On scaling limits of planar maps with stable face-degrees

TL;DR

This paper investigates the scaling limits of random planar maps with face degrees in the domain of attraction of a stable law, showing convergence to the Brownian map for  and to -stable maps for <, extending previous results.

Contribution

It extends the understanding of scaling limits of planar maps with stable face degrees, removing regularity assumptions in prior work and identifying new stable map limits.

Findings

Convergence to the Brownian map for  face degrees.

Convergence to -stable maps for < after subsequence extraction.

Improves on previous results by Le Gall & Miermont with fewer regularity assumptions.

Abstract

We discuss the asymptotic behaviour of random critical Boltzmann planar maps in which the degree of a typical face belongs to the domain of attraction of a stable law with index $\alpha \in (1,2]$. We prove that when conditioning such maps to have $n$ vertices, or $n$ edges, or $n$ faces, the vertex-set endowed with the graph distance suitably rescaled converges in distribution towards the celebrated Brownian map when $\alpha=2$, and, after extraction of a subsequence, towards another `$\alpha$-stable map' when $\alpha <2$, which improves on a first result due to Le Gall & Miermont who assumed slightly more regularity.