Mixing times of Markov chains of 2-orientations

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 updown Markov chain on this distributive lattice corresponds to reversals of directed facial cycles in the α-orientation. 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.