15 research outputs found
Determinant Formulae for some Tiling Problems and Application to Fully Packed Loops
We present determinant formulae for the number of tilings of various domains
in relation with Alternating Sign Matrix and Fully Packed Loop enumeration
A Bijection between classes of Fully Packed Loops and Plane Partitions
It has recently been observed empirically that the number of FPL
configurations with 3 sets of a, b and c nested arches equals the number of
plane partitions in a box of size a x b x c. In this note, this result is
proved by constructing explicitly the bijection between these FPL and plane
partitions
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
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
On some ground state components of the O(1) loop model
We address a number of conjectures about the ground state O(1) loop model,
computing in particular two infinite series of partial sums of its entries and
relating them to the enumeration of plane partitions. Our main tool is the use
of integral formulae for a polynomial solution of the quantum
Knizhnik-Zamolodchikov equation.Comment: v4: fixed problem with refs appearing as
Correlation Functions of Harish-Chandra Integrals over the Orthogonal and the Symplectic Groups
The Harish-Chandra correlation functions, i.e. integrals over compact groups
of invariant monomials prod tr{X^{p_1} Omega Y^{q_1} Omega^dagger X^{p_2} ...
with the weight exp tr{X Omega Y Omega^dagger} are computed for the orthogonal
and symplectic groups. We proceed in two steps. First, the integral over the
compact group is recast into a Gaussian integral over strictly upper triangular
complex matrices (with some additional symmetries), supplemented by a summation
over the Weyl group. This result follows from the study of loop equations in an
associated two-matrix integral and may be viewed as the adequate version of
Duistermaat-Heckman's theorem for our correlation function integrals. Secondly,
the Gaussian integration over triangular matrices is carried out and leads to
compact determinantal expressions.Comment: 58 pages; Acknowledgements added; small corrections in appendix A;
minor changes & Note Adde
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
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
A Matrix model for plane partitions
We construct a matrix model equivalent (exactly, not asymptotically), to the
random plane partition model, with almost arbitrary boundary conditions.
Equivalently, it is also a random matrix model for a TASEP-like process with
arbitrary boundary conditions. Using the known solution of matrix models, this
method allows to find the large size asymptotic expansion of plane partitions,
to ALL orders. It also allows to describe several universal regimes.Comment: Latex, 41 figures. Misprints and corrections. Changing the term TASEP
to self avoiding particle porces
Determinant Formulae for some Tiling Problems and Application to Fully Packed Loops
We present a number of determinant formulae for the number of tilings of various domain