Random irreducible quadrangulations

Thesis

In the thesis we study random irreducible quadrangulations. We aim to extend known results on local limits of random planar maps and the perceived universality to the class of irreducible quadrangulations.

The first part of the thesis establishes a multitude of enumeration results using standard generating function techniques. We introduce various classes of irreducible quadrangulations and we make a detailed study of their generating functions and asymptotics. The material of this chapter forms the backbone of the subsequent calculations.

The last chapter is conceptually divided into two parts. The first part extends known sharp concentration results for maximum vertex degree and vertex degree distribution to the class of irreducible quadrangulations. In the second part we describe the skeleton decomposition of irreducible hulls and introduce a formal operator to evaluate statistical sums over the semi-layers of the decomposition.

The concluding section of the thesis exploits the properties of the hull operator to provide a new proof of the existence of the Uniform Infinite Irreducible Planar Quadrangulation (UIIPQ). We also explore the asymptotics of the iterates of the hull operator to derive limiting probabilities for the perimeter and volume of r-hulls in the UIIPQ. Our results are (unsurprisingly) in agreement with the conjectured universal behaviour observed in other random models such as the UIPT and UIPQ.

Authors: Bergant, S

• DOI: 10.5287/ora-9ezpx0b6q
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: volume distribution; irreducible quadrangulations; quadrangulation hulls; random planar maps; uniform infinite irreducible planar quadrangulation (UIIPQ)
• Subjects: Geometry; Discrete mathematics; Mathematics