621 research outputs found
Enumeration of quarter-turn symmetric alternating-sign matrices of odd order
It was shown by Kuperberg that the partition function of the square-ice model
related to the quarter-turn symmetric alternating-sign matrices of even order
is the product of two similar factors. We propose a square-ice model whose
states are in bijection with the quarter-turn symmetric alternating-sign
matrices of odd order, and show that the partition function of this model can
be also written in a similar way. This allows to prove, in particular, the
conjectures by Robbins related to the enumeration of the quarter-turn symmetric
alternating-sign matrices.Comment: 11 pages, 13 figures; minor correction
Spin chains and combinatorics: twisted boundary conditions
The finite XXZ Heisenberg spin chain with twisted boundary conditions was
considered. For the case of even number of sites , anisotropy parameter -1/2
and twisting angle the Hamiltonian of the system possesses an
eigenvalue . The explicit form of the corresponding eigenvector was
found for . Conjecturing that this vector is the ground state of the
system we made and verified several conjectures related to the norm of the
ground state vector, its component with maximal absolute value and some
correlation functions, which have combinatorial nature. In particular, the
squared norm of the ground state vector is probably coincides with the number
of half-turn symmetric alternating sign matrices.Comment: LaTeX file, 7 page
UV manifold and integrable systems in spaces of arbitrary dimension
The dimensional manifold with two mutually commutative operators of
differentiation is introduced. Nontrivial multidimensional integrable systems
connected with arbitrary graded (semisimple) algebras are constructed. The
general solution of them is presented in explicit form
Polynomial solutions of qKZ equation and ground state of XXZ spin chain at Delta = -1/2
Integral formulae for polynomial solutions of the quantum
Knizhnik-Zamolodchikov equations associated with the R-matrix of the six-vertex
model are considered. It is proved that when the deformation parameter q is
equal to e^{+- 2 pi i/3} and the number of vertical lines of the lattice is
odd, the solution under consideration is an eigenvector of the inhomogeneous
transfer matrix of the six-vertex model. In the homogeneous limit it is a
ground state eigenvector of the antiferromagnetic XXZ spin chain with the
anisotropy parameter Delta equal to -1/2 and odd number of sites. The obtained
integral representations for the components of this eigenvector allow to prove
some conjectures on its properties formulated earlier. A new statement relating
the ground state components of XXZ spin chains and Temperley-Lieb loop models
is formulated and proved.Comment: v2: cosmetic changes, new section on refined TSSCPPs vs refined ASM
On the symmetry of the partition function of some square ice models
We consider the partition function Z(N;x_1,...,x_N,y_1,...,y_N) of the square
ice model with domain wall boundary. We give a simple proof of the symmetry of
Z with respect to all its variables when the global parameter a of the model is
set to the special value a=exp(i\pi/3). Our proof does not use any
determinantal interpretation of Z and can be adapted to other situations (for
examples to some symmetric ice models).Comment: 8 page
Sum rules for the ground states of the O(1) loop model on a cylinder and the XXZ spin chain
The sums of components of the ground states of the O(1) loop model on a
cylinder or of the XXZ quantum spin chain at Delta=-1/2 (of size L) are
expressed in terms of combinatorial numbers. The methods include the
introduction of spectral parameters and the use of integrability, a mapping
from size L to L+1, and knot-theoretic skein relations.Comment: final version to be publishe
Bethe roots and refined enumeration of alternating-sign matrices
The properties of the most probable ground state candidate for the XXZ spin
chain with the anisotropy parameter equal to -1/2 and an odd number of sites is
considered. Some linear combinations of the components of the considered state,
divided by the maximal component, coincide with the elementary symmetric
polynomials in the corresponding Bethe roots. It is proved that those
polynomials are equal to the numbers providing the refined enumeration of the
alternating-sign matrices of order M+1 divided by the total number of the
alternating-sign matrices of order M, for the chain of length 2M+1.Comment: LaTeX 2e, 12 pages, minor corrections, references adde
Quantum Knizhnik-Zamolodchikov equation: reflecting boundary conditions and combinatorics
We consider the level 1 solution of quantum Knizhnik-Zamolodchikov equation
with reflecting boundary conditions which is relevant to the Temperley--Lieb
model of loops on a strip. By use of integral formulae we prove conjectures
relating it to the weighted enumeration of Cyclically Symmetric Transpose
Complement Plane Partitions and related combinatorial objects
Different facets of the raise and peel model
The raise and peel model is a one-dimensional stochastic model of a
fluctuating interface with nonlocal interactions. This is an interesting
physical model. It's phase diagram has a massive phase and a gapless phase with
varying critical exponents. At the phase transition point, the model exhibits
conformal invariance which is a space-time symmetry. Also at this point the
model has several other facets which are the connections to associative
algebras, two-dimensional fully packed loop models and combinatorics.Comment: 29 pages 17 figure
Raise and Peel Models of fluctuating interfaces and combinatorics of Pascal's hexagon
The raise and peel model of a one-dimensional fluctuating interface (model A)
is extended by considering one source (model B) or two sources (model C) at the
boundaries. The Hamiltonians describing the three processes have, in the
thermodynamic limit, spectra given by conformal field theory. The probability
of the different configurations in the stationary states of the three models
are not only related but have interesting combinatorial properties. We show
that by extending Pascal's triangle (which gives solutions to linear relations
in terms of integer numbers), to an hexagon, one obtains integer solutions of
bilinear relations. These solutions give not only the weights of the various
configurations in the three models but also give an insight to the connections
between the probability distributions in the stationary states of the three
models. Interestingly enough, Pascal's hexagon also gives solutions to a
Hirota's difference equation.Comment: 33 pages, an abstract and an introduction are rewritten, few
references are adde
- …