Geometry and percolation on half planar triangulations

2013/12/11 by Gourab Ray, Ray, Gourab · 1 citation
Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1312.3055

openalex publication_date 2013/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We analyze the geometry of domain Markov half planar triangulations. In \citeAR13 it is shown that there exists a one-parameter family of measures 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 in view of their geometry at α= 2/3. For α<2/3, the maps form a tree-like stricture with infinitely many small cut-sets. For α> 2/3, we obtain maps of hyperbolic nature with exponential growth and anchored expansion. Some results about the geometry of percolation clusters on such maps and random walk on them are also obtained.

Citations

• Scaling of Percolation on Infinite Planar Maps, I

Cited by

• Planar stochastic hyperbolic infinite triangulations

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