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 obtained by Shor and Movassagh.