Open Access
2017 Percolation and convergence properties of graphs related to minimal spanning forests
Christian Hirsch, Tim Brereton, Volker Schmidt
Electron. J. Probab. 22: 1-21 (2017). DOI: 10.1214/17-EJP129


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.


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Christian Hirsch. Tim Brereton. Volker Schmidt. "Percolation and convergence properties of graphs related to minimal spanning forests." Electron. J. Probab. 22 1 - 21, 2017.


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

zbMATH: 06827082
MathSciNet: MR3742402
Digital Object Identifier: 10.1214/17-EJP129

Primary: 60D05
Secondary: 82B43

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

Vol.22 • 2017
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