Geometry and percolation on half planar triangulations

Gourab Ray (University of British Columbia)

Abstract


We analyze the geometry of domain Markov half planar triangu-lations. 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.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-28

Publication Date: May 31, 2014

DOI: 10.1214/EJP.v19-3238

References

  • Ambjørn, Jan. Quantization of geometry. Géométries fluctuantes en mécanique statistique et en théorie des champs (Les Houches, 1994), 77--193, North-Holland, Amsterdam, 1996. MR1426135
  • Angel, O. Growth and percolation on the uniform infinite planar triangulation. Geom. Funct. Anal. 13 (2003), no. 5, 935--974. MR2024412
  • 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. MR3053754
  • Benjamini, Itai; Schramm, Oded. Percolation beyond $\Bbb Z^ d$, many questions and a few answers [ MR1423907 (97j:60179)]. Selected works of Oded Schramm. Volume 1, 2, 679--690, Sel. Works Probab. Stat., Springer, New York, 2011. MR2883387
  • 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. MR3083919
  • 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 MR2722836
  • 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. MR0270403
  • 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 MR0702512
  • Kesten, Harry. Subdiffusive behavior of random walk on a random cluster. Ann. Inst. H. Poincaré Probab. Statist. 22 (1986), no. 4, 425--487. MR0871905
  • Kumagai, Takashi; Misumi, Jun. Heat kernel estimates for strongly recurrent random walk on random media. J. Theoret. Probab. 21 (2008), no. 4, 910--935. MR2443641
  • Kuratowski, K. Topology. Vol. I. New edition, revised and augmented. Translated from the French by J. Jaworowski Academic Press, New York-London; Państwowe Wydawnictwo Naukowe, Warsaw 1966 xx+560 pp. MR0217751
  • 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 MR2466937
  • 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. MR1175279
  • Tutte, W. T. A census of planar triangulations. Canad. J. Math. 14 1962 21--38. MR0130841
  • Virág, B. Anchored expansion and random walk. Geom. Funct. Anal. 10 (2000), no. 6, 1588--1605. MR1810755
  • 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. MR1329946
Bookmark and Share


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