Electronic Journal of Probability

Geometry of infinite planar maps with high degrees

Timothy Budd and Nicolas Curien

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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

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Zentralblatt MATH 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

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