## Electronic Journal of Probability

### Percolation and convergence properties of graphs related to minimal spanning forests

#### Abstract

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

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

Dates
Accepted: 27 November 2017
First available in Project Euclid: 28 December 2017

https://projecteuclid.org/euclid.ejp/1514430041

Digital Object Identifier
doi:10.1214/17-EJP129

Mathematical Reviews number (MathSciNet)
MR3742402

Zentralblatt MATH identifier
06827082

#### Citation

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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