Electronic Journal of Probability

Geometry and percolation on half planar triangulations

Gourab Ray

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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

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

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

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Zentralblatt MATH identifier

Primary: 60B05: Probability measures on topological spaces

half planar maps volume growth anchored expansion percolation

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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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  • Ambjørn, Jan. Quantization of geometry. Géométries fluctuantes en mécanique statistique et en théorie des champs (Les Houches, 1994), 77–193, North-Holland, Amsterdam, 1996.
  • Angel, O. Growth and percolation on the uniform infinite planar triangulation. Geom. Funct. Anal. 13 (2003), no. 5, 935–974.
  • O. Angel. Scaling of percolation on infinite planar maps, I. arXiv:math/0501006, 2005.
  • O. Angel and N. Curien. Percolations on random maps I: half-plane models. Ann. Inst. H. Poincaré, 2013. To appear.
  • O. Angel and G. Ray. Classification of half planar maps. Ann. Probab., 2013. To appear.
  • Benjamini, Itai; Curien, Nicolas. Simple random walk on the uniform infinite planar quadrangulation: subdiffusivity via pioneer points. Geom. Funct. Anal. 23 (2013), no. 2, 501–531.
  • Benjamini, Itai; Schramm, Oded. Percolation beyond $\Bbb Z^ d$, many questions and a few answers [ (97j:60179)]. Selected works of Oded Schramm. Volume 1, 2, 679–690, Sel. Works Probab. Stat., Springer, New York, 2011.
  • N. Curien. A glimpse of the conformal structure of random planar maps. ArXiv:1308.1807, 2013.
  • N. Curien and I. Kortchemski. Percolation on random triangulations and stable looptrees. ArXiv:1307.6818, 2013.
  • N. Curien and J.-F. Le Gall. The Brownian plane. Journal of Theoretical Probability, pages 1–43, 2012.
  • Curien, N.; Ménard, L.; Miermont, G. A view from infinity of the uniform infinite planar quadrangulation. ALEA Lat. Am. J. Probab. Math. Stat. 10 (2013), no. 1, 45–88.
  • Durrett, Rick. Probability: theory and examples. Fourth edition. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2010. x+428 pp. ISBN: 978-0-521-76539-8
  • Feller, William. An introduction to probability theory and its applications. Vol. II. Second edition John Wiley & Sons, Inc., New York-London-Sydney 1971 xxiv+669 pp.
  • R. G. Gallager. Discrete stochastic processes, volume 101. Kluwer Academic Publishers Boston, 1996.
  • Goulden, I. P.; Jackson, D. M. Combinatorial enumeration. With a foreword by Gian-Carlo Rota. A Wiley-Interscience Publication. Wiley-Interscience Series in Discrete Mathematics. John Wiley & Sons, Inc., New York, 1983. xxiv+569 pp. ISBN: 0-471-86654-7
  • Kesten, Harry. Subdiffusive behavior of random walk on a random cluster. Ann. Inst. H. Poincaré Probab. Statist. 22 (1986), no. 4, 425–487.
  • Kumagai, Takashi; Misumi, Jun. Heat kernel estimates for strongly recurrent random walk on random media. J. Theoret. Probab. 21 (2008), no. 4, 910–935.
  • Kuratowski, K. Topology. Vol. I. New edition, revised and augmented. Translated from the French by J. Jaworowski Academic Press, New York-London; PaÅ„stwowe Wydawnictwo Naukowe, Warsaw 1966 xx+560 pp.
  • Levin, David A.; Peres, Yuval; Wilmer, Elizabeth L. Markov chains and mixing times. With a chapter by James G. Propp and David B. Wilson. American Mathematical Society, Providence, RI, 2009. xviii+371 pp. ISBN: 978-0-8218-4739-8
  • R. Lyons and Y. Peres. Probability on Trees and Networks. Cambridge University Press. In preparation. Current version available at hfilbreak http://mypage.iu.edu/string rdlyons/.
  • L. Ménard and P. Nolin. Percolation on uniform infinite planar maps. 2013. ArXiv:1302.2851.
  • Thomassen, Carsten. Isoperimetric inequalities and transient random walks on graphs. Ann. Probab. 20 (1992), no. 3, 1592–1600.
  • Tutte, W. T. A census of planar triangulations. Canad. J. Math. 14 1962 21–38.
  • Virág, B. Anchored expansion and random walk. Geom. Funct. Anal. 10 (2000), no. 6, 1588–1605.
  • Watabiki, Yoshiyuki. Construction of non-critical string field theory by transfer matrix formalism in dynamical triangulation. Nuclear Phys. B 441 (1995), no. 1-2, 119–163.