Limit theorems for random spatial drainage networks

Penrose, Mathew D.; Wade, Andrew R.

doi:10.1017/s0001867800050394

Cited by 4 publications

(4 citation statements)

References 38 publications

(174 reference statements)

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“…As for minimal spanning trees, one of the main objects of interest is the total α-powered length, which is the Euclidean length raised to the power α > 0, of all the edges. Distributional approximation results for the sum of α-powered length of all the edges in an MDST with a different partial ordering on the points than the one considered here was proved in [PW10], where it was shown that for d ≥ 2 one obtains a Gaussian limit for small α while for large α, one has an additional independent non-Gaussian part in the limit. In this paper, we consider a related statistic, the total α-powered length of all the rooted edges, i.e., all the edges starting at the origin.…”

Section: Introduction and Main Resultsmentioning

confidence: 81%

Gaussian approximation for rooted edges in a random minimal directed spanning tree

Bhattacharjee¹

2021

Preprint

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We study the total α-powered length of rooted edges in a random minimal directed spanning tree -first introduced in [BR04] -in the unit cube [0, 1] d for d ≥ 3. In the case of d = 2, a Dickman limit was proved in [PW04]. In dimensions three and higher, [BLP06] showed a Gaussian central limit theorem when α = 1. In this paper, we extend the results and prove a central limit theorem for d ≥ 3 for any α > 0. Moreover, by utilizing recent results from [BM21] originating in Stein's method, we provide presumably optimal non-asymptotic bounds on the Wasserstein and the Kolmogorov distances between the distributions of total α-powered length of rooted edges, suitably normalized, and a standard Gaussian random variable.

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Section: Introduction and Main Resultsmentioning

confidence: 81%

Gaussian approximation for rooted edges in a random minimal directed spanning tree

Bhattacharjee¹

2021

Preprint

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“…First, directed navigations, where the flow of transport is steered towards a certain direction. This type of navigation has been studied rigorously in recent years and includes the directed spanning tree [1], [20], Delaunay routeing [2], the directed minimum spanning tree [3], [19], and the Poisson tree [9]. Second, navigations where the network nodes have a single site as their transport destination.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Traffic flow densities in large transport networks

2017

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We consider transport networks with nodes scattered at random in a large domain. At certain local rates, the nodes generate traffic flowing according to some navigation scheme in a given direction. In the thermodynamic limit of a growing domain, we present an asymptotic formula expressing the local traffic flow density at any given location in the domain in terms of three fundamental characteristics of the underlying network: the spatial intensity of the nodes together with their traffic generation rates, and of the links induced by the navigation. This formula holds for a general class of navigations satisfying a link-density and a sub-ballisticity condition. As a specific example, we verify these conditions for navigations arising from a directed spanning tree on a Poisson point process with inhomogeneous intensity function.

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“…More complex models assign edges between vertices in a randomized fashion (see, e.g., [12,21,51]). Some models introduce an ordering on the sampled points (see, e.g., [1,52,53,61,67]) which results in a directed graph that resembles the RRT tree [35].…”

Section: Random Geometric Graphsmentioning

confidence: 99%

New perspective on sampling-based motion planning via random geometric graphs

Solovey¹,

Salzman²,

Halperin³

Robotics: Science and Systems XII

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Abstract-Roadmaps constructed by many sampling-based motion planners coincide, in the absence of obstacles, with standard models of random geometric graphs (RGGs). Those models have been studied for several decades and by now a rich body of literature exists analyzing various properties and types of RGGs. In their seminal work on optimal motion planning Karaman and Frazzoli [31] conjectured that a sampling-based planner has a certain property if the underlying RGG has this property as well. In this paper we settle this conjecture and leverage it for the development of a general framework for the analysis of sampling-based planners. Our framework, which we call localization-tessellation, allows for easy transfer of arguments on RGGs from the free unit-hypercube to spaces punctured by obstacles, which are geometrically and topologically much more complex. We demonstrate its power by providing alternative and (arguably) simple proofs for probabilistic completeness and asymptotic (near-)optimality of probabilistic roadmaps (PRMs). Furthermore, we introduce two variants of PRMs, analyze them using our framework, and discuss the implications of the analysis.

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Limit theorems for random spatial drainage networks

Cited by 4 publications

References 38 publications

Gaussian approximation for rooted edges in a random minimal directed spanning tree

Gaussian approximation for rooted edges in a random minimal directed spanning tree

Traffic flow densities in large transport networks

New perspective on sampling-based motion planning via random geometric graphs

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