Electronic Journal of Probability

Percolation and convergence properties of graphs related to minimal spanning forests

Christian Hirsch, Tim Brereton, and Volker Schmidt

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Lyons, Peres and Schramm have shown that minimal spanning forests on randomly weighted lattices exhibit a critical geometry in the sense that adding or deleting only a small number of edges results in a radical change of percolation properties. We show that these results can be extended to a Euclidean setting by considering families of stationary super- and subgraphs that approximate the Euclidean minimal spanning forest arbitrarily closely, but whose percolation properties differ decisively from those of the minimal spanning forest. Since these families can be seen as generalizations of the relative neighborhood graph and the nearest-neighbor graph, respectively, our results provide a new perspective on known percolation results from literature. We argue that the rates at which the approximating families converge to the minimal spanning forest are closely related to geometric characteristics of clusters in critical continuum percolation, and we show that convergence occurs at a polynomial rate.

Article information

Electron. J. Probab., Volume 22 (2017), paper no. 105, 21 pp.

Received: 10 January 2017
Accepted: 27 November 2017
First available in Project Euclid: 28 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 82B43: Percolation [See also 60K35]

Euclidean minimal spanning forest percolation nearest-neighbor graph relative neighborhood graph rate of convergence

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Hirsch, Christian; Brereton, Tim; Schmidt, Volker. Percolation and convergence properties of graphs related to minimal spanning forests. Electron. J. Probab. 22 (2017), paper no. 105, 21 pp. doi:10.1214/17-EJP129. https://projecteuclid.org/euclid.ejp/1514430041

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