Event at Galileo Galilei Institute

Focus Week

Lattice models: exact methods and combinatorics

May 18, 2015 - May 22, 2015


This event is one of the activities taking place at Galileo Galilei Institute, in Florence, in the framework of the two month workshop on Statistical Mechanics, Integrability and Combinatorics. It aims at bringing together mathematicians and physicists interested in problems at the interface between statistical mechanics, combinatorics and probability theory.

In this Focus Week, emphasis is put on lattice models, and their behaviour at a macroscopic level (scaling limits at criticality) and especially at a microscopic level (combinatorial and algebraic structures).


Filippo Colomo (INFN, Florence)
Yacine Ikhlef (CNRS, UPMC-Paris 6)
Paul Zinn-Justin (CNRS, UPMC-Paris 6)


Main event
Statistical mechanics, integrability and combinatorics (Workshop) - May 11, 2015

Date Speaker Title Type Useful Links
May 18, 2015 - 10:30 Guttmann Tony Integrability, Solvability and Enumeration Seminar

Integrability, Solvability and Enumeration

There are a number of seminal two-dimensional lattice models that are integrable, but have only
been partially solved, in the sense that only some properties are fully known (e.g. the two-
dimensional Ising model, where the free-energy is known, but not the susceptibility). Alternatively,
critical properties are known for some lattices but not others. For example, the critical point of the
self-avoiding walk model is known rigorously for the honeycomb lattice, but not for other lattices.
Similarly for the q-state Potts model and both bond and site percolation. The critical manifold of the
former is known only for some lattices, likewise the percolation threshold is known only for some
A range of numerical procedures exist, based on exact enumeration, or other numerical work, such
as calculating the eigenvalues of transfer matrices, which, when combined with various structural
invariants seem to give exact results in those cases that are known to be exact, but can be used to
give increasingly precise estimates in those cases which are not exactly known. Reasons for this
partial success are not well understood. In this talk I will describe four such procedures, and
demonstrate their performance, and speculate on their partial success

May 18, 2015 - 11:30 Biane Philippe Covering trees of the graph of covering trees Seminar

Covering trees of the graph of covering trees

The set of covering trees of a directed graph can be endowed with a natural graph structure covering
the original graph. We show that the generating function of covering trees of this new graph exhibits
a remarkable factorization property. This is joint work with Guillaume Chapuy.

May 18, 2015 - 15:00 Johansson Kurt Random matrices and Aztec diamonds Seminar

Random matrices and Aztec diamonds

Random tilings, or dimer models on bipartite graphs, give rise to determinantal point processes.
Natural scaling limits of these processes give rise to limiting processes and distributions of the same
type as we see in random matrix theory and also in random growth models in the KPZ universality
class, e.g the Airy process. I will give some background on this and in particular discuss it in

connection with the two-periodic Aztec diamond, which is recent joint work with Sunil Chhita

May 18, 2015 - 16:30 Striker Jessica Posets of alternating sign matrices and totally symmetric self-complementary plane partitions Seminar

Posets of alternating sign matrices and totally symmetric self-complementary plane partitions

Alternating sign matrices and totally symmetric self-complementary plane partitions are
mysteriously equinumerous combinatorial objects with interesting connections to physics. In this
talk, we discuss how encoding these objects as order ideals of partially ordered sets, or posets,
yields a unifying framework for investigation. We study several actions on these objects from the
poset perspective, including gyration on fully-packed loops, and show some surprising periodicity
and orbit-average properties. We also give a permutation-case bijection and study the poset
structures on both sides of this bijection.

