Open Access
2017 Geometry of infinite planar maps with high degrees
Timothy Budd, Nicolas Curien
Electron. J. Probab. 22: 1-37 (2017). DOI: 10.1214/17-EJP55


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.


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Timothy Budd. Nicolas Curien. "Geometry of infinite planar maps with high degrees." Electron. J. Probab. 22 1 - 37, 2017.


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

zbMATH: 1360.05151
MathSciNet: MR3646061
Digital Object Identifier: 10.1214/17-EJP55

Primary: 05C12 , 05C80 , 05C81 , 60G52

Keywords: graph distance , Peeling process , Random planar map , Scaling limit , Stable processes

Vol.22 • 2017
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