The Annals of Probability

Random planar maps and growth-fragmentations

Jean Bertoin, Nicolas Curien, and Igor Kortchemski

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We are interested in the cycles obtained by slicing at all heights random Boltzmann triangulations with a simple boundary. We establish a functional invariance principle for the lengths of these cycles, appropriately rescaled, as the size of the boundary grows. The limiting process is described using a self-similar growth-fragmentation process with explicit parameters. To this end, we introduce a branching peeling exploration of Boltzmann triangulations, which allows us to identify a crucial martingale involving the perimeters of cycles at given heights. We also use a recent result concerning self-similar scaling limits of Markov chains on the nonnegative integers. A motivation for this work is to give a new construction of the Brownian map from a growth-fragmentation process.

Article information

Source
Ann. Probab., Volume 46, Number 1 (2018), 207-260.

Dates
Received: February 2016
Revised: January 2017
First available in Project Euclid: 5 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.aop/1517821222

Digital Object Identifier
doi:10.1214/17-AOP1183

Mathematical Reviews number (MathSciNet)
MR3758730

Zentralblatt MATH identifier
06865122

Subjects
Primary: 60F17: Functional limit theorems; invariance principles 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 60G51: Processes with independent increments; Lévy processes 60G18: Self-similar processes

Keywords
Planar maps growth-fragmentations self-similar Markov processes scaling limits

Citation

Bertoin, Jean; Curien, Nicolas; Kortchemski, Igor. Random planar maps and growth-fragmentations. Ann. Probab. 46 (2018), no. 1, 207--260. doi:10.1214/17-AOP1183. https://projecteuclid.org/euclid.aop/1517821222