May 19, 2015 - 09:00 de Gier Jan A matrix product formula for Macdonald polynomials Seminar

A matrix product formula for Macdonald polynomials

I will discuss how an inhomogeneous version of the multi-species asymmetric simple exclusion
process gives rise to a matrix product formula for Macdonald polynomials. This is work in

collaboration with Luigi Cantini and Michael Whee

May 19, 2015 - 10:30 Corteel Sylvie Combinatorics of exclusion processes with open boundaries Seminar

Combinatorics of exclusion processes with open boundaries

We study the combinatorics of the two-species asymmetric simple exclusion process with open
boundaries studied among others by Masaru Uchiyama. We give a combinatorial interpretation of
the partition function thanks to the Matrix Ansatz and the combinatorics of lattice paths. This brings
us to a general positivity conjecture on Koornwinder moments. We also present a generalization
of the staircase tableaux and TAT tableaux of Mandelshtam and Viennot which gives a
combinatorial interpretation of the stationary distribution. This is joint work with Lauren Williams

(Berkeley) and also with Olya Mandelshtam (Berkeley) for the tableaux parts

May 19, 2015 - 11:30 Cantini Luigi Inhomogenous Multispecies TASEP on a ring Seminar

Inhomogenous Multispecies TASEP on a ring

In this talk I will present some results about a multispecies version of the TASEP, a model which
describes the stochastic evolution of a system of particles of different species on a periodic oriented
one dimensional lattice, where two neighboring particles exchange their positions with a rate which
depends on their species. For some choice of these rates the Markov matrix turns out to be
integrable and for the same choice the (unnormalized) stationary probability is conjectured to show
remarkable positivity and combinatorial properties related to Schubert polynomials. I will discuss
how integrability leads to an interesting algebraic structure underlying the problem, and allowing to
prove some remarkable properties of the stationary measure and to give exact formulas for the
stationary probability of some classes of configurations.

May 19, 2015 - 15:00 Beffara Vincent Smirnov s parafermionic observable, at and away from criticality Seminar

Smirnov s parafermionic observable, at and away from criticality

Smirnov s parafermionic observable, together with the theory of discrete holomorphicity, was
instrumental in proving the conformal invariance of the interfaces of the Ising model at its critical
point; it exhibits a flavor of integrability at the critical temperature for all the random-cluster
models, and in a few cases (namely the Ising model itself and the RC models for
) it can be
studied directly at non-critical temperatures, where it provides precise information about the two-
point function. I will present a few results in this direction, joint with H. Duminil-Copin and S.


May 19, 2015 - 16:30 Weston Robert Discrete Holomorphicity and Perturbed Conformal Field Theory Seminar

Discrete Holomorphicity and Perturbed Conformal Field Theory

I will introduce the concept of discrete holomorphicity for functions on a 2D lattice embedded into
the complex plane. I will briefly discuss the important role that functions with this property have
played in proving the existence and uniqueness of the continuum limit of certain models of
statistical mechanics. I will then go on to discuss a systematic procedure for constructing discretely
holomorphic operators in solvable statistical mechanical models in terms of the underlying quantum
group symmetry of these models.
This procedure will be illustrated for the Chiral Potts model. I
will show how the resulting discrete holomorphicity conditions for non-critical models leads
directly to an identification of these models with a corresponding perturbed conformal field theory.

May 20, 2015 - 09:30 Bleher Pavel Dimer Model: Full Asymptotic Expansion of the Partition Function Seminar

Dimer Model: Full Asymptotic Expansion of the Partition Function

We obtain the full asymptotic expansion of the partition function of the 2D dimer model on the m x
n lattice with free and cylindrical boundary conditions. We show that the asymptotic expansion goes
over powers of
, where
for the free boundary conditions and
for the
cylindrical boundary conditions. The coefficients of the asymptotic expansion are expressed in
terms of the Kronecker double series, the Dedekind eta function, and the Jacobi theta functions. Our
calculations use the technique of the work of Ivashkevich, Izmailian, and Hu, in which the full
asymptotic expansion is developed for the periodic boundary conditions.
This is a joint work with Brad Elwood and Drazen Petrovic.

