Polynomial mixing time of edge flips on quadrangulations

Abstract

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.

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

“…We provide the first polynomial upper bound for the mixing time of this "edge rotation" chain on planar maps: we show that the spectral gap of the edge 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 the same bound applies to the spectral gap of edge flips on quadrangulations as defined in [8], which makes it possible to generalise the result of [8] to a variant of the edge flip chain related to edge rotations via Tutte's bijection.…”

“…But triangulations of the n-gon are not the only structures that are well-suited to supporting an edge flip chain, though they provide perhaps the simplest possible example; edge flip dynamics have been considered for example on lattice triangulations [6,7,13] and rectangular dissections [5,4]. Recently, Alexandre Stauffer and the author proved a polynomial upper bound for the mixing time of edge flips on quadrangulations of the sphere [8].…”

“…); as a consequence, EJP 25 (2020), paper 118. opportunities arise for a number of possible Markov chain comparisons. One such comparison, made with a "leaf translation" Markov chain on labelled plane trees, is what made it possible to show the main result of [8], namely an upper bound of order n 11/2 for the relaxation time of the edge flip Markov chain on quadrangulations of the sphere.…”

“…It should now be mentioned that, in order to have the Schaeffer bijection with labelled plane trees and to have Catalan numbers emerge when enumerating quadrangulations, one considers pointed, rooted quadrangulations of the sphere -that is, quadrangulations endowed with a distinguished vertex and a distinguished oriented edge. Redefining the dynamics to take the pointing and rooting into account poses no difficulties; the choice made in [8] is that of performing edge flips exactly as described, preserving the pointing and the orientation of the root edge when flipped (Figure 8). It seems quite reasonable that pointing and rooting should not be truly relevant, and indeed the pointing can be quickly dealt with and does not appear in the results of [8].…”

“…Redefining the dynamics to take the pointing and rooting into account poses no difficulties; the choice made in [8] is that of performing edge flips exactly as described, preserving the pointing and the orientation of the root edge when flipped (Figure 8). It seems quite reasonable that pointing and rooting should not be truly relevant, and indeed the pointing can be quickly dealt with and does not appear in the results of [8]. As for the rooting, however, it is worth noting that its role is more central.…”