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

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 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.