7,886 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
On the asymptotics of dimers on tori
We study asymptotics of the dimer model on large toric graphs. Let be a weighted -periodic planar graph, and let
be a large-index sublattice of . For bipartite we
show that the dimer partition function on the quotient
has the asymptotic expansion , where is the area of ,
is the free energy density in the bulk, and is a finite-size
correction term depending only on the conformal shape of the domain together
with some parity-type information. Assuming a conjectural condition on the zero
locus of the dimer characteristic polynomial, we show that an analogous
expansion holds for non-bipartite. The functional form of the
finite-size correction differs between the two classes, but is universal within
each class. Our calculations yield new information concerning the distribution
of the number of loops winding around the torus in the associated double-dimer
models.Comment: 48 pages, 18 figure
On oriented graphs with minimal skew energy
Let be the skew-adjacency matrix of an oriented graph
. The skew energy of is defined as the sum of all singular
values of its skew-adjacency matrix . In this paper, we first
deduce an integral formula for the skew energy of an oriented graph. Then we
determine all oriented graphs with minimal skew energy among all connected
oriented graphs on vertices with arcs, which is an
analogy to the conjecture for the energy of undirected graphs proposed by
Caporossi {\it et al.} [G. Caporossi, D. Cvetkovi, I. Gutman, P.
Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs
with external energy, J. Chem. Inf. Comput. Sci. 39 (1999) 984-996.]Comment: 15 pages. Actually, this paper was finished in June 2011. This is an
updated versio
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