Electronic Journal of Probability

The incipient infinite cluster of the uniform infinite half-planar triangulation

Loïc Richier

Full-text: Open access

Abstract

We introduce the Incipient Infinite Cluster ($\mathsf{IIC} $) in the critical Bernoulli site percolation model on the Uniform Infinite Half-Planar Triangulation ($\mathsf{UIHPT} $), which is the local limit of large random triangulations with a boundary. The $\mathsf{IIC} $ is defined from the $\mathsf{UIHPT} $ by conditioning the open percolation cluster of the origin to be infinite. We prove that the $\mathsf{IIC} $ can be obtained by adding within the $\mathsf{UIHPT} $ an infinite triangulation with a boundary whose distribution is explicit.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 89, 38 pp.

Dates
Received: 29 September 2017
Accepted: 28 August 2018
First available in Project Euclid: 12 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1536717748

Digital Object Identifier
doi:10.1214/18-EJP218

Mathematical Reviews number (MathSciNet)
MR3858917

Zentralblatt MATH identifier
1398.05183

Subjects
Primary: 05C80: Random graphs [See also 60B20] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K37: Processes in random environments

Keywords
random planar maps local limits percolation incipient infinite cluster

Rights
Creative Commons Attribution 4.0 International License.

Citation

Richier, Loïc. The incipient infinite cluster of the uniform infinite half-planar triangulation. Electron. J. Probab. 23 (2018), paper no. 89, 38 pp. doi:10.1214/18-EJP218. https://projecteuclid.org/euclid.ejp/1536717748


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