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

• Arno Kuijlaars (KU Leuven)

Room: CYCL01 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.

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

• Max van Horssen (KU Leuven)

Room: CYCL01 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.

Invited lecture 3 : The combinatorics of alternating sign matrices

• Roger Behrend (Cardiff University)

Room: CYCL01 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.

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

• Panagiotis Zografos (University of Leipzig)

Room: CYCL01 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.

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

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

Room: CYCL01 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.

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

• Wenkui Liu

Room: CYCL01 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.