, and Alfred O. Hero III

Full-text: Open access

Abstract

Let $(M,g_{1})$ be a complete $d$-dimensional Riemannian manifold for $d>1$. Let $\mathcal{X}_{n}$ be a set of $n$ sample points in $M$ drawn randomly from a smooth Lebesgue density $f$ supported in $M$. Let $x,y$ be two points in $M$. We prove that the normalized length of the power-weighted shortest path between $x,y$ through $\mathcal{X}_{n}$ converges almost surely to a constant multiple of the Riemannian distance between $x,y$ under the metric tensor $g_{p}=f^{2(1-p)/d}g_{1}$, where $p>1$ is the power parameter.

Article information

Source
Ann. Appl. Probab., Volume 26, Number 5 (2016), 2791-2823.

Dates
Received: February 2013
Revised: November 2015
First available in Project Euclid: 19 October 2016

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1476884304

Digital Object Identifier
doi:10.1214/15-AAP1162

Mathematical Reviews number (MathSciNet)
MR3563194

Zentralblatt MATH identifier
1353.60028

Subjects
Primary: 60F15: Strong theorems
Secondary: 60C05: Combinatorial probability 53B21: Methods of Riemannian geometry

Keywords
Shortest path power-weighted graph Riemannian geometry conformal metric

Citation

Hwang, Sung Jin; Damelin, Steven B.; Hero III, Alfred O. Shortest path through random points. Ann. Appl. Probab. 26 (2016), no. 5, 2791--2823. doi:10.1214/15-AAP1162. https://projecteuclid.org/euclid.aoap/1476884304


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