Polynomial mixing time of edge flips on quadrangulations

We establish the first polynomial upper bound for the mixing time of random edge flips on rooted quadrangulations: we show that the spectral gap of the edge flip Markov chain on quadrangulations with n faces admits, up to constants, an upper bound of n ^{- 5 / 4} and a lower bound of n ^{- 11 / 2} . In order to obtain the lower bound, we also consider a very natural Markov chain on plane trees—or, equivalently, on Dyck paths—and improve the previous lower bound for its spectral gap by Shor and Movassagh.