## Electronic Journal of Probability

### Geometry and percolation on half planar triangulations

Gourab Ray

#### 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

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

Rights

#### 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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