Decomposition theorem on matchable distributive lattices

A distributive lattice structure ${\mathbf M}(G)$ has been established on the set of perfect matchings of a plane bipartite graph $G$. We call a lattice {\em matchable distributive lattice} (simply MDL) if it is isomorphic to such a distributive lattice. It is natural to ask which lattices are MDLs. We show that if a plane bipartite graph $G$ is elementary, then ${\mathbf M}(G)$ is irreducible. Based on this result, a decomposition theorem on MDLs is obtained: a finite distributive lattice $\mathbf{L}$ is an MDL if and only if each factor in any cartesian product decomposition of $\mathbf{L}$ is an MDL. Two types of MDLs are presented: $J(\mathbf{m}\times \mathbf{n})$ and $J(\mathbf{T})$, where $\mathbf{m}\times \mathbf{n}$ denotes the cartesian product between $m$-element chain and $n$-element chain, and $\mathbf{T}$ is a poset implied by any orientation of a tree.Comment: 19 pages, 7 figure

Kasteleyn cokernels

We consider Kasteleyn and Kasteleyn-Percus matrices, which arise in enumerating matchings of planar graphs, up to matrix operations on their rows and columns. If such a matrix is defined over a principal ideal domain, this is equivalent to considering its Smith normal form or its cokernel. Many variations of the enumeration methods result in equivalent matrices. In particular, Gessel-Viennot matrices are equivalent to Kasteleyn-Percus matrices. We apply these ideas to plane partitions and related planar of tilings. We list a number of conjectures, supported by experiments in Maple, about the forms of matrices associated to enumerations of plane partitions and other lozenge tilings of planar regions and their symmetry classes. We focus on the case where the enumerations are round or $q$-round, and we conjecture that cokernels remain round or $q$-round for related ``impossible enumerations'' in which there are no tilings. Our conjectures provide a new view of the topic of enumerating symmetry classes of plane partitions and their generalizations. In particular we conjecture that a $q$-specialization of a Jacobi-Trudi matrix has a Smith normal form. If so it could be an interesting structure associated to the corresponding irreducible representation of \SL(n,\C). Finally we find, with proof, the normal form of the matrix that appears in the enumeration of domino tilings of an Aztec diamond.Comment: 14 pages, 19 in-line figures. Very minor copy correction

The critical Z-invariant Ising model via dimers: the periodic case

We study a large class of critical two-dimensional Ising models namely critical Z-invariant Ising models on periodic graphs, example of which are the classical square, triangular and honeycomb lattice at the critical temperature. Fisher introduced a correspondence between the Ising model and the dimer model on a decorated graph, thus setting dimer techniques as a powerful tool for understanding the Ising model. In this paper, we give a full description of the dimer model corresponding to the critical Z-invariant Ising model. We prove that the dimer characteristic polynomial is equal (up to a constant) to the critical Laplacian characteristic polynomial, and defines a Harnack curve of genus 0. We prove an explicit expression for the free energy, and for the Gibbs measure obtained as weak limit of Boltzmann measures.Comment: 35 pages, 8 figure

A graph polynomial for independent sets of bipartite graphs

We introduce a new graph polynomial that encodes interesting properties of graphs, for example, the number of matchings and the number of perfect matchings. Most importantly, for bipartite graphs the polynomial encodes the number of independent sets (#BIS). We analyze the complexity of exact evaluation of the polynomial at rational points and show that for most points exact evaluation is #P-hard (assuming the generalized Riemann hypothesis) and for the rest of the points exact evaluation is trivial. We conjecture that a natural Markov chain can be used to approximately evaluate the polynomial for a range of parameters. The conjecture, if true, would imply an approximate counting algorithm for #BIS, a problem shown, by [Dyer et al. 2004], to be complete (with respect to, so called, AP-reductions) for a rich logically defined sub-class of #P. We give a mild support for our conjecture by proving that the Markov chain is rapidly mixing on trees. As a by-product we show that the "single bond flip" Markov chain for the random cluster model is rapidly mixing on constant tree-width graphs