On scaling limits of planar maps with stable face-degrees

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.

arXiv comment31 pages, 5 figures

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Cited by in corpus (7)

• On scaling limits of random trees and maps with a prescribed degree sequence
• Large deviation Local Limit Theorems and limits of biconditioned Trees and Maps
• Planar maps and random partitions
• Scaling limits of random looptrees and bipartite plane maps with prescribed large faces
• On the growth of random planar maps with a prescribed degree sequence
• Markovian explorations of random planar maps are roundish
• Distances on the ${\mathrm{CLE}}_4$, critical Liouville quantum gravity and $3/2$-stable maps