May 20, 2015 - 10:30 Panova Greta Lozenge tilings and other lattice models from the viewpoint of symmetric functions Seminar

Lozenge tilings and other lattice models from the viewpoint of symmetric functions

What do lozenge tilings (a.k.a. plane partitions, dimer covers of the hexagonal lattice), alternating
sign matrices (or the six-vertex model) and the dense loop model have in common? For one, their
limiting behavior can be studied with the help of some "asymptotic" algebraic combinatorics.
We develop methods to analyze normalized symmetric functions (Schur functions and more general
Lie group characters), as the indexing partition converges to a limiting profile. We apply this
analysis together with some combinatorial interpretations to study the limiting behavior of the
integrable models listed above. In particular, we show that the positions of horizontal lozenges near
a vertical flat boundary are distributed like the eigenvalues of GUE matrices, and this holds for a
wide class of domains (including such with free boundary). These methods can also be used to
establish the existence of limit shapes also for free boundary domains. We discover Gaussian
distribution for some observables of the Alternating Sign Matrices, leading again to GUE
eigenvalues for the positions of 1s near the border (result of V. Gorin). We also find the asymptotics
for the [conjectural] expected value of the mean total current between two adjacent points in the
dense loop model. Based on joint work with Vadim Gorin.
We will also discuss limit behavior under non-uniform (
^volume) distributions, based on ongoing

work with Jonathan Novak

May 20, 2015 - 11:40 Ciucu Mihai Lozenge tilings with gaps in a 90 degree wedge domain with mixed boundary conditions Seminar

Lozenge tilings with gaps in a 90 degree wedge domain with mixed boundary conditions

We consider a triangular gap of side two in a 90 degree angle on the triangular lattice with mixed
boundary conditions: a constrained, zig-zag boundary along one side, and a free lattice line
boundary along the other. We study the interaction of the gap with the corner as the rest of the angle
is completely filled with lozenges. We show that the resulting correlation is governed by the product
of the distances between the gap and its three images in the sides of the angle. This provides
evidence for a unified way of understanding the interaction of gaps with the boundary under mixed
boundary conditions, which we present as a conjecture. Our conjecture is phrased in terms of the
steady state heat flow problem in a uniform block of material in which there are a finite number of
heat sources and sinks. This new physical analogy is equivalent in the bulk to the electrostatic
analogy we developed in previous work, but arises as the correct one for the correlation with the
The starting point for our analysis is an exact formula we prove for the number of lozenge tilings of
certain trapezoidal regions with mixed boundary conditions, which is equivalent to a new, multi-
parameter generalization of a classical plane partition enumeration problem (that of enumerating
symmetric, self-complementary plane partitions).

May 21, 2015 - 09:00 Di Francesco Philippe Physics and combinatorics of the octahedron equation: from cluster algebras to arctic curves Seminar

Physics and combinatorics of the octahedron equation: from cluster algebras to arctic curves

The octahedron equation is a non-linear system of recursion relations in discrete 2+1 dimensions,
that first appeared in the context of generalized Heisenberg quantum spin chains (T-system). Some
solutions of this equation also have a purely combinatorial interpretation as generalized Coxeter-
Conway Frieze patterns.
This same equation plays a central role in the seemingly unrelated purely combinatorial problem of
enumeration of domino tilings of a plane square-shaped domain called the Aztec diamond. For large
size, the tiling configurations display some drastic change of typical behavior between the corners
of the domain, where it freezes in certain tiling patterns, and a central region in the bulk, with
disorder. In the continuum limit of large size and small mesh, the separation of phases is along the
so-called "arctic circle" inscribed inside the square domain.
In this talk, we shall show that the octahedron equation is part of a combinatorial structure called
cluster algebra, and how its exact solution may be rephrased in terms of non-intersecting lattice
paths, and eventually domino tilings. Using the octahedron equation explicitly, we will proceed and
determine various arctic curves depending on initial data. As predicted by Kenyon and Okounkov,
we find in general new "facet" phases of the dimer model within some connected components of
these curves.
(Joint works with R. Kedem and R. Soto Garrido).

May 21, 2015 - 10:30 Kedem Rinat Fusion products and q -Whittaker functions Seminar

Fusion products and q -Whittaker functions

The solution of the generalized Heisenberg model via Bethe ansatz comes with generalized
fermionic combinatorics for the spectrum. Using this to compute linearized partition functions
relates this model to conformal blocks of WZW theories and graded finite-dimensional
representations of affine algebras, which are tensor products. In the simplest example, these are
affine Demazure modules, and the limit as the size of the system becomes infinite are the well-
known level-1 representations associated with the CFT. In this talk, I concentrate on the finite
system, and explain why the linearized partition functions in the level-1 case are Whittaker
functions for the quantum (finite) algebra, which satisfy the quantum difference Toda equation.

