Electronic Journal of Probability

Universal aspects of critical percolation on random half-planar maps

Loïc Richier

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We study a large class of Bernoulli percolation models on random lattices of the half-plane, obtained as local limits of uniform planar triangulations or quadrangulations. We first compute the exact value of the site percolation threshold in the quadrangular case using the so-called peeling techniques. Then, we generalize a result of Angel about the scaling limit of crossing probabilities, that are a natural analogue to Cardy’s formula in (non-random) plane lattices. Our main result is that those probabilities are universal, in the sense that they do not depend on the percolation model neither on the degree of the faces of the map.

Article information

Electron. J. Probab., Volume 20 (2015), paper no. 129, 45 pp.

Received: 6 January 2015
Accepted: 7 December 2015
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: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K37: Processes in random environments

Random planar maps Percolation Critical threshold Crossing probabilities

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Richier, Loïc. Universal aspects of critical percolation on random half-planar maps. Electron. J. Probab. 20 (2015), paper no. 129, 45 pp. doi:10.1214/EJP.v20-4041. https://projecteuclid.org/euclid.ejp/1465067235

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