Workshop on integrable combinatorics. Chaire de la Vallée Poussin 2025

18 Nov 2025, 16:15 → 20 Nov 2025, 12:30 Europe/Brussels

CYCL01 (Marc de Hemptinne (chemin du Cyclotron, 2, Louvain-la-Neuve))

Chaire de la Vallée Poussin 2025 : Philippe Di Francesco

The workshop aims to bring together researchers working on combinatorial models through methods inspired by integrable systems. A highlight of the program will be a lecture series by Prof. Philippe Di Francesco, recipient of the 2025 de la Vallée Poussin Chair. The inaugural lecture is designed for a broad audience of mathematicians and physicists, while the subsequent lectures and invited research talks will remain pedagogical and accessible, though more specialized, with a focus on topics of interest to the mathematical physics community.

 

We have limited funding available to cover the accommodation of young researchers who wish to attend the entire workshop. Please indicate it in the registration form if you wish to apply for funding.

 

Invited speakers:

   Roger Behrend (Cardiff University)
   Philippe Di Francesco (CEA Saclay and University of Illinois)
   Arno Kuijlaars (KU Leuven)
   Wenkui Liu (UCLouvain)
   Max van Horssen (KU Leuven) 
   Andrea Sportiello (CNRS and LIPN, Université Sorbonne Paris Nord)
   Panagiotis Zografos (University of Leipzig)

Chaire de la Vallée Poussin 2025

Invitation.pdf

Poster.pdf

Tuesday 18 November

Inaugural lecture: Integrable combinatorics

Physics has always provided insights and inspiration into new mathematics. We will concentrate here on Combinatorics, namely the art of counting objects

in classes, and follow guidance from Statistical physics that attaches probability weights to those objects and tries to tackle fundamental questions such as correlations or thermodynamic behavior. Symmetries of the systems studied can sometimes drastically simplify them, and in the best cases lead to exact solutions (i.e. exact counting, or exact asymptotics of such counts). Discrete or continuous integrable systems have enough symmetries to guarantee the existence of exact solutions. In this lecture, we will explore and solve a number of combinatorial integrable problems, in relation to various areas of mathematics: random geometry, random surfaces, cluster algebras, etc.

Convener: Prof. Philippe Di Francesco

CdVP25_DiFrancesco_0.pdf

17:15 Reception

Wednesday 19 November

Lecture 1: Tilings, Vertex Models and other remarkable combinatorial objects

The study of Random tilings was initiated and first developed in physics, from the Dimer models to quasi-crystals, before it became a pure mathematics subject. On the other hand, Vertex models of statistical physics describe the long range effect of local interactions in crystals. In this lecture, we start from a remarkable coincidence between the number of configurations of the so-called six vertex model, describing an ideal square lattice crystal of ice in two dimensions and a rhombus tiling model of a suitable hexagon. This is a small part of an extraordinary sequence of combinatorial coincidences, none of which is yet understood in terms of natural bijections. We show how the integrability of the six-vertex model allows to count its weighted configurations via a determinant. The computation of this determinant and some of its limits allows to identify the counting with that of so-called Descending Plane Partitions, in bijection with some specific rhombus tiling problem. We then generalize the problem to triangular lattice ice, in the form of the twenty vertex model, and identify a suitable non-bijective domino tilings counting problem. Again, integrability allows to compute the numbers of configurations exactly, and to prove new coincidences, unsupported by natural bijections.

Convener: Philippe Di Francesco

CdVP25_DiFrancesco_1.pdf

10:00 Coffee break

1 Invited lecture 1 : Large random tilings of a hexagon with periodic weightings

Large random tilings of a hexagon have the fascinating behavior of distinct asymptotic phases (frozen and rough; also called solid and liquid) separated by a well-defined Arctic curve. In a weighted tiling model with periodically varying weights a third phase (smooth; or gaseous) appears where correlations between tiles decay at an exponential rate.

I will discuss an approach towards a rigorous analysis of a three periodic hexagon tiling model, that includes matrix valued orthogonal polynomials, Riemann-Hilbert problems, and steepest descent analysis on a Harnack curve.

Speaker: Arno Kuijlaars (KU Leuven)

CdVP25_Kuijlaars.pdf

2 Invited lecture 2 : Large random tilings of a hexagon with periodic weightings: steepest descent analysis on the double amoeba

We continue the study of the 3x3-periodic random hexagon tiling model from the previous talk by Arno Kuijlaars. There is a determinantal point process on the hexagon that is equivalent to the tiling model. The correlation kernel of this process has been expressed as a double contour integral by Maurice Duits and Arno Kuijlaars. This representation of the correlation kernel allows us to use steepest descent analysis to obtain large-N asymptotics. It turns out that the position of the dominant saddle point determines the phase (solid, liquid, or gas) for a given asymptotic coordinate. We show that the liquid phase is homeomorphic to the (double) amoeba of a genus-one Harnack curve. This double amoeba is used to prove the existence of steepest descent (and ascent) paths when the asymptotic coordinate lies in the liquid region.

