Electronic Journal of Probability

Geometry of infinite planar maps with high degrees

Abstract

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

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

Dates
Accepted: 5 April 2017
First available in Project Euclid: 19 April 2017

https://projecteuclid.org/euclid.ejp/1492588824

Digital Object Identifier
doi:10.1214/17-EJP55

Citation

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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