484 research outputs found
From Cycle Rooted Spanning Forests to the Critical Ising Model: an Explicit Construction
Fisher established an explicit correspondence between the 2-dimensional Ising
model defined on a graph and the dimer model defined on a decorated version
\GD of this graph \cite{Fisher}. In this paper we explicitly relate the dimer
model associated to the critical Ising model and critical cycle rooted spanning
forests (CRSFs). This relation is established through characteristic
polynomials, whose definition only depends on the respective fundamental
domains, and which encode the combinatorics of the model. We first show a
matrix-tree type theorem establishing that the dimer characteristic polynomial
counts CRSFs of the decorated fundamental domain \GD_1. Our main result
consists in explicitly constructing CRSFs of \GD_1 counted by the dimer
characteristic polynomial, from CRSFs of where edges are assigned
Kenyon's critical weight function \cite{Kenyon3}; thus proving a relation on
the level of configurations between two well known 2-dimensional critical
models.Comment: 51 pages, 24 figures. To appear, Comm. Math. Phys. Revised version:
title has changed. The terminology `correspondence' has been changed to that
of `explicit construction' and `mapping
Logarithmic corrections in the free energy of monomer-dimer model on plane lattices with free boundaries
Using exact computations we study the classical hard-core monomer-dimer
models on m x n plane lattice strips with free boundaries. For an arbitrary
number v of monomers (or vacancies), we found a logarithmic correction term in
the finite-size correction of the free energy. The coefficient of the
logarithmic correction term depends on the number of monomers present (v) and
the parity of the width n of the lattice strip: the coefficient equals to v
when n is odd, and v/2 when n is even. The results are generalizations of the
previous results for a single monomer in an otherwise fully packed lattice of
dimers.Comment: 4 pages, 2 figure
Tetromino tilings and the Tutte polynomial
We consider tiling rectangles of size 4m x 4n by T-shaped tetrominoes. Each
tile is assigned a weight that depends on its orientation and position on the
lattice. For a particular choice of the weights, the generating function of
tilings is shown to be the evaluation of the multivariate Tutte polynomial
Z\_G(Q,v) (known also to physicists as the partition function of the Q-state
Potts model) on an (m-1) x (n-1) rectangle G, where the parameter Q and the
edge weights v can take arbitrary values depending on the tile weights.Comment: 8 pages, 6 figure
Generation of folk song melodies using Bayes transforms
The paper introduces the `Bayes transform', a mathematical procedure for putting data into a hierarchical representation. Applicable to any type of data, the procedure yields interesting results when applied to sequences. In this case, the representation obtained implicitly models the repetition hierarchy of the source. There are then natural applications to music. Derivation of Bayes transforms can be the means of determining the repetition hierarchy of note sequences (melodies) in an empirical and domain-general way. The paper investigates application of this approach to Folk Song, examining the results that can be obtained by treating such transforms as generative models
Loop models and their critical points
Loop models have been widely studied in physics and mathematics, in problems
ranging from polymers to topological quantum computation to Schramm-Loewner
evolution. I present new loop models which have critical points described by
conformal field theories. Examples include both fully-packed and dilute loop
models with critical points described by the superconformal minimal models and
the SU(2)_2 WZW models. The dilute loop models are generalized to include
SU(2)_k models as well.Comment: 20 pages, 15 figure
Exact solution of A-D Temperley-Lieb Models
We solve for the spectrum of quantum spin chains based on representations of
the Temperley-Lieb algebra associated with the quantum groups {\cal U}_q(X_n }
for X_n = A_1,B_n,C_nD_n$. We employ a generalization of the coordinate
Bethe-Ansatz developed previously for the deformed biquadratic spin one chain.
As expected, all these models have equivalent spectra, i.e. they differ only in
the degeneracy of their eigenvalues. This is true for finite length and open
boundary conditions. For periodic boundary conditions the spectra of the lower
dimensional representations are containded entirely in the higher dimensional
ones. The Bethe states are highest weight states of the quantum group, except
for some states with energy zero
Some Exact Results for Spanning Trees on Lattices
For -vertex, -dimensional lattices with , the number
of spanning trees grows asymptotically as
in the thermodynamic limit. We present an exact closed-form result for the
asymptotic growth constant for spanning trees on the
-dimensional body-centered cubic lattice. We also give an exact integral
expression for on the face-centered cubic lattice and an exact
closed-form expression for on the lattice.Comment: 7 pages, 1 tabl
Phase Transition in a Self-repairing Random Network
We consider a network, bonds of which are being sequentially removed; that is
done at random, but conditioned on the system remaining connected
(Self-Repairing Bond Percolation SRBP). This model is the simplest
representative of a class of random systems for which forming of isolated
clusters is forbidden. It qualitatively describes the process of fabrication of
artificial porous materials and degradation of strained polymers. We find a
phase transition at a finite concentration of bonds , at which the
backbone of the system vanishes; for all the network is a dense
fractal.Comment: 4 pages, 4 figure
On the duality relation for correlation functions of the Potts model
We prove a recent conjecture on the duality relation for correlation
functions of the Potts model for boundary spins of a planar lattice.
Specifically, we deduce the explicit expression for the duality of the n-site
correlation functions, and establish sum rule identities in the form of the
M\"obius inversion of a partially ordered set. The strategy of the proof is by
first formulating the problem for the more general chiral Potts model. The
extension of our consideration to the many-component Potts models is also
given.Comment: 17 pages in RevTex, 5 figures, submitted to J. Phys.
Vacancy localization in the square dimer model
We study the classical dimer model on a square lattice with a single vacancy
by developing a graph-theoretic classification of the set of all configurations
which extends the spanning tree formulation of close-packed dimers. With this
formalism, we can address the question of the possible motion of the vacancy
induced by dimer slidings. We find a probability 57/4-10Sqrt[2] for the vacancy
to be strictly jammed in an infinite system. More generally, the size
distribution of the domain accessible to the vacancy is characterized by a
power law decay with exponent 9/8. On a finite system, the probability that a
vacancy in the bulk can reach the boundary falls off as a power law of the
system size with exponent 1/4. The resultant weak localization of vacancies
still allows for unbounded diffusion, characterized by a diffusion exponent
that we relate to that of diffusion on spanning trees. We also implement
numerical simulations of the model with both free and periodic boundary
conditions.Comment: 35 pages, 24 figures. Improved version with one added figure (figure
9), a shift s->s+1 in the definition of the tree size, and minor correction
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