Export citation

References

  • [1] Aldous, D. (1991). The continuum random tree. II. An overview. In Stochastic Analysis (Durham, 1990). London Mathematical Society Lecture Note Series 167 23–70. Cambridge Univ. Press, Cambridge.
  • [2] Angel, O. (2005). Scaling of percolation on infinite planar maps, I. Preprint, available on arXiv, arXiv:0501006.
  • [3] Angel, O. (2003). Growth and percolation on the uniform infinite planar triangulation. Geom. Funct. Anal. 13 935–974.
  • [4] Angel, O. and Curien, N. (2015). Percolations on random maps I: Half-plane models. Ann. Inst. Henri Poincaré B, Probab. Stat. 51 405–431.
  • [5] Angel, O. and Ray, G. (2015). Classification of half-planar maps. Ann. Probab. 43 1315–1349.
  • [6] Angel, O. and Schramm, O. (2003). Uniform infinite planar triangulations. Comm. Math. Phys. 241 191–213.
  • [7] Benjamini, I. and Curien, N. (2013). Simple random walk on the uniform infinite planar quadrangulation: Subdiffusivity via pioneer points. Geom. Funct. Anal. 23 501–531.
  • [8] Bertoin, J. (2002). Self-similar fragmentations. Ann. Inst. Henri Poincaré B, Probab. Stat. 38 319–340.
  • [9] Bertoin, J. (2016). Compensated fragmentation processes and limits of dilated fragmentations. Ann. Probab. 44 1254–1284.
  • [10] Bertoin, J. (2017). Markovian growth-fragmentation processes. Bernoulli 23 1082–1101.
  • [11] Bertoin, J. and Kortchemski, I. (2016). Self-similar scaling limits of Markov chains on the positive integers. Ann. Appl. Probab. 26 2556–2595.
  • [12] Bertoin, J. and Yor, M. (2005). Exponential functionals of Lévy processes. Probab. Surv. 2 191–212.
  • [13] Bettinelli, J. (2015). Scaling limit of random planar quadrangulations with a boundary. Ann. Inst. Henri Poincaré B, Probab. Stat. 51 432–477.
  • [14] Bettinelli, J. and Miermont, G. (2017). Compact Brownian surfaces I: Brownian disks. Probab. Theory Related Fields 167 555–614.
  • [15] Budd, T. (2016). The peeling process of infinite Boltzmann planar maps. Electron. J. Combin. 23 Paper 1.28, 37.
  • [16] Budzinski, T. (2016). The hyperbolic Brownian plan. Preprint, available on arXiv, arXiv:1604.06622.
  • [17] Curien, N. (2015). A glimpse of the conformal structure of random planar maps. Comm. Math. Phys. 333 1417–1463.
  • [18] Curien, N. (2016). Planar stochastic hyperbolic triangulations. Probab. Theory Related Fields 165 509–540.
  • [19] Curien, N. and Kortchemski, I. (2015). Percolation on random triangulations and stable looptrees. Probab. Theory Related Fields 163 303–337.
  • [20] Curien, N. and Le Gall, J.-F. (2015). First-passage percolation and local modifications of distances in random triangulations. Available at arXiv:1511.04264.
  • [21] Curien, N. and Le Gall, J.-F. (2014). The Brownian plane. J. Theoret. Probab. 27 1249–1291.
  • [22] Curien, N. and Le Gall, J.-F. (2017). Scaling limits for the peeling process on random maps. Ann. Inst. Henri Poincaré Probab. Stat. 53 322–357.
  • [23] Curien, N. and Le Gall, J.-F. (2016). The hull process of the Brownian plane. Probab. Theory Related Fields 166 187–231.
  • [24] Curien, N., Le Gall, J.-F. and Miermont, G. (2013). The Brownian cactus I. Scaling limits of discrete cactuses. Ann. Inst. Henri Poincaré B, Probab. Stat. 49 340–373.
  • [25] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
  • [26] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin.
  • [27] Krikun, M. (2007). Explicit enumeration of triangulations with multiple boundaries. Electron. J. Combin. 14 Research Paper 61, 14.
  • [28] Krikun, M. A. (2004). A uniformly distributed infinite planar triangulation and a related branching process. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 307 141–174, 282–283.
  • [29] Lamperti, J. (1972). Semi-stable Markov processes. I. Z. Wahrsch. Verw. Gebiete 22 205–225.
  • [30] Le Gall, J.-F. (2013). Uniqueness and universality of the Brownian map. Ann. Probab. 41 2880–2960.
  • [31] Le Gall, J.-F. (2014). Random geometry on the sphere. In Proceedings of the ICM.
  • [32] Lieb, E. H. and Loss, M. (2001). Analysis, 2nd ed. Graduate Studies in Mathematics 14. Amer. Math. Soc., Providence, RI.
  • [33] Lyons, R., Pemantle, R. and Peres, Y. (1995). Conceptual proofs of $L\log L$ criteria for mean behavior of branching processes. Ann. Probab. 23 1125–1138.
  • [34] Ménard, L. and Nolin, P. (2014). Percolation on uniform infinite planar maps. Electron. J. Probab. 19 no. 79, 27.
  • [35] Miermont, G. (2014). Aspects of random maps. In Saint-Flour Lecture Notes. (Preliminary version).
  • [36] Miermont, G. (2003). Self-similar fragmentations derived from the stable tree. I. Splitting at heights. Probab. Theory Related Fields 127 423–454.
  • [37] Miermont, G. (2013). The Brownian map is the scaling limit of uniform random plane quadrangulations. Acta Math. 210 319–401.
  • [38] Miller, J. and Sheffield, S. (2015). An axiomatic characterization of the Brownian map. arXiv:1506.03806.
  • [39] Miller, J. and Sheffield, S. (2016). Liouville quantum gravity and the Brownian map I: The QLE(8/3, 0) metric. Preprint, available on arXiv, arXiv:1507.00719.
  • [40] Miller, J. and Sheffield, S. (2016). Quantum Loewner evolution. Duke Math. J. 165 3241–3378.
  • [41] Schaeffer, G. (1998). Conjugaison d’arbres et cartes combinatoires aléatoires. PhD thesis.
  • [42] Stephenson, R. (2016). Local convergence of large critical multitype Galton–Watson trees and applications to random maps. J. Theoret. Probab. To appear.
  • [43] Uribe Bravo, G. (2009). The falling apart of the tagged fragment and the asymptotic disintegration of the Brownian height fragmentation. Ann. Inst. Henri Poincaré B, Probab. Stat. 45 1130–1149.
  • [44] Watabiki, Y. (1995). Construction of non-critical string field theory by transfer matrix formalism in dynamical triangulation. Nuclear Phys. B 441 119–163.