# Week 1

• C : Dmitry Chelkak (École Normale Supérieure, Paris, France)
Planar Ising model: from combinatorics to CFT and s-embeddingscourse page
In theoretical physics, the critical planar Ising model serves as a toy example, in which many precursors of Conformal Field Theory objects and structures exist and can be studied directly in discrete, before passing to the small mesh size limit. Mathematically, a number of results on convergence and conformal invariance of such limits were established during the last decade, both for correlation functions and for interfaces (domain walls) arising in the model. In this mini-course we plan to discuss
• discrete fermions and the Kadanoff-Ceva spin-disorder formalism - crucial tools that allow one to analyse the planar Ising model;
• streamlined version of the classical computation of the magnetization via orthogonal polynomials;
• results on convergence of critical correlation functions (energy densities, spins, ...) in bounded domains to CFT limits;
• recent ideas on appropriate embeddings of weighted planar graphs that play the same role for the planar Ising model as Tutte’s barycentric embeddings do for random walks, allowing one to use discrete complex analysis techniques beyond "regular" lattices.
• W : Ole Warnaar (University of Queensland, Brisbane, Australia)
Schur functions and Schur processescourse page
##### Monday, May 27
• 08:55 - 09:00 Opening remarks
• 9:00 - 10:30 Lecture (C1)
• 10:30 - 11:00 Break
• 11:00 - 12:00 TA session (C1)
• 12:00 - 2:00 Lunch time
• 2:00 - 3:30 Lecture (W1)
• 3:30 - 4:00 Break
• 4:00 - 5:00 TA session (W1)
##### Tuesday, May 28
• 9:00 - 10:30 Lecture (W2)
• 10:30 - 11:00 Break
• 11:00 - 12:00 TA session (W2)
• 12:00 - 2:00 Lunch time
• 2:00 - 3:30 Lecture (C2)
• 3:30 - 4:00 Break
• 4:00 - 5:00 TA session (C2)
##### Wednesday, May 29
• 9:00 - 10:30 Lecture (C3)
• 10:30 - 11:00 Break
• 11:00 - 12:00 TA session (C3)
• Free time / collaboration
##### Thursday, May 30
• 9:00 - 10:30 Lecture (W3)
• 10:30 - 11:00 Break
• 11:00 - 12:00 TA session (W3)
• 12:00 - 2:00 Lunch time
• 2:00 - 3:00 Research talk: Marianna Russkikh (University of Geneva) - Dimers and embeddings
One of the main questions in the context of the universality and conformal invariance of a critical 2D lattice model is to find an embedding which geometrically encodes the weights of the model and that admits “nice” discretizations of Laplace and Cauchy-Riemann operators. We establish a correspondence between dimer models on a bipartite graph and circle patterns with the combinatorics of that graph. We describe how to construct a circle pattern of a dimer planar graph using its Kasteleyn weights. We also introduce the definition of discrete holomorphicity on such an embedding. We discuss the link between these functions and actual continuous holomorphic functions.
Based on:
• “Dimers and Circles” joint with R. Kenyon, W. Lam, S. Ramassamy;
• “Holomorphic functions on t-embeddings of planar graphs” joint with D. Chelkak, B. Laslier.
• 3:00 - 3:30 Break
• 3:30 - 5:00 Lecture (C4)
##### Friday, May 31
• 9:00 - 10:30 Lecture (W4)
• 10:30 - 11:00 Break with snacks
• 11:00 - 12:00 Research talk: Guillaume Barraquand (ENS) - Diffusions in random environment
We will consider the effect of adding a space-time white noise drift to a collection of independent Brownian motions. Using an integrable discretization of the model, we will see that the extreme value behavior for these diffusions is governed by the Kardar-Parisi-Zhang universality class which arises in random growth models and random matrix theory.
This talk is based on joint works with Ivan Corwin and Mark Rychnovsky.
• 12:15 - 1:15 Research talk: Cesar Cuenca (MIT) - Probability measures of representation theoretic origin
We introduce the BC type Z-measures as members of a 4-parameter family of point processes, with origins in the representation theory of the infinite-dimensional orthogonal and symplectic groups. The main result we present is that the BC type Z-measures are determinantal point processes with explicit correlation kernels, in terms of hypergeometric functions. In joint work with Grigori Olshanski, we have defined natural q-analogues of the BC type Z-measures. Our construction is based on the theory of q-hypergeometric orthogonal polynomials, though we hope that these measures can also be constructed from the representation theory of quantum groups. The last part of the talk is a brief overview of the quantization of the BC type Z-measures.
• Free time / collaboration

# Week 2

• S : Tomohiro Sasamoto (Tokyo Institute of Technology, Tokyo, Japan)
Fluctuations of 1D exclusion processes: exact analysis and hydrodynamic approachcourse page
One dimensional exclusion processes are stochastic processes in which many particles perform random walks under exclusion constraint. They have been playing important role in the fields of stochastic interacting systems in probability theory and non-equilibrium statistical mechanics in physics. For the last two decades, fluctuations of the processes have been studied quite intensively, since the seminal work by Johansson[1-1] on totally asymmetric simple exclusion process (TASEP) showing that the current fluctuation of TASEP with step initial condition is described by the GUE Tracy-Widom distribution. There have been a vast accumulation of generalizations and related results, but there are still many intriguing questions and problems to be solved.