At the end of the talk, we give an outlook for the 3x3-periodic random tiling model on hexagons that are stretched in one direction. Depending on the direction, we observe two distinct cases of a splitting of the gas phase.

Speaker: Max van Horssen (KU Leuven)

CdVP25_VanHorssen.pdf

12:30 Lunch Break

Lecture 2: Limit shapes

A natural question in physics regards the thermodynamic limit, of large size and small mesh, where one expects to capture only the essential properties of a model, independently of its details. When it comes to tilings of specific domains a whole theory of limit shapes was developed in mathematics by Kenyon, Okounkov, Sheffield and others. These are generalizations of the famous arctic circle theorem of Cohn, Kenyon, Propp for the domino tilings of the so-called Aztec diamond. From the physics point of view, tiling problems are free fermion models, namely they can be expressed in terms of ``fermionic" degrees of freedom, generally in the form non-intersecting paths. However Vertex models such as the 6 vertex model are models of interacting fermions, where paths are allowed to touch thus violating the fermionic condition. In this lecture, we explore limit shapes of 6 and 20 vertex models, establishing their arctic phenomena by using their integrability. We find some remarkable relations between limit shapes of equinumerous yet non-bijective problems explored in lecture 1.

Convener: Prof. Philippe Di Francesco

CdVP25_DiFrancesco_2.pdf

15:00 Coffee break

3 Invited lecture 3 : The combinatorics of alternating sign matrices

An alternating sign matrix is a square matrix in which each entry is -1, 0 or 1, and along each row and column the nonzero entries alternate in sign, starting and ending with a 1. In this talk, I will discuss various intriguing combinatorial results involving these matrices and related objects.

Speaker: Roger Behrend (Cardiff University)

CdVP25_Behrend.pdf

4 Invited lecture 4 : Domino tilings of the Aztec diamond in random environment

It is well-known that under a certain choice of weights random domino tilings of the Aztec diamond can be analyzed via Schur measures on partitions. In this talk we will discuss the model in which the parameters of the Schur measure (or, equivalently, some weights of dominoes) are random themselves. We establish the limit shape and global fluctuations (quenched and annealed) via the technique of Schur generating functions.

Speaker: Panagiotis Zografos (University of Leipzig)

CdVP25_Zografos.pdf

Thursday 20 November

Lecture 3: Meanders

This lecture is devoted entirely to the "Problème des Timbres poste" or stamp-folding problem first posed by Emile Lemoine in 1891, revisited by Poincaré in 1912 in purely geometric terms and later rebaptized "meander problem" by Arnold in 1991. In the latter language, we wish to count inequivalent configurations of a non-self-intersecting road circuit crossing a straight infinite river through 2N bridges.

We show how the question relates to deep and fundamental problems in algebra and geometry, and how the physics theory of two-dimensional quantum gravity has allowed us to make remarkable predictions on the asymptotic count of meander configurations.

Convener: Philippe Di Francesco

CdVP25_DiFrancesco_3.pdf

10:00 Coffee break

5 Invited lecture 5 : Towards a black+white Razumov-Stroganov conjecture

In June 2024 the conference "Philippe60" was held in Saclay. I was a speaker. I talked about some conjectures on how the enumerations of VSASM's according to black and white link patterns appear to possibly coincide with the structure constants of some (so far unknown) deformations of canonical Grothendieck polynomials.
I mentioned in the introduction of that talk the disappointment for the fact that we have no Razumov-Stroganov conjectures involving the black and white link patterns, despite the fact that the Wieland lemma, instrumental to the proof, deals with the two.
The disappointment turned into a new impulse to try that again, and finally something sensible came out. The results of this research will be our subject today !Spoiler! you will see: FPL configurations with cuts, cylindrical link patterns, loop models on finite cylinders...
Work in collaboration with Luigi Cantini.

Speaker: Andrea Sportiello (CNRS and LIPN, Université Sorbonne Paris Nord)

CdVP25_Sportiello.pdf

6 Invited lecture 6: Limit Shape of Random Lozenge tilings of a Hexagon with $q$-Volume weights

We will introduce the $q$-Volume weight for random lozenge tilings in a hexagon. The probability of each configuration is associated with its total volume. More generally we can assign $q$-Racah weight to the tiling model. These are natural generalisation of the random tilings with uniform weights. However, one can also fine tune the parameters to see singular behaviours.

One fascinating object to study about these models is the counting statistics which is called the height function of the tilings. In non-singular cases, it concentrates near a deterministic limit shape and that the global fluctuations are described by the Gaussian Free Field.

There are some remarkable relations between limit shapes and variational problems. A particular motivation for studying the $q$-Volume/Racah model is that the variational characterisation of the limiting height function has an inhomogeneous term. The study of the regularity properties of the minimizer for general variational problems with such inhomogeneous terms is a challenging open problem.

Speaker: Wenkui Liu

CdVP25_Liu.pdf