Electronic Journal of Probability

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

Loïc Richier

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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

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

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

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Zentralblatt MATH identifier

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

random planar maps local limits percolation incipient infinite cluster

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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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