Growing uniform planar maps face by face

We provide “growth schemes” for inductively generating uniform random (Figure presented.) -angulations of the sphere with (Figure presented.) faces, as well as uniform random simple triangulations of the sphere with (Figure presented.) faces. In the case of (Figure presented.) -angulations, we provide a way to insert a new face at a random location in a uniform (Figure presented.) -angulation with (Figure presented.) faces in such a way that the new map is precisely a uniform (Figure presented.) -angulation with (Figure presented.) faces. Similarly, given a uniform simple triangulation of the sphere with (Figure presented.) faces, we describe a way to insert two new adjacent triangles so as to obtain a uniform simple triangulation of the sphere with (Figure presented.) faces. The latter is based on a new bijective presentation of simple triangulations that relies on a construction by Poulalhon and Schaeffer.