Francisco
Alcaraz
Free fermion branches in some
quantum spin models
Extensive numerical analysis of the
eigenspectra of the $SU_q(N)$ invariant Perk-Schultz
Hamiltonian shows some simple regularities for a significant part of the
eigenspectrum. Inspired by those results we have found two set of
solutions of the associated nested Bethe-ansatz
equations. The first set is obtained at a special value of the
anisotropy ($q = \exp(i2\pi (N-1)/N)$)
and describes in particular the ground state and nearby
excitations as a sum of free-fermion
quasienergies. The second set of solutions provides the energies in
the sectors whose number $n_i$
of particles of distinct species ($i=0, \ldots, N-1$) are less or
equal to the unity except for one of
the species. For this last set we obtain the eigenspectra of a free
fermion model for arbitrary values
of the anisotropy.
Cyril
Banderier
A considerable number of asymptotic
distributions arising in random combinatorics and
analysis of algorithms are of the exponential-quadratic type, that
is, Gaussian. In a joint
work with Ph. Flajolet, G. Schaeffer, and M. Soria,
we exhibit a class of ``universal'' phenomena that are of the
exponential-cubic type,
corresponding to distributions that
involve the Airy function. Such Airy phenomena are related to the
coalescence of saddle
points and the confluence of singularities of generating functions.
We established that a common Airy distribution
(equivalently, a stable law of exponent
$3/2$,
sometimes also called Holtsmark distribution) describes the
size of the largest (multi)connected
components of about a dozen types of random planar maps
and seems also to be fundamental
for distribution functions of random matrix theory
It also appears in the Seppalainen-Johansson
model (~the oriented digital boiling model of Gravner/Tracy/Widom).
Based on an extension of the singularity analysis framework
suggested by the Airy case, we
present a general classification of compositional schemas in analytic
combinatorics.
Murray Batchelor
I'll review the way in which the well known combinatorial
numbers associated with alternating sign matrices and their
cousins have arisen in the groundstate properties of the
XXZ and O(n) loop model hamiltonians. I'll also discuss
the related Bethe Ansatz equations and some attempts at
proving the various conjectures.
Giovanni
Feverati
We consider minimal theories of CFT from a lattice combinatorial
perspective. To each allowed one-dimensional configuration path of
the $A_L$ Restricted Solid-on-Solid (RSOS) models we associate a
physical state and a monomial in a finite fermionic algebra. The
orthonormal states produced by the action of these monomials on the
primary states $|h\rangle$ generate finite Verma modules with
dimensions given by the finitized Virasoro characters
$\chi^{(N)}_h(q)$. These finitized characters are the generating
functions for the double row transfer matrix spectra of the critical
RSOS models. We argue that an energy preserving bijection exists
between the one-dimensional configuration paths and eigenstates of
these transfer matrices. We also propose a general algorithm to
obtain matrix representations of the Virasoro generators and primary
fields in a fermionic basis and illustrate it for the critical Ising,
tricritical Ising, 3-state Potts and Yang-Lee theories.
Peter
Forrester
Recently a new class of solvable directed percolation
models have been uncovered, which have close ties with
combinatorics through non-intersecting lattice paths and
the Robinson-Schensted-Knuth correspondence. These models
also have close ties with random matrices. Some details of
these inter-relationships will be given, and their
consequences discussed.
Tony Guttmann
I will briefly describe standard scaling theory, and discuss
recent work on the susceptibility of the two-dimensional Ising
model, and on the perimeter-area generating function of the
two-dimensional self-avoiding walk model, with a view to elucidating
their scaling behaviour.
Nikolai
Kitanine
In this talk I give a brief description of a method of
calculation of the correlation functions
of the XXZ model, based on the algebraic Bethe ansatz. Using this
method for a special
case of the XXZ model with $\Delta=1/2$ one can calculate some correlation functions,
such as emptiness formation probability, explicitely. It can be shown
that this function can
be expressed in terms of the corresponding number of alternating sign matrices.
