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2014 Percolation on uniform infinite planar maps
Laurent Ménard, Pierre Nolin
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Electron. J. Probab. 19: 1-27 (2014). DOI: 10.1214/EJP.v19-2675

Abstract

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 $p^{\textrm{bond}}_c=1/3$.

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Laurent Ménard. Pierre Nolin. "Percolation on uniform infinite planar maps." Electron. J. Probab. 19 1 - 27, 2014. https://doi.org/10.1214/EJP.v19-2675

Information

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

zbMATH: 1300.60114
MathSciNet: MR3256879
Digital Object Identifier: 10.1214/EJP.v19-2675

Subjects:
Primary: 05C80
Secondary: 82B43

Keywords: Peeling process , percolation threshold , Random map , UIPQ

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