Local structure of random quadrangulations

TL;DR

This paper studies the local structure of random quadrangulations, proving convergence properties and linking growth behavior to branching processes, with a new generating function for quadrangulations with boundary.

Contribution

It adapts a method to quadrangulations, establishing local weak convergence and connecting growth to branching processes, along with deriving a new generating function.

Findings

Proves local weak convergence of uniform quadrangulations.

Shows growth governed by a critical time-reversed branching process.

Rescaled profile converges to a reversed continuous-state branching process.

Abstract

This paper is an adaptation of a method used in \cite{K} to the model of random quadrangulations. We prove local weak convergence of uniform measures on quadrangulations and show that the local growth of quadrangulation is governed by certain critical time-reversed branching process and the rescaled profile converges to the reversed continuous-state branching process. As an intermediate result we derieve a biparametric generating function for certain class of quadrangulations with boundary.