Mixing Times of Markov Chains on Degree Constrained Orientations of Planar Graphs

TL;DR

This paper investigates the mixing times of Markov chains on degree-constrained orientations of planar graphs, revealing conditions for rapid or slow mixing and providing new examples of slow mixing in specific graph classes.

Contribution

It introduces new results on the mixing times of Markov chains for $eta$-orientations, including simpler slow-mixing examples and the first slow-mixing case for certain triangulations.

Findings

Slow mixing for 2-orientations of some quadrangulations.

Rapid mixing for 2-orientations with maximum degree ≤ 4.

New examples of slow mixing with smaller maximum degree.

Abstract

We study Markov chains for $\alpha$-orientations of plane graphs, these are orientations where the outdegree of each vertex is prescribed by the value of a given function $\alpha$. The set of $\alpha$-orientations of a plane graph has a natural distributive lattice structure. The moves of the up-down Markov chain on this distributive lattice corresponds to reversals of directed facial cycles in the $\alpha$-orientation. We have a positive and several negative results regarding the mixing time of such Markov chains. A 2-orientation of a plane quadrangulation is an orientation where every inner vertex has outdegree 2. We show that there is a class of plane quadrangulations such that the up-down Markov chain on the 2-orientations of these quadrangulations is slowly mixing. On the other hand the chain is rapidly mixing on 2-orientations of quadrangulations with maximum degree at most 4. Regarding examples for slow mixing we also revisit the case of 3-orientations of triangulations which has been studied before by Miracle et al.. Our examples for slow mixing are simpler and have a smaller maximum degree, Finally we present the first example of a function $\alpha$ and a class of plane triangulations of constant maximum degree such that the up-down Markov chain on the $\alpha$-orientations of these graphs is slowly mixing.