Christian Krattenthaler
I shall present the recent results on rhombus tiling enumeration
which I have obtained with several coworkers (M. Ciucu, T. Eisenk\"olbl,
M. Fulmek, D. Zare). This includes results on the number of rhombus
tilings of a hexagon which contain a fixed rhombus, and results on the
number of rhombus tilings of incomplete hexagons. Typically, the
solution of such problems requires to evaluate some determinant.
I shall explain how I go about to accomplish the evaluation of such
determinants.
Jim
McGuire
If the state function of a system in delta interaction is expressed
as the determinant of a square matrix the algebraic properties of the
determinant make symmetry properties of the state function
transparent. These algebraic properties are topological in nature,
and can only be exploited for delta interaction on a ring. The state
function for scattering on a line, for example, cannot be expressed
as a determinant in all regions of state space. The determinant form
leads to a form of separation and very simple spectral properties.
Soichi
Okada
G. Kuperberg uses square ice models to enumerate several symmetry classes
of alternating sign matrices. His main tool is a set of determinant and
Pfaffian formulas for the partition functions of square ice models.
In this talk, I will show that these partition functions are written as
irreducible characters of classical groups. And I enumerate VHSASM
(vertically and horizontally symmetric ASMs) of odd size.
Paul Pearce
Jim Propp
(telephone lecture)
The discrete Hirota equation plays an important
role in the theory of integrable systems, but it also lurks incognito in classical
algebra and combinatorics. By discussing the particular case of Dodgson condensation
and domino tilings of Aztec diamonds, I hope to indicate how the Hirota equation may
point towards a host of exactly solvable two-dimensional lattice models.
David
Wilson
Consider random walk on a very fine grid
in the plane. As the grid size tends to zero, the random walk
converges to a process (Brownian motion) that is rotationally
invariant, and which is also invariant under conformal
maps. Other random processes on the 2D grid
also converge to conformally invariant
limits, but until recently this convergence was
known only non-rigorously. In this
talk we present the SLE process that was introduced
by Schramm, and subsequent work of Lawler,
Schramm, Werner, Smirnov, and Rohde
that relates SLE to different random combinatorial
objects such as loop-erased random walk (SLE(2)), Brownian frontier
(SLE(8/3)), self-avoiding walk (SLE(8/3)?), superposition
of two domino tilings (SLE(4)?), percolation (SLE(6)), and the
spanning tree Peano curve (SLE(8)).
Paul
Zinn-Justin
We shall first briefly review the standard connection between the one-matrix
model, orthogonal polynomials and the 1D Toda lattice / KP hierarchy, and its
usefulness in taking the large N limit. Then we shall consider the more
complicated case of the Harish Chandra--Itzykson--Zuber integral. Even though
this matrix integral can be computed exactly, its large N limit is most
elegantly studied in the framework of the 2D Toda lattice. We shall show how
the associated bi-orthogonal polynomials allow to do some explicit
calculations and discuss their combinatorial meaning.
The Airy phenomena
in analytic combinatorics
The XXZ/O(n) loop model
Hamiltonians and combinatorics
Physical Combinatorics: Fermionic
Paths and Virasoro Algebra I
Maximum increasing subsequences
and non-intersecting paths.
Scaling theory for Ising and SAW
models
I will also show how the theory of generalised lattice-lattice
scaling can be used to extend the Ising susceptibility results to other
lattices.
Correlation functions of the XXZ chain and alternating sign
matrices
Enumeration of rhombus tilings and
determinant evaluations
Determinants in Delta Interaction
Alternating sign matrices, square ice
models and characters of classical groups.
Physical Combinatorics: Fermionic
Paths and Virasoro Algebra II
Integrability, Exact Solvability, and
Algebraic Combinatorics: A Three-Way Bridge?
A new theory of conformally invariant
fractals: SLE and its connection to combinatorics
Matrix models and integrability:
the HCIZ integral