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In this talk, based on joint work with Alexandre Stauffer, we will discuss the problem of providing “uniform growth schemes” for various types of planar maps — namely, of coupling a uniform map with n faces with a uniform map with n+1 faces in such a way that the smaller map is always obtained from the larger by collapsing a single face. We show that uniform growth schemes exist for rooted 2p-angulations of the sphere and for rooted simple triangulations, and briefly touch on some applications to mixing time questions for edge flip chains.
