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Percolation probability and critical exponents for site percolation on the UIPT

Part of: Geometric probability and stochastic geometry Graph theory Combinatorial probability Combinatorics Special processes

Published online by Cambridge University Press:  20 October 2022

Laurent Ménard
Laurent Ménard*
Affiliation:
New York University Shanghai, Shanghai, China and Laboratoire Modal’X, Université Paris Nanterre, Nanterre, France
*
e-mail: laurent.menard@normalesup.org
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Abstract

We derive three critical exponents for Bernoulli site percolation on the uniform infinite planar triangulation (UIPT). First, we compute explicitly the probability that the root cluster is infinite. As a consequence, we show that the off-critical exponent for site percolation on the UIPT is $\beta = 1/2$. Then we establish an integral formula for the generating function of the number of vertices in the root cluster. We use this formula to prove that, at criticality, the probability that the root cluster has at least n vertices decays like $n^{-1/7}$. Finally, we also derive an expression for the law of the perimeter of the root cluster and use it to establish that, at criticality, the probability that the perimeter of the root cluster is equal to n decays like $n^{-4/3}$. Among these three exponents, only the last one was previously known. Our main tools are the so-called gasket decomposition of percolation clusters, generic properties of random Boltzmann maps, and analytic combinatorics.

Keywords

Random planar mapstriangulationsUIPTpercolationcritical exponents

MSC classification

Primary: 05A15: Exact enumeration problems, generating functions 05A16: Asymptotic enumeration 05C12: Distance in graphs 05C30: Enumeration in graph theory 60C05: Combinatorial probability 60D05: Geometric probability and stochastic geometry 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Article
Information
Canadian Journal of Mathematics , Volume 75 , Issue 6 , December 2023 , pp. 1869 - 1903
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

This work is partially supported by the ANR grant ProGraM (Projet-ANR-19-CE40-0025) and the Labex MME-DII (ANR11-LBX-0023-01).

References

Albenque, M. and Ménard, L., Geometric properties of spin clusters in random triangulations coupled with an Ising model. Preprint, 2022. arXiv:2201.11922 Google Scholar
Angel, O., Growth and percolation on the uniform infinite planar triangulation . Geom. Funct. Anal. 13(2003), no. 5, 935–974.10.1007/s00039-003-0436-5CrossRefGoogle Scholar
Angel, O. and Curien, N., Percolations on random maps I: half-plane models . Ann. Inst. Henri Poincaré 51(2015), no. 2, 405–431.CrossRefGoogle Scholar
Angel, O. and Schramm, O., Uniform infinite planar triangulations . Comm. Math. Phys. 241(2003), nos. 2–3, 191–213.CrossRefGoogle Scholar
Beffara, V., The dimension of the SLE curves . Ann. Probab. 36(2008), no. 4, 1421–1452.10.1214/07-AOP364CrossRefGoogle Scholar
Bernardi, O., Curien, N., and Miermont, G., A Boltzmann approach to percolation on random triangulations . Canad. J. Math. 71(2019), no. 1, 1–43.10.4153/CJM-2018-009-xCrossRefGoogle Scholar
Bernardi, O., Holden, N., and Sun, X., Percolation on triangulations: a bijective path to Liouville quantum gravity. To appear in Mem. Am. Math. Soc. Preprint, 2021. arXiv:1807.01684 Google Scholar
Borot, G., Bouttier, J., and Duplantier, B., Nesting statistics in the O(n) loop model on random planar maps. Preprint, 2018. arXiv:1605.02239 Google Scholar
Borot, G., Bouttier, J., and Guitter, E., A recursive approach to the O(n) model on random maps via nested loops . J. Phys. A 45(2011), no. 4, 045002.10.1088/1751-8113/45/4/045002CrossRefGoogle Scholar
Borot, G., Bouttier, J., and Guitter, E., Loop models on random maps via nested loops: the case of domain symmetry breaking and application to the Potts model . J. Phys. A 45(2012), no. 49, 494017.10.1088/1751-8113/45/49/494017CrossRefGoogle Scholar
Borot, G., Bouttier, J., and Guitter, E., More on the O(n) model on random maps via nested loops: loops with bending energy . J. Phys. A 45(2012), no. 27, 275206.10.1088/1751-8113/45/27/275206CrossRefGoogle Scholar
Borot, G. and Garcia-Failde, E., Nesting statistics in the O(n) loop model on random maps of arbitrary topologies. Preprint, 2018. arXiv:1609.02074 Google Scholar
Bouttier, J., Di Francesco, P., and Guitter, E., Planar maps as labeled mobiles . Electron. J. Combin. 11(2004), no. 1, Article no. 69, 27 pp.10.37236/1822CrossRefGoogle Scholar
Budd, T., The peeling process of infinite Boltzmann planar maps . Electron. J. Combin. 23(2016), no. 1, Article no. 1.28, 37 pp.10.37236/5428CrossRefGoogle Scholar
Budd, T. and Curien, N., Simple peeling of planar maps with application to site percolation . Canad. J. Math. 74(2022), 907–941.10.4153/S0008414X21000146CrossRefGoogle Scholar
Curien, N. and Kortchemski, I., Percolation on random triangulations and stable looptrees . Probab. Theory Related Fields 163(2015), nos. 1–2, 303–337.10.1007/s00440-014-0593-5CrossRefGoogle Scholar
Curien, N. and Richier, L., Duality of random planar maps via percolation . Ann. Inst. Fourier (Grenoble) 70(2020), no. 6, 2425–2471.CrossRefGoogle Scholar
Eynard, B., Counting surfaces, Progress in Mathematical Physics, 70, Birkhäuser/Springer, Cham, 2016. CRM Aisenstadt chair lectures.10.1007/978-3-7643-8797-6CrossRefGoogle Scholar
Flajolet, P. and Sedgewick, R., Analytic combinatorics, Cambridge University Press, Cambridge, 2009.CrossRefGoogle Scholar
Garban, C., Pete, G., and Schramm, O., Pivotal, cluster, and interface measures for critical planar percolation . J. Amer. Math. Soc. 26(2013), no. 4, 939–1024.10.1090/S0894-0347-2013-00772-9CrossRefGoogle Scholar
Gorny, M., Maurel-Segala, E., and Singh, A., The geometry of a critical percolation cluster on the UIPT . Ann. Inst. Henri Poincaré 54(2018), no. 4, 2203–2238.CrossRefGoogle Scholar
Holden, N. and Sun, X., Convergence of uniform triangulations under the Cardy embedding. To appear in Acta Math. Preprint, 2021. arXiv:1905.13207 Google Scholar
Knizhnik, V. G., Polyakov, A. M., and Zamolodchikov, A. B., Fractal structure of $2$ D-quantum gravity. Modern Phys. Lett. A 3(1988), no. 8, 819–826.10.1142/S0217732388000982CrossRefGoogle Scholar
Ménard, L. and Nolin, P., Percolation on uniform infinite planar maps . Electron. J. Probab. 19(2014), no. 79, 1–27.10.1214/EJP.v19-2675CrossRefGoogle Scholar
Miermont, G., An invariance principle for random planar maps . Discrete Math. Theor. Comput. Sci. Proc. (2006), 39–58.Google Scholar
Normalesup.org. Maple companion file. Available on the author’s webpage: https://www.normalesup.org/menard/research.html Google Scholar
Richier, L., Universal aspects of critical percolation on random half-planar maps . Electron. J. Probab. 20(2015), Article no. 129, 45 pp.10.1214/EJP.v20-4041CrossRefGoogle Scholar