The Scaling Limit of Random Two-Connected Series–Parallel Maps

Abstract

A finite graph embedded in the plane is called a series–parallel map if it can be obtained from a finite tree by repeatedly subdividing and doubling edges. We study the scaling limit of weighted random two-connected series–parallel maps with n edges and show that under fairly general integrability conditions on these weights, the maps with distances rescaled by a factor n-1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n^{-1/2}$$\end{document} converge to a constant multiple of Aldous’ continuum random tree (CRT) in the Gromov–Hausdorff sense. The proof relies on a bijection between a set of trees with n leaves and a set of series–parallel maps with n edges, together with a novel blob decomposition of the maps.