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2014 Geometry and percolation on half planar triangulations
Gourab Ray
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Electron. J. Probab. 19: 1-28 (2014). DOI: 10.1214/EJP.v19-3238


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.


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Gourab Ray. "Geometry and percolation on half planar triangulations." Electron. J. Probab. 19 1 - 28, 2014.


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

zbMATH: 1360.60034
MathSciNet: MR3217335
Digital Object Identifier: 10.1214/EJP.v19-3238

Primary: 60B05

Keywords: anchored expansion , half planar maps , percolation , volume growth


Vol.19 • 2014
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