Quadrangulations with no pendant vertices

TL;DR

This paper proves that the metric space of a uniformly distributed planar quadrangulation with no pendant vertices converges to the Brownian map after rescaling, advancing understanding of random planar maps with local constraints.

Contribution

It establishes the convergence of quadrangulations with no pendant vertices to the Brownian map, extending previous results to graphs with local constraints.

Findings

Convergence of quadrangulations with no pendant vertices to the Brownian map.

First step towards extending convergence results to constrained random planar maps.

Abstract

We prove that the metric space associated with a uniformly distributed planar quadrangulation with n faces and no pendant vertices converges modulo a suitable rescaling to the Brownian map. This is a first step towards the extension of recent convergence results for random planar maps to the case of graphs satisfying local constraints.