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A Boltzmann Approach to Percolation on Random Triangulations

  • Olivier Bernardi (a1), Nicolas Curien (a2) and Grégory Miermont (a3)

We study the percolation model on Boltzmann triangulations using a generating function approach. More precisely, we consider a Boltzmann model on the set of finite planar triangulations, together with a percolation configuration (either site-percolation or bond-percolation) on this triangulation. By enumerating triangulations with boundaries according to both the boundary length and the number of vertices/edges on the boundary, we are able to identify a phase transition for the geometry of the origin cluster. For instance, we show that the probability that a percolation interface has length $n$ decays exponentially with $n$ except at a particular value $p_{c}$ of the percolation parameter $p$ for which the decay is polynomial (of order $n^{-10/3}$ ). Moreover, the probability that the origin cluster has size $n$ decays exponentially if $p<p_{c}$ and polynomially if $p\geqslant p_{c}$ .

The critical percolation value is $p_{c}=1/2$ for site percolation, and $p_{c}=(2\sqrt{3}-1)/11$ for bond percolation. These values coincide with critical percolation thresholds for infinite triangulations identified by Angel for site-percolation, and by Angel and Curien for bond-percolation, and we give an independent derivation of these percolation thresholds.

Lastly, we revisit the criticality conditions for random Boltzmann maps, and argue that at $p_{c}$ , the percolation clusters conditioned to have size $n$ should converge toward the stable map of parameter $\frac{7}{6}$ introduced by Le Gall and Miermont. This enables us to derive heuristically some new critical exponents.

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We thank the Newton institute for hospitality during the Random Geometry program in 2015 where part of this work was completed. We acknowledge the support of the NSF grant DMS-1400859, and of the Agence Nationale de la Recherche via the grants ANR Liouville (ANR-15-CE40-0013) and ANR GRAAL (ANR-14-CE25-0014).

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[1] Angel, O., Growth and percolation on the uniform infinite planar triangulation . Geom. Funct. Anal. 13(2003), no. 5, 935974.
[2] Angel, O. and Curien, N., Percolations on infinite random maps, half-plane models . Ann. Inst. H. Poincaré Probab. Stat. 51(2015), no. 2, 405431.
[3] Angel, O. and Schramm, O., Uniform infinite planar triangulation . Comm. Math. Phys. 241(2003), no. 2–3, 191213.
[4] Bertoin, J., Curien, N., and Kortchemski, I., Random planar maps and growth-fragmentations . Ann. of Probab. 46(2018), no. 1, 207260.
[5] Borot, G., Bouttier, J., and Guitter, E., Loop models on random maps via nested loops: case of domain symmetry breaking and application to the potts model . J. Phys. A 45(2012), no. 49, 494017.
[6] 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(2012), no. 4, 045002.
[7] Bousquet-Mélou, M. and Jehanne, A., Polynomial equations with one catalytic variable, algebraic series and map enumeration . J. Combin. Theory Ser. B 96(2006), no. 5, 623672.
[8] Bouttier, J., Di Francesco, P., and Guitter, E., Planar maps as labeled mobiles . Electron. J. Combin. 11(2004), no. 1, Research Paper 69.
[9] Budd, T., The peeling process of infinite Boltzmann planar maps . Electron. J. Combin. 23(2016), no. 1, Paper 1.28.
[10] Curien, N., Peeling random planar maps.∼curien/.
[11] Curien, N. and Kortchemski, I., Percolation on random triangulations and stable looptrees . Probab. Theory Related Fields 163(2015), no. 1–2, 303337.
[12] Curien, N. and Le Gall, J.-F., Scaling limits for the peeling process on random maps . Ann. Inst. Henri Poincaré Probab. Stat. 53(2017), no. 1, 322357.
[13] Curien, N., Le Gall, J.-F., and Miermont, G., The Brownian cactus I. Scaling limits of discrete cactuses . Ann. Inst. Henri Poincaré Probab. Stat. 49(2013), no. 2, 340373.
[14] Flajolet, P. and Sedgewick, R., Analytic combinatorics. Cambridge University Press, Cambridge, 2009.
[15] Gorny, M., Maurel-Segala, E., and Singh, A., The geometry of a critical percolation cluster on the UIPT. 2017. arxiv:1701.01667.
[16] Goulden, I. and Jackson, D., Combinatorial enumeration . Wiley-Interscience Series in Discrete Mathematics. John Wiley and Sons, New York, 1983.
[17] Le Gall, J.-F. and Miermont, G., Scaling limits of random planar maps with large faces . Ann. Probab. 39(2011), no. 1, 169.
[18] Marckert, J.-F. and Miermont, G., Invariance principles for random bipartite planar maps . Ann. Probab. 35(2007), no. 5, 16421705.
[19] Ménard, L., Volumes in the uniform infinite planar triangulation: from skeletons to generating functions. 2016. arxiv:1604.00908.
[20] Ménard, L. and Nolin, P., Percolation on uniform infinite planar maps . Electron. J. Probab. 19(2014), no. 79.
[21] Miermont, G., An invariance principle for random planar maps. In: Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities, Discrete Math. Theor. Comput. Sci. Proc., AG, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2006, pp. 39–57.
[22] Miermont, G., Invariance principles for spatial multitype Galton–Watson trees . Ann. Inst. Henri Poincaré Probab. Stat. 44(2008), no. 6, 11281161.
[23]Maple worksheet: site-percolation-triangulations.mws; see
[24]Maple worksheet: bond-percolation-triangulations.mws; see
[25] Richier, L., Universal aspects of critical percolation on random half-planar maps . Electron. J. Probab. 20(2015), Paper No. 129.
[26] Richier, L., Limits of the boundary of random planar maps. 2017. arxiv:1704.01950.
[27] Stephenson, R., Local convergence of large critical multi-type Galton–Watson trees and applications to random maps . J. Theoret. Probab. 31(2018), no. 1, 159205.
[28] Tutte, W., A census of slicings . Canad. J. Math. 14(1962), 708722.
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
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