A bijection between rigid and integer-labeled quadrangulations

Abstract

In this talk I will introduce the combinatorial class of rigid quadrangulations, which form a subclass of flat quadrangulations of the disk, meaning that all non-boundary vertices are of degree 4. Rigid quadrangulations are shown to be in bijection with certain integer-labeled quadrangulations of the sphere, that were enumerated recently by Bousquet-Mélou and Elvey Price. The bijection relates several natural statistics on one side to equally natural, but rather different, statistics on the other. Finally, I will touch upon the question of scaling limits of large rigid quadrangulations and similar models of random flat metrics on the disk, and their physics motivation.

Date
Location
IPhT, CEA Paris-Saclay