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

Abstract

We study Markov chains for α-orientations of plane graphs, these are orientations where the outdegree of each vertex is prescribed by the value of a given function α. The set of α-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 α-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 α and a class of plane triangulations of constant maximum degree such that the up-down Markov chain on the α-orientations of these graphs is slowly mixing.