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We consider a natural local dynamic on the set of all rooted planar maps with$n$ edges that is in some sense analogous to "edge flip" Markov chains, whichhave been considered before on a variety of combinatorial structures(triangulations of the $n$-gon and quadrangulations of the sphere, amongothers). We provide the first polynomial upper bound for the mixing time ofthis "edge rotation" chain on planar maps: we show that the spectral gap of theedge rotation chain is bounded below by an appropriate constant times$n^{-11/2}$. In doing so, we provide a partially new proof of the fact that thesame bound applies to the spectral gap of edge flips on quadrangulations, whichmakes it possible to generalise a recent result of the author and Stauffer to achain that relates to edge rotations via Tutte's bijection.

A polynomial upper bound for the mixing time of edge rotations on planar maps