Electronic Journal of Probability

Universal aspects of critical percolation on random half-planar maps

Loïc Richier

Full-text: Open access

Abstract

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

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

Dates
Received: 6 January 2015
Accepted: 7 December 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067235

Digital Object Identifier
doi:10.1214/EJP.v20-4041

Mathematical Reviews number (MathSciNet)
MR3438743

Zentralblatt MATH identifier
1329.05267

Subjects
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

Keywords
Random planar maps Percolation Critical threshold Crossing probabilities

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

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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