In these lectures, we discuss a few new directions in the studies of fluctuations of exclusion processes. We also stress that such studies provide valuable insight to other methods based on hydrodynamic ideas which can be applied to a wider class of interacting particle systems. In the first lecture we review the basics of the subject. After introducing a few models such as the asymmetric simple exclusion process(ASEP) and the Kardar-Parisi-Zhang (KPZ) equation, we explain how one can study their fluctuations for the case of TASEP[1-2]. In the second lecture, we show that an approach introduced in [2] using Frobenius determinant can be applied to a large class of models in a unified manner. In the third lecture we explain our recent result on a two-species exclusion process and connection to the nonlinear fluctuating hydrodynamics[3]. In the last lecture we will consider an application of the techniques to study the large derivation in the symmetric exclusion process[4-1,2].

References
• [1-1] K. Johansson, Shape fluctuations and random matrices, Commun. Math. Phys. (2009) 437-476. [arXiv:math/9903134]
• [1-2] T. Sasamoto, Fluctuations of the one-dimensional asymmetric exclusion process using random matrix techniques, J. Stat. Mech. (2007) P07007. [arXiv:0705.2942]
• [2] T. Imamura, T. Sasamoto, Fluctuations for stationary q- TASEP, to appear in Prob. Th. Rel. Fields. [arXiv:1701.05991]
• [3] Z. Chen, J. de Gier, I. Hiki, T. Sasamoto, Exact confirmation of 1D nonlinear fluctuating hydrodynamics for a two-species exclusion process, Phys. Rev. Lett. 120, 240601 (2018). [arXiv:1803.06829]
• [4-1] T. Imamura, K. Mallick, T. Sasamoto, Large deviations of a tracer in the symmetric exclusion process, Phys. Rev. Lett. 118, 160601 (2017). [arXiv:1701.05991]
• [4-2] T. Imamura, K. Mallick, T. Sasamoto, Distribution of a tagged particle position in the one-dimensional symmetric simple exclusion process with two-sided Bernoulli initial condition, arXiv:1810.06131.
• Z : Paul Zinn-Justin (University of Melbourne, Melbourne, Australia)
Quantum integrability and symmetric polynomialscourse page
##### Monday, June 3
• 9:00 - 10:30 Lecture (S1)
• 10:30 - 11:00 Break
• 11:00 - 12:00 TA session (S1)
• 12:00 - 2:00 Lunch time
• 2:00 - 3:30 Lecture (Z1)
• 3:30 - 4:00 Break
• 4:00 - 5:00 TA session (Z1)
##### Tuesday, June 4
• 9:00 - 10:30 Lecture (Z2)
• 10:30 - 11:00 Break
• 11:00 - 12:00 TA session (Z2)
• 12:00 - 2:00 Lunch time
• 2:00 - 3:30 Lecture (S2)
• 3:30 - 4:00 Break
• 4:00 - 5:00 TA session (S2)
##### Wednesday, June 5
• 9:00 - 10:30 Lecture (S3)
• 10:30 - 11:00 Break
• 11:00 - 12:30 Lecture (Z3)
• Free time / collaboration
##### Thursday, June 6
• 9:00 - 10:30 Lecture (Z4)
• 10:30 - 11:00 Break
• 11:00 - 12:00 TA session (Z3)
• 12:00 - 2:00 Lunch time
• 2:00 - 3:30 Lecture (S4)
• 3:30 - 4:00 Break
• 4:00 - 5:00 TA session (S3)
• 6:30 - 9:00 Banquet at Himalayan Fusion ($20/person) # Research talks ##### Friday, June 7 • 9:00 - 9:35 Andrew Ahn (MIT) - Macdonald Plane Partitions and Product Processes The Macdonald plane partitions are a two-parameter family of deformation of the q^vol measure on random plane partitions. It was shown that, under a suitable limit, the Macdonald plane partitions degenerate to products of beta Jacobi ensembles, where the notion of product for arbitrary beta is an extension of free multiplication for unitarily invariant random matrices. We discuss a difference operators method which can be used to access global asymptotics of the Macdonald plane partitions. Under a suitable limit transition, we discuss how this method can be applied to access global asymptotics of products of beta Jacobi ensembles. • 9:40 - 10:15 Jules Lamers (University of Melbourne) - q-deformed Haldane-Shastry spin chain Abstract TBA • 10:15 - 10:50 Break • 10:50 - 11:25 Costanza Benassi (Northumbria University) - Thermodynamic Limit and Dispersive Regularisation in Matrix Models We show that Hermitian Matrix Models support the occurrence of a new type of phase transition characterised by dispersive regularisation of the order parameter near the critical point. Using the identification of the partition function with a solution of a reduction of the Toda hierarchy, known as Volterra system, we argue that the singularity is resolved via the onset of a multi-dimensional dispersive shock of the order parameter in the space of coupling constants. This analysis explains the origin and mechanism leading to the emergence of chaotic behaviours observed in M6 matrix models and extends its validity to even nonlinearity of arbitrary order. Based on a joint work with A. Moro (arXiv:1903.11473). • 11:30 - 12:05 Shalin Parekh (Columbia) - Brownian