On scaling limits of planar maps with stable face-degrees

Cyril Marzouk
2018 Latin American Journal of Probability and Mathematical Statistics  
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 α∈ (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 α=2, and, after extraction of a subsequence, towards another 'α-stable map' when α <2, which
more » ... mproves on a first result due to Le Gall & Miermont who assumed slightly more regularity.
doi:10.30757/alea.v15-40 fatcat:gx3fcyczlnhuvezyj2xn4deh6y