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Description
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 large-scale geometry of random two-connected series--parallel maps with $n$ edges. Our main result shows that, when graph distances are rescaled by a factor $n^{-1/2}$, these maps converge to a constant multiple of Aldous' continuum random tree (CRT). This identifies the CRT as the universal scaling limit governing the macroscopic metric structure of this model.
The proof relies on a bijection between series--parallel maps with $n$ edges and a class of trees with $n$ leaves. This correspondence allows us to compare geodesics in the maps with paths in the associated trees which enables us to transfer metric information from trees to maps and thereby establish convergence to the continuum limit.