Geometry of infinite planar maps with high degrees

TL;DR

This paper investigates the geometric properties of infinite planar maps with high-degree vertices, revealing a phase transition at a critical parameter and analyzing volume growth, scaling limits, and percolation properties.

Contribution

It establishes a phase transition at a = 2 for the geometry of these maps and provides detailed volume growth and scaling limit results in both phases.

Findings

Volume of balls scales polynomially in the dilute phase

Volume of balls grows exponentially in the dense phase

FPP distance is finite in the dense phase

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 $a \in ( 3/2, 5/2)$ 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 $a \in (2, 5/2)$ we prove that the volume of the ball of radius $r$ (for the graph distance) is of order $r^{\mathsf{d}}$ with $\mathsf{d}= (a-1/2)/(a-2)$, and we provide distributional scaling limits for the volume and perimeter process. In the dense phase corresponding to $a \in (3/2,2)$ 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 infinity 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 arXiv:1506.01590 and use ideas of arXiv:1412.5509 in the dilute phase.