The Structure of Unicellular Maps, and a Connection between Maps of Positive Genus and Planar Labelled Trees

A unicellular map is a map which has only one face. We give a bijection between a dominant subset of rooted unicellular maps of given genus and a set of rooted plane trees with distinguished vertices. The bijection applies as well to the case of labelled unicellular maps, which are related to all rooted maps by Marcus and Schaeffer’s bijection. This gives an immediate derivation of the asymptotic number of unicellular maps of given genus, and a simple bijective proof of a formula of Lehman and Walsh on the number of triangulations with one vertex. From the labelled case, we deduce an expression of the asymptotic number of maps of genus g with n edges involving the ISE random measure, and an explicit characterization of the limiting profile and radius of random bipartite quadrangulations of genus g in terms of the ISE.