10,179 research outputs found
Open boundary Quantum Knizhnik-Zamolodchikov equation and the weighted enumeration of Plane Partitions with symmetries
We propose new conjectures relating sum rules for the polynomial solution of
the qKZ equation with open (reflecting) boundaries as a function of the quantum
parameter and the -enumeration of Plane Partitions with specific
symmetries, with . We also find a conjectural relation \`a la
Razumov-Stroganov between the limit of the qKZ solution and refined
numbers of Totally Symmetric Self Complementary Plane Partitions.Comment: 27 pages, uses lanlmac, epsf and hyperbasics, minor revision
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
Loop model with mixed boundary conditions, qKZ equation and alternating sign matrices
The integrable loop model with mixed boundary conditions based on the
1-boundary extended Temperley--Lieb algebra with loop weight 1 is considered.
The corresponding qKZ equation is introduced and its minimal degree solution
described. As a result, the sum of the properly normalized components of the
ground state in size L is computed and shown to be equal to the number of
Horizontally and Vertically Symmetric Alternating Sign Matrices of size 2L+3. A
refined counting is also considered
The Razumov-Stroganov conjecture: Stochastic processes, loops and combinatorics
A fascinating conjectural connection between statistical mechanics and
combinatorics has in the past five years led to the publication of a number of
papers in various areas, including stochastic processes, solvable lattice
models and supersymmetry. This connection, known as the Razumov-Stroganov
conjecture, expresses eigenstates of physical systems in terms of objects known
from combinatorics, which is the mathematical theory of counting. This note
intends to explain this connection in light of the recent papers by Zinn-Justin
and Di Francesco.Comment: 6 pages, 4 figures, JSTAT News & Perspective
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
Inhomogeneous loop models with open boundaries
We consider the crossing and non-crossing O(1) dense loop models on a
semi-infinite strip, with inhomogeneities (spectral parameters) that preserve
the integrability. We compute the components of the ground state vector and
obtain a closed expression for their sum, in the form of Pfaffian and
determinantal formulas.Comment: 42 pages, 31 figures, minor corrections, references correcte
Quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials
We prove higher rank analogues of the Razumov--Stroganov sum rule for the
groundstate of the O(1) loop model on a semi-infinite cylinder: we show that a
weighted sum of components of the groundstate of the A_{k-1} IRF model yields
integers that generalize the numbers of alternating sign matrices. This is done
by constructing minimal polynomial solutions of the level 1 U_q(\hat{sl(k)})
quantum Knizhnik--Zamolodchikov equations, which may also be interpreted as
quantum incompressible q-deformations of fractional quantum Hall effect wave
functions at filling fraction nu=1/k. In addition to the generalized
Razumov--Stroganov point q=-e^{i pi/k+1}, another combinatorially interesting
point is reached in the rational limit q -> -1, where we identify the solution
with extended Joseph polynomials associated to the geometry of upper triangular
matrices with vanishing k-th power.Comment: v3: misprint fixed in eq (2.1
Integrability of graph combinatorics via random walks and heaps of dimers
We investigate the integrability of the discrete non-linear equation
governing the dependence on geodesic distance of planar graphs with inner
vertices of even valences. This equation follows from a bijection between
graphs and blossom trees and is expressed in terms of generating functions for
random walks. We construct explicitly an infinite set of conserved quantities
for this equation, also involving suitable combinations of random walk
generating functions. The proof of their conservation, i.e. their eventual
independence on the geodesic distance, relies on the connection between random
walks and heaps of dimers. The values of the conserved quantities are
identified with generating functions for graphs with fixed numbers of external
legs. Alternative equivalent choices for the set of conserved quantities are
also discussed and some applications are presented.Comment: 38 pages, 15 figures, uses epsf, lanlmac and hyperbasic
Crossed spin-1/2 Heisenberg chains as a quantum impurity problem
Using equivalencies between different models we reduce the model of two
spin-1/2 Heisenberg chains crossed at one point to the model of free fermions.
The spin-spin correlation function is calculated by summing the perturbation
series in the interchain coupling. The result reveals a power law decay with a
nonuniversal exponent.Comment: 3 pages, the background information is adde
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