A refined Razumov-Stroganov conjecture II

We extend a previous conjecture [cond-mat/0407477] relating the Perron-Frobenius eigenvector of the monodromy matrix of the O(1) loop model to refined numbers of alternating sign matrices. By considering the O(1) loop model on a semi-infinite cylinder with dislocations, we obtain the generating function for alternating sign matrices with prescribed positions of 1's on their top and bottom rows. This seems to indicate a deep correspondence between observables in both models.Comment: 21 pages, 10 figures (3 in text), uses lanlmac, hyperbasics and epsf macro

A refined Razumov-Stroganov conjecture

We extend the Razumov-Stroganov conjecture relating the groundstate of the O(1) spin chain to alternating sign matrices, by relating the groundstate of the monodromy matrix of the O(1) model to the so-called refined alternating sign matrices, i.e. with prescribed configuration of their first row, as well as to refined fully-packed loop configurations on a square grid, keeping track both of the loop connectivity and of the configuration of their top row. We also conjecture a direct relation between this groundstate and refined totally symmetric self-complementary plane partitions, namely, in their formulation as sets of non-intersecting lattice paths, with prescribed last steps of all paths.Comment: 20 pages, 4 figures, uses epsf and harvmac macros a few typos correcte

On FPL configurations with four sets of nested arches

The problem of counting the number of Fully Packed Loop (FPL) configurations with four sets of a,b,c,d nested arches is addressed. It is shown that it may be expressed as the problem of enumeration of tilings of a domain of the triangular lattice with a conic singularity. After reexpression in terms of non-intersecting lines, the Lindstr\"om-Gessel-Viennot theorem leads to a formula as a sum of determinants. This is made quite explicit when min(a,b,c,d)=1 or 2. We also find a compact determinant formula which generates the numbers of configurations with b=d.Comment: 22 pages, TeX, 16 figures; a new formula for a generating function adde

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

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

Exact expressions for correlations in the ground state of the dense O(1) loop model

Conjectures for analytical expressions for correlations in the dense O$(1)$ loop model on semi infinite square lattices are given. We have obtained these results for four types of boundary conditions. Periodic and reflecting boundary conditions have been considered before. We give many new conjectures for these two cases and review some of the existing results. We also consider boundaries on which loops can end. We call such boundaries ''open''. We have obtained expressions for correlations when both boundaries are open, and one is open and the other one is reflecting. Also, we formulate a conjecture relating the ground state of the model with open boundaries to Fully Packed Loop models on a finite square grid. We also review earlier obtained results about this relation for the three other types of boundary conditions. Finally, we construct a mapping between the ground state of the dense O$(1)$ loop model and the XXZ spin chain for the different types of boundary conditions.Comment: 25 pages, version accepted by JSTA

A_k Generalization of the O(1) Loop Model on a Cylinder: Affine Hecke Algebra, q-KZ Equation and the Sum Rule

We study the A_k generalized model of the O(1) loop model on a cylinder. The affine Hecke algebra associated with the model is characterized by a vanishing condition, the cylindric relation. We present two representations of the algebra: the first one is the spin representation, and the other is in the vector space of states of the A_k generalized model. A state of the model is a natural generalization of a link pattern. We propose a new graphical way of dealing with the Yang-Baxter equation and $q$-symmetrizers by the use of the rhombus tiling. The relation between two representations and the meaning of the cylindric relations are clarified. The sum rule for this model is obtained by solving the q-KZ equation at the Razumov-Stroganov point.Comment: 43 pages, 22 figures, LaTeX, (ver 2) Introduction rewritten and Section 4.3 adde

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