1,312 research outputs found
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
Forest matrices around the Laplacian matrix
We study the matrices Q_k of in-forests of a weighted digraph G and their
connections with the Laplacian matrix L of G. The (i,j) entry of Q_k is the
total weight of spanning converging forests (in-forests) with k arcs such that
i belongs to a tree rooted at j. The forest matrices, Q_k, can be calculated
recursively and expressed by polynomials in the Laplacian matrix; they provide
representations for the generalized inverses, the powers, and some eigenvectors
of L. The normalized in-forest matrices are row stochastic; the normalized
matrix of maximum in-forests is the eigenprojection of the Laplacian matrix,
which provides an immediate proof of the Markov chain tree theorem. A source of
these results is the fact that matrices Q_k are the matrix coefficients in the
polynomial expansion of adj(a*I+L). Thereby they are precisely Faddeev's
matrices for -L.
Keywords: Weighted digraph; Laplacian matrix; Spanning forest; Matrix-forest
theorem; Leverrier-Faddeev method; Markov chain tree theorem; Eigenprojection;
Generalized inverse; Singular M-matrixComment: 19 pages, presented at the Edinburgh (2001) Conference on Algebraic
Graph Theor
- …