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 $n^{-1/2}$ 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.