Meanders in Directed Polymers Stochastic partial differential equations (SPDEs) such as the KPZ equation arise naturally as scaling limits of various probabilistic and physical models which are driven or directed by i.i.d. weights. However, obtaining precise information about the behavior of solutions to these SPDEs poses tremendous difficulties. So far, the most fruitful approach has been to look at exactly solvable models which converge to these SPDEs, and then extract information about the SPDE from the limiting models. One such exactly solvable model is the Log-Gamma directed polymer. In this talk, we will realize a multiplicative-noise stochastic heat equation on a half space as a limit of these Log-Gamma polymers, and we will prove a surprising identity in distribution for such equations using the exact solvability. Our analysis involves obtaining intricate estimates for random walks conditioned to stay positive. • 12:05 - 2:00 Lunch time • 2:00 - 2:35 Yier Lin (Columbia) - KPZ equation limit of the stochastic higher spin vertex model We consider the stochastic higher spin six vertex model introduced by Corwin and Petrov with general integer spin parameter$I, J$. Starting from near stationary initial condition, we prove that the stochastic higher spin six vertex model converges to the KPZ equation under weakly asymmetric scaling. This generalizes a result of Corwin et al. from$I = J =1$(stochastic six vertex model) to general$I, J$. • 2:40 - 3:15 Ryosuke Sato (Nagoya University) - Asymptotic representation theory of quantum groups Asymptotic representation theory means studies of characters and unitary representations of inductive limit groups, for instance, the infinite-dimensional unitary group. A fundamental idea of asymptotic representation theory is to correspond characters to probability measures on a graph giving from branching rules of representations. In this talk, we discuss a natural quantization of this framework, that is, natural character theory of inductive systems of (compact) quantum groups. In particular, we give serious thought when a given inductive system consists of quantum unitary groups. • 3:15 - 3:50 Break • 3:50 - 4:25 Elia Bisi (University College Dublin) - Transition between characters of classical groups, decomposition of Gelfand-Tsetlin patterns and last passage percolation We introduce two families of symmetric polynomials that interpolate between irreducible characters of Sp(2n,C) and SO(2n+1,C) and between irreducible characters of SO(2n,C) and SO(2n+1,C). We then study the last passage percolation model with various symmetries via a number of identities that involve orthogonal/symplectic characters and our interpolating polynomials, thus going beyond the link with classical Schur polynomials originally found by Baik and Rains. We achieve this by applying the Robinson-Schensted-Knuth correspondence to triangular arrays and using a decomposition procedure for Gelfand-Tsetlin patterns. As an application, we provide an explanation of why the Tracy-Widom GOE and GSE distributions from random matrix theory admit formulations in terms of both Fredholm determinants and Fredholm Pfaffians. • 4:30 - 5:05 Yi Sun (Columbia) - Gaussian fluctuations for products of random matrices This talk concerns singular values of M-fold products of i.i.d. right-unitarily invariant N x N random matrix ensembles. As N tends to infinity, the height function of the Lyapunov exponents converges to a deterministic limit by work of Voiculescu and Nica-Speicher for M fixed and by work of Newman and Isopi-Newman for M tending to infinity with N. In this talk, I will show for a variety of ensembles that fluctuations of these height functions about their mean converge to explicit Gaussian fields which are log-correlated for M fixed and have a white noise component for M tending to infinity with N. These ensembles include rectangular Ginibre matrices, truncated Haar-random unitary matrices, and right-unitarily invariant matrices with fixed singular values. I will sketch our technique, which derives a central limit theorem for global fluctuations via certain conditions on the multivariate Bessel generating function, a Laplace-transform-like object associated to the spectral measures of these matrix products. This is joint work with Vadim Gorin. ##### Saturday, June 8 • 9:00 - 9:35 Matteo Mucciconi (Tokyo Tech) - Yang-Baxter Random Fields and Stochastic Vertex Models Starting from the notion of bijectivization of the Yang-Baxter equation [BP] we construct random fields of Young diagrams whose measure is described by spin Hall-Littlewood functions (sHL) and spin q-Whittaker functions (sqW). These are