Electronic Journal of Probability

Geometry of infinite planar maps with high degrees

Timothy Budd and Nicolas Curien

Full-text: Open access


We study the geometry of infinite random Boltzmann planar maps with vertices of high degree. These correspond to the duals of the Boltzmann maps associated to a critical weight sequence $(q_{k})_{ k \geq 0}$ for the faces with polynomial decay $k^{-a}$ with $a \in ( \frac{3} {2}, \frac{5} {2})$ which have been studied by Le Gall & Miermont as well as by Borot, Bouttier & Guitter. We show the existence of a phase transition for the geometry of these maps at $a = 2$. In the dilute phase corresponding to $a \in (2, \frac{5} {2})$ we prove that the volume of the ball of radius $r$ (for the graph distance) is of order $r^{ \mathsf{d} }$ with $ \mathsf{d} = (a-\frac 12)/(a-2)$, and we provide distributional scaling limits for the volume and perimeter process. In the dense phase corresponding to $ a \in ( \frac{3} {2},2)$ the volume of the ball of radius $r$ is exponential in $r$. We also study the first-passage percolation (fpp) distance with exponential edge weights and show in particular that in the dense phase the fpp distance between the origin and $\infty $ is finite. The latter implies in addition that the random lattices in the dense phase are transient. The proofs rely on the recent peeling process introduced in [16] and use ideas of [22] in the dilute phase.

Article information

Electron. J. Probab. Volume 22 (2017), paper no. 35, 37 pp.

Received: 11 April 2016
Accepted: 5 April 2017
First available in Project Euclid: 19 April 2017

Permanent link to this document

Digital Object Identifier

Primary: 05C80: Random graphs [See also 60B20] 05C12: Distance in graphs 60G52: Stable processes 05C81: Random walks on graphs

Random planar map scaling limit peeling process graph distance stable processes

Creative Commons Attribution 4.0 International License.


Budd, Timothy; Curien, Nicolas. Geometry of infinite planar maps with high degrees. Electron. J. Probab. 22 (2017), paper no. 35, 37 pp. doi:10.1214/17-EJP55. https://projecteuclid.org/euclid.ejp/1492588824

Export citation


  • [1] L. Alili, L. Chaumont, and R. A. Doney,On a fluctuation identity for random walks and Lévy processes, Bull. London Math. Soc., 37 (2005), pp. 141–148.
  • [2] J. Ambjørn and T. Budd,Multi-point functions of weighted cubic maps, Ann. Inst. H. Poincaré D, 3 (2016), pp. 1–44.
  • [3] J. Ambjørn, T. Budd, and Y. Makeenko,Generalized multicritical one-matrix models, Nucl. Phys. B, 913 (2016), pp. 357–380.
  • [4] J. Ambjørn, B. Durhuus, and T. Jonsson,Quantum geometry: A statistical field approach, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1997.
  • [5] J. Ambjørn and Y. Watabiki,Scaling in quantum gravity, Nucl. Phys. B, 445 (1995), pp. 129–142.
  • [6] O. Angel,Growth and percolation on the uniform infinite planar triangulation, Geom. Funct. Anal., 13 (2003), pp. 935–974.
  • [7] O. Angel and N. Curien,Percolations on infinite random maps, half-plane models, Ann. Inst. H. Poincaré Probab. Statist., 51 (2014), pp. 405–431.
  • [8] O. Angel and G. Ray,Classification of half planar maps, Ann. of Probab., 43 (2015), pp. 1315–1349.
  • [9] O. Angel and O. Schramm,Uniform infinite planar triangulation, Comm. Math. Phys., 241 (2003), pp. 191–213.
  • [10] I. Benjamini and N. Curien,Simple random walk on the uniform infinite planar quadrangulation: Subdiffusivity via pioneer points, Geom. Funct. Anal., 23 (2013), pp. 501–531.
  • [11] J. Bertoin, T. Budd, N. Curien, and I. Kortchemski,Martingales in self-similar growth-fragmentations and their applications to random planar maps,arXiv:1605.00581(2016).
  • [12] J. Bertoin, N. Curien, and I. Kortchemski,Random planar maps & growth-fragmentations,arXiv:1507.02265(2015).
  • [13] N. H. Bingham, C. M. Goldie, and J. L. Teugels,Regular variation, vol. 27 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1989.
  • [14] J. E. Björnberg and S. O. Stefansson,Recurrence of bipartite planar maps, Electron. J. Probab., 19 (2014), pp. 1–40.
  • [15] G. Borot, J. Bouttier, and E. Guitter,A recursive approach to the O(N) model on random maps via nested loops, J. Phys. A: Math. Theor., 45 (2012).
  • [16] T. Budd,The peeling process of infinite Boltzmann planar maps, Electron. J. Comb., 23(1) (2016), #P1.28.
  • [17] T. Budd,The peeling process on random planar maps coupled to an O(n) loop model, In preparation, (2016).
  • [18] F. Caravenna and L. Chaumont,Invariance principles for random walks conditioned to stay positive, Ann. Inst. Henri Poincaré Probab. Stat., 44 (2008), pp. 170–190.
  • [19] N. Curien,Planar stochastic hyperbolic triangulations, Probab. Theory Relat. Fields, 165 (2016), pp. 509–540.
  • [20] N. Curien,A glimpse of the conformal structure of random planar maps, Commun. Math. Phys., 333 (2015), pp. 1417–1463.
  • [21] N. Curien, L. Chen, and P. Maillard,The perimeter cascade in critical Boltzmann quadrangulations decorated by an $O(n)$ loop model,arXiv:1702.06916(2017).
  • [22] N. Curien and J.-F. Le Gall,Scaling limits for the peeling process on random maps, Ann. Inst. H. Poincaré Probab. Statist., 53 (2017), pp. 322–357.
  • [23] N. Curien and J.-F. Le Gall,First-passage percolation and local perturbations on random planar maps, arXiv:1511.04264, (2015).
  • [24] D. Denisov, A. B. Dieker, and V. Shneer,Large deviations for random walks under subexponentiality: the big-jump domain., Ann. Probab., 36 (2008), pp. 1946–1991.
  • [25] O. Gurel-Gurevich and A. Nachmias,Recurrence of planar graph limits, Ann. Maths, 177 (2013), pp. 761–781.
  • [26] I. A. Ibragimov and Y. V. Linnik,Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen, 1971. With a supplementary chapter by I. A. Ibragimov and V. V. Petrov, Translation from the Russian edited by J. F. C. Kingman.
  • [27] J.-F. Le Gall and G. Miermont,Scaling limits of random planar maps with large faces, Ann. Probab., 39 (2011), pp. 1–69.
  • [28] R. Lyons and Y. Peres,Probability on Trees and Networks, Current version available athttp://mypage.iu.edu/~rdlyons/, In preparation.
  • [29] J.-F. Marckert and G. Miermont,Invariance principles for random bipartite planar maps, Ann. Probab., 35 (2007), pp. 1642–1705.
  • [30] L. Ménard and P. Nolin,Percolation on uniform infinite planar maps, Electron. J. Probab., 19 (2014), pp. 1–27.
  • [31] R. Stephenson,Local convergence of large critical multi-type galton-watson trees and applications to random maps, J. Theor. Probab., (2016), pp. 1–47.
  • [32] H. Tanaka,Time reversal of random walks in one-dimension, Tokyo J. Math., 12 (1989), pp. 159–174.
  • [33] Y. Watabiki,Construction of non-critical string field theory by transfer matrix formalism in dynamical triangulation, Nuclear Phys. B, 441 (1995), pp. 119–163.