Electronic Journal of Probability

Geometry and percolation on half planar triangulations

Gourab Ray

Full-text: Open access

Abstract

We analyze the geometry of domain Markov half planar triangulations. In [5] it is shown thatthere exists a one-parameter family ofmeasures supported on half planar triangulations satisfying translation invariance and domain Markov property. We study the geometry of these maps and show that they exhibit a sharp phase-transition inview of their geometry atα = 2/3. For α < 2/3, the maps form atree-like stricture with infinitely many small cut-sets.For α > 2/3,we obtain maps of hyperbolic nature with exponential growth andanchoredexpansion. Some results about the geometry of percolation clusters on such maps and random walk on them are also obtained.

Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 47, 28 pp.

Dates
Accepted: 31 May 2014
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465065689

Digital Object Identifier
doi:10.1214/EJP.v19-3238

Mathematical Reviews number (MathSciNet)
MR3217335

Zentralblatt MATH identifier
1360.60034

Subjects
Primary: 60B05: Probability measures on topological spaces

Keywords
half planar maps volume growth anchored expansion percolation

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Ray, Gourab. Geometry and percolation on half planar triangulations. Electron. J. Probab. 19 (2014), paper no. 47, 28 pp. doi:10.1214/EJP.v19-3238. https://projecteuclid.org/euclid.ejp/1465065689


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