Integrability and conformal data of the dimer model

The central charge of the dimer model on the square lattice is still being debated in the literature. In this paper, we provide evidence supporting the consistency of a $c=-2$ description. Using Lieb's transfer matrix and its description in terms of the Temperley-Lieb algebra $TL_n$ at $\beta = 0$, we provide a new solution of the dimer model in terms of the model of critical dense polymers on a tilted lattice and offer an understanding of the lattice integrability of the dimer model. The dimer transfer matrix is analysed in the scaling limit and the result for $L_0-\frac c{24}$ is expressed in terms of fermions. Higher Virasoro modes are likewise constructed as limits of elements of $TL_n$ and are found to yield a $c=-2$ realisation of the Virasoro algebra, familiar from fermionic $bc$ ghost systems. In this realisation, the dimer Fock spaces are shown to decompose, as Virasoro modules, into direct sums of Feigin-Fuchs modules, themselves exhibiting reducible yet indecomposable structures. In the scaling limit, the eigenvalues of the lattice integrals of motion are found to agree exactly with those of the $c=-2$ conformal integrals of motion. Consistent with the expression for $L_0-\frac c{24}$ obtained from the transfer matrix, we also construct higher Virasoro modes with $c=1$ and find that the dimer Fock space is completely reducible under their action. However, the transfer matrix is found not to be a generating function for the $c=1$ integrals of motion. Although this indicates that Lieb's transfer matrix description is incompatible with the $c=1$ interpretation, it does not rule out the existence of an alternative, $c=1$ compatible, transfer matrix description of the dimer model.Comment: 54 pages. v2: minor correction

Ising Model Observables and Non-Backtracking Walks

This paper presents an alternative proof of the connection between the partition function of the Ising model on a finite graph $G$ and the set of non-backtracking walks on $G$. The techniques used also give formulas for spin-spin correlation functions in terms of non-backtracking walks. The main tools used are Viennot's theory of heaps of pieces and turning numbers on surfaces.Comment: 33 pages, 11 figures. Typos and errors corrected, exposition improved, results unchange

Enumeration of maximum matchings of graphs

Counting maximum matchings in a graph is of great interest in statistical mechanics, solid-state chemistry, theoretical computer science, mathematics, among other disciplines. However, it is a challengeable problem to explicitly determine the number of maximum matchings of general graphs. In this paper, using Gallai-Edmonds structure theorem, we derive a computing formula for the number of maximum matching in a graph. According to the formula, we obtain an algorithm to enumerate maximum matchings of a graph. In particular, The formula implies that computing the number of maximum matchings of a graph is converted to compute the number of perfect matchings of some induced subgraphs of the graph. As an application, we calculate the number of maximum matchings of opt trees. The result extends a conclusion obtained by Heuberger and Wagner[C. Heuberger, S. Wagner, The number of maximum matchings in a tree, Discrete Math. 311 (2011) 2512--2542]