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

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

The Hintermann-Merlini-Baxter-Wu and the infinite-coupling-limit Ashkin-Teller models

We show how the Hintermann–Merlini–Baxter–Wu model (which is a generalization of the well-known Baxter–Wu model to a general Eulerian triangulation) can be mapped onto a particular infinite-coupling-limit of the Ashkin–Teller model. We work out some mappings among these models, also including the standard and mixed Ashkin–Teller models. Finally, we compute the phase diagram of the infinite-coupling-limit Ashkin–Teller model on the square, triangular, hexagonal, and kagome lattices.The research of Y.H. and Y.D. is supported by National Nature Science Foundation of China under grants Nos. 11275185 and 10975127, and the Chinese Academy of Sciences. The work of J.L.J. was supported by the Agence Nationale de la Recherche (grant ANR-10-BLAN-0414: DIME), and the Institut Universitaire de France. The research of J.S. was supported in part by Spanish MEC grants FPA2009-08785 and MTM2011-24097 and by U.S. National Science Foundation grant PHY-0424082