A Tauberian approach to metric scaling limits of random discrete structures, with an application to random planar maps

We prove sandwich theorems and a Tauberian theorem in the space of compact metric measure spaces, endowed with the Gromov-Hausdorff-Prokhorov (GHP) topology. These results hold with respect to a close relative of Gromov’s Lipschitz order. As a proof-of-concept of a general method to prove metric scaling limits of random discrete structures, we give an application to the theory of random planar maps: the Brownian sphere is the scaling limit in the GHP topology of irreducible quandrangulations. Our main inputs are (i) the convergence of general quadrangulations to the Brownian sphere (Le Gall, 2013; Miermont, 2013); and (ii) couplings where irreducible quadrangulations of the hexagon are “grown” by face-openings (Addario-Berry, 2014).