two families of special symmetric functions recently introduced in [B], [BW] that generalize Hall-Littlewood and q-Whittaker functions. The bijectivization formalism uncovers a Schur processes like structure for a number of stochastic integrable vertex models that are obtained as marginals of the fields of Young diagram. Among these we have the six vertex model, the higher spin vertex model or a rather complicated push-type system that generalizes the q-Hahn pushTASEP [CMP]. We also discover q-difference operators acting diagonally on the sHL and sqW functions and we use them to write formulas for observables of the vertex models. The talk is based on collaboration with A. Bufetov and L. Petrov. References: *[B] *A. Borodin, "On a family of symmetric rational functions" *[BP] *A. Bufetov and L. Petrov, "Yang-Baxter field for spin Hall-Littlewood symmetric functions" *[BW] *A. Borodin and M. Wheeler, "Spin q-Whittaker polynomials" *[CMP] *I. Corwin, K. Matveev and L. Petrov, "The q-Hahn pushTASEP" • 9:40 - 10:15 Mikhail Basok (Saint Petersburg State University) - Tau-functions a la Dubedat and cylindrical events in the double-dimer model Double-dimer model on a given graph is a random loop ensemble that is obtained by sampling two independent dimer configurations taken uniformly at random and removing double edges. Given a simplpy-conected domain and a sequence of "discrete" domains, drawn on a square grid, that approximate this domain (we assume that the step of the grid tends to zero) one can consider the corresponding sequence of random loop ensembles induced by the double dimer model in each discrete domain (seen as a subgraph of the square grid). It was predicted by R. Kenyon that this sequence of random loop ensembles converges to Conformal Loop Ensemble with parameter 4 (CLE(4)) sampled in the original domain. Recently his conjecture was deeply supported by a breakthrough work of J. Dubedat: in this work a large family of observables called topological correlators is introduced and it is shown that given sequence of a Temperley discretizations of a simply connected domain topological correlators converge to the corresponding observables for CLE(4). As a biproduct Dubedat showed that topological correlators for CLE(4) coinside with Jimbo-Miwa isomonodromic tau functions; this correspondens seems to be interesting in its own side. It turns out that these results of Dubedat acturally characterize the limit of double-dimer loop ensembles, i.e. the following corollary holds: if the sequence of measures induced by double-dimer ensembles in discrete domains is tight then it converges to CLE(4). We will discuss these results of J. Dubedat and the machinary developed to extract this corollary. Based on a joint work with Dmitry Chelkak (Paris). • 10:15 - 10:50 Break • 10:50 - 11:25 Mark Rychnovsky (Columbia) - Tracy-Widom Fluctuations for a river delta We consider an exactly solvable directed first passage percolation model for a river delta. We prove that asymptotically the width of the river delta of length L is order$L^{2/3}$with Tracy-Widom fluctuations of order$L^{4/9}$. We can also reformulate this result as a Tracy-Widom limit theorem for an interacting particle system sitting above pushTASEP. • 11:30 - 12:05 Guilherme Silva (Michigan) - Properties of the Limiting Distribution for the Periodic TASEP It is now a classical result that the one-point fluctuations of the height function in the TASEP (with step initial data) converge, in a suitable scaling, to the Tracy-Widom distribution$F_2$. In addition to the remarkable universality feature of$F_2\$, appearing in dozens of different models that are seemingly unrelated, this distribution also enjoys several nice different characterizations, for instance it can be given in terms of a somewhat simple Fredholm determinant, or in terms of a solution to the Painlev\'e II equation, or yet expressed via a Riemann-Hilbert problem.

Recently, Baik and Liu found an expression for the limiting distribution for the periodic TASEP. Their formula, somewhat complicated, gives this limiting distribution in terms of a Fredholm determinant. In this talk, after reviewing the just mentioned facts, we plan to explain how to obtain other expressions for Baik-Zhipeng's distribution in terms of a Riemann-Hilbert problem, or yet in terms of a nonlocal PDE, along very much the same spirit of the aforementioned properties of the Tracy-Widom distribution.

This is based on work in progress with Jinho Baik (University of Michigan) and Zhipeng Liu (University of Kansas)

Organizers: Leo Petrov, Axel Saenz

Scientific Committee: Jinho Baik, Alexei Borodin, Ivan Corwin, Vadim Gorin, Leo Petrov