May 21, 2015 - 11:40 Okada Soichi Generalized Cauchy determinant and Schur Pfaffian, and their applications Seminar

Generalized Cauchy determinant and Schur Pfaffian, and their applications

The Cauchy determinant and the Schur Pfaffian, together with their generalizations, play a
fundamental role in combinatorics and representation theory. For example, such
determinant/Pfaffian identities are used to evaluate Izergin-Korepin-type determinants/Pfaffians in
enumeration problems of alternating sign matrices. In this talk, we present yet other generalizations
of the Cauchy determinant and the Schur Pfaffian, and give their applications to symmetric function

May 21, 2015 - 15:00 Smirnov Fedor On the space of quasi-local operators for the Fateev-Zamolodchikov model Seminar

On the space of quasi-local operators for the Fateev-Zamolodchikov model

We consider the Fateev-Zamolodchikov model (integrable anisotropic spin chain of spin 1). We
show that the "good" basis of quasi-local operators is created by two doublets of fermions and one
triplet of bosons. The latter constitute the Kac-Moody algebra of level one. This is to be compared
with the spin 1/2 case for which we had one doublet of fermions.

May 21, 2015 - 16:30 Gohmann Frank Low-temperature spectrum of correlation lengths of the XXZ chain in the anti-ferromagnetic massive regime Seminar Slides
May 22, 2015 - 09:00 Rosengren Hjalmar Towards combinatorics of elliptic lattice models Seminar

Towards combinatorics of elliptic lattice models

Much of the recent progress on combinatorics of solvable lattice models is concerned with the six-
vertex model and XXZ spin chain. For the more general eight-vertex model and XYZ spin chain,
which are naturally parametrized by elliptic functions,
progress has been slower. Various quantities related to the models are described in terms of
polynomials with positive integer coefficients, but a combinatorial explanation for this phenomenon
is still lacking. On the other hand, new intriguing
features appear in the elliptic case, such as relations to the Painlevé VI equation already at the finite
lattice. In the present talk, I will try to give an overview of the current knowledge, discussing the
work of myself and others (V. Bazhanov, V. Mangazeev, A. Razumov, Yu. Stroganov, P. Zinn-

May 22, 2015 - 10:30 Kytola Kalle Boundary correlation functions with a hidden quantum group Seminar

Boundary correlation functions with a hidden quantum group

In the scaling limit, critical lattice model correlation functions satisfy partial differential equations
of conformal field theories. In this talk, we present a method of constructing boundary correlation
functions using a hidden quantum group. Solutions with desired asymptotic behavior are found by
solving a problem in representations of the quantum group. Two applications to random curves will
be considered: multiple SLE pure partition functions and boundary visit amplitudes of chordal
The talk is based on joint work with Eveliina Peltola ([arXiv:1408.1384] and in preparation), and

with Niko Jokela and Matti Järvinen ([arXiv:1311.2297])

May 22, 2015 - 11:30 Delfino Gesualdo Phase separation, interfaces and wetting in two dimensions Seminar

Phase separation, interfaces and wetting in two dimensions

Separation of phases with finite correlation length and the same free energy is a common
phenomenon. For long time the lattice integrability of the Ising model in two dimensions has
provided the only exact result for the scaling limit of the order parameter profile. It has been
recently realized that field theory yields a series of exact results for the different universality classes
in two dimensions, including order parameter profiles, interface structure and wetting properties,
both in the bulk and at boundaries.
This is joint work with J. Viti (
Phase separation and interface structure in two dimensions from
field theory
, J. Stat. Mech. (2012) P10009) and with A. Squarcini (
Exact theory of intermediate
phases in two dimensions,
Annals of Physics 342 (2014) 171, and
Phase separation in a wedge.
Exact results
, PRL 113 (2014) 066101