Non-bijective scaling limits and phase transitions of planar maps

Abstract

We prove that the uniform random non-separable planar map with $n$ edges admits the Brownian sphere as Gromov--Hausdorff--Prokhorov scaling limit as $n$ tends to infinity. Our proof introduces a non-bijective ``common-core transfer method'' that constitutes a novel and universal proof strategy for scaling limits of random discrete structures. As an application, we complete the phase diagram for limiting shapes of block-weighted planar maps by Stufler~(2019). We describe phases with limits given by the Brownian sphere, stable trees, and Brownian sphere decorated stable trees recently introduced by S{\'e}nizergues, Stef{\'a}nsson and Stufler~(2023).