On the distance-profile of random rooted plane graphs

Olivier Bernardi (1) , Gwendal Collet (2) , Eric Fusy (2)

1 Brandeis University
2 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]

We study the distance-profile of the random rooted plane graph Gn with n edges (by a plane graph we mean a planar map with no loops nor multiple edges). Our main result is that the profile and radius of Gn (with respect to the root-vertex), rescaled by (2n) 1/4 , converge to explicit distributions related to the Brownian snake. A crucial ingredient of our proof is a bijection we have recently introduced between rooted outer-triangular plane graphs and rooted eulerian triangulations, combined with ingredients from Chassaing and Schaeffer (2004), Bousquet-Mélou and Schaeffer (2000), and Addario-Berry and Albenque (2013). We also show that the result for plane graphs implies similar results for random rooted loopless maps and general maps.

Mots clés: Planar maps; bijections; ISE; Brownian snake; distance-profile

Domaines: Combinatoire [math.CO]

Olivier Bernardi, Gwendal Collet, Eric Fusy. On the distance-profile of random rooted plane graphs. AofA'2014, Jun 2014, Paris, France. pp.37-48. ⟨hal-01188334⟩