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, which have been considered before on a variety of combinatorial structures (triangulations of the n-gon and quadrangulations of the sphere, among others). 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.

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

Caraceni A.
2020

Abstract

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, which have been considered before on a variety of combinatorial structures (triangulations of the n-gon and quadrangulations of the sphere, among others). 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.
2020
Settore MAT/06 - Probabilita' e Statistica Matematica
Edge flips; Edge rotations; Markov chain; Mixing time; Planar maps
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11384/125544
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