Electronic Journal of Probability

Percolation on uniform infinite planar maps

Laurent Ménard and Pierre Nolin

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We construct the uniform infinite planar map (UIPM), obtained as the $n \to \infty$ local limit of planar maps with $n$ edges, chosen uniformly at random. We then describe how the UIPM can be sampled using a "peeling" process, in a similar way as for uniform triangulations. This process allows us to prove that for bond and site percolation on the UIPM, the percolation thresholds are $p^{\textrm{bond}}_c=1/2$ and $p^{\textrm{site}}_c=2/3$ respectively. This method also works for other classes of random infinite planar maps, and we show in particular that for bond percolation on the uniform infinite planar quadrangulation, the percolation threshold is $p^{\textrm{bond}}_c=1/3$.

Article information

Electron. J. Probab., Volume 19 (2014), paper no. 79, 27 pp.

Accepted: 2 September 2014
First available in Project Euclid: 4 June 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20]
Secondary: 82B43: Percolation [See also 60K35]

random map UIPQ percolation threshold peeling process

This work is licensed under a Creative Commons Attribution 3.0 License.


Ménard, Laurent; Nolin, Pierre. Percolation on uniform infinite planar maps. Electron. J. Probab. 19 (2014), paper no. 79, 27 pp. doi:10.1214/EJP.v19-2675. https://projecteuclid.org/euclid.ejp/1465065721

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