##### Integrable Probability Working Group

This is an informal seminar run by Alexei Borodin, Jimmy He, and Roger Van Peski and devoted to recent developments related to the integrable probability in the wide sense.

Talks in Fall 2022:

• THURSDAY, September 15. 3:00pm in 4-153 (note: not building 2!)

Guilherme Silva (ICMC-USP): Differential equations for KPZ fixed points.

Abstract: We discuss how multipoint distributions of the KPZ fixed point with narrow wedge initial condition relate to matrix versions of NLS and mKdV systems, and also with matrix KP equation and KP hierarchy. Our approach is via integrable operators, and covers multipoint distributions for the KPZ fixed point, its periodic counterpart, and unravels common features of both of them in an unified way. As we will discuss, our results also contain previously found connections by Tracy and Widom, Adler and van Moerbeke, and others, in the context of the Airy2 process.

• THURSDAY, September 22. 3:00pm in 4-153 (note: not building 2!)

Matteo Mucciconi (Warwick): A bijective approach to solvable KPZ models.

I will provide a detailed proof of the bijection at the core of our new approach to study KPZ models. A fundamental ingredient here is the notion of Kashiwara's crystals and their relation to the RSK correspondence, but also to discrete integrable systems such as the box-ball system.

• THURSDAY, September 29. 3:00pm in 4-153

Cesar Cuenca (Harvard): Random matrices and random partitions at varying temperatures

Abstract: We discuss the global-scale behavior of random matrix eigenvalues and random partitions which depend on the "inverse temperature" parameter beta. The goal is to convince the audience of the effectiveness of the moment method via Fourier-like transforms in characterizing the Law of Large Numbers and Central Limit Theorems. We focus on the regimes of high and low temperatures, i.e., when the parameter beta converges to zero and to infinity, respectively. This talk is based on joint works with Florent Benaych-Georges -- Vadim Gorin, and Maciej Dolega -- Alex Moll.

• THURSDAY, October 20. 3:00pm in 4-153

Misha Goltsblat (Yale): Variations of Schur functions and classical group characters

• THURSDAY, November 3. 3:00pm in 4-153

Ryan Mickler (Singulariti): New results on the structure of Jack Littlewood-Richardson coefficients.

Abstract: In 2013, Nazarov-Skylanin introduced a quantization of the Lax operator $L$ of the Benjamin-Ono integrable hierarchy on the torus, a 1+1 dimensional classical integrable system of non-linear dispersive waves. They showed that a particular matrix element of the powers of $L$ provide an infinite family of commuting quantum Hamiltonians which are diagonalised on Jack symmetric functions. We prove a spectral theorem for $L$, showing that its eigenvectors are polynomials which refine Jack symmetric functions. As a consequence, the products of these eigenfunctions refine the algebra of Jack functions. We study these refined products through the introduction of several of trace-like functionals, which reveal a surprising simplicity to the structure of the algebra. Using this technique, we find explicit closed-form relations between Jack Littlewood-Richardson coefficients and prove several new cases of a conjecture of Stanley on the structure of these coefficients. [ In part joint work with Alexander Moll and Per Alexandersson (papers to appear in October) ]

• THURSDAY, November 17. 3:00pm in 4-153

Zhengye Zhou (Texas A&M): TBA

Abstract: TBA

Talks in Spring 2022:

• THURSDAY, Feb 24. 4:00pm in 4-261 (note: not building 2!)

Tomas Berggren (MIT): Domino tilings of the Aztec diamond with doubly periodic weightings

Abstract: In this talk we will discuss random domino tilings of the Aztec diamond with doubly periodic weights. More precisely, asymptotic results of the $2 \times k$-periodic Aztec diamond will be discussed, both in the macroscopic and microscopic scale. The starting point of the asymptotic analysis is a double integral formula for the associated correlation kernel, which is expressed in terms of a matrix Wiener--Hopf factorization. We will also see a close connection to an associated Riemann surface, with which we will describe the global picture.

• THURSDAY, March 3. 4:00pm in 4-261

Alexei Borodin (MIT): sl(1|1)-vertex models: boson-fermion correspondence and determinantal point processes

• THURSDAY, March 10. 4:00pm in 4-261

Sergei Korotkikh (MIT): Spin q-Whittaker functions and vertex models

Abstract: I will tell about a new family of symmetric functions I've been working on, called inhomogeneous spin q-Whittaker functions. They originate from a certain vertex model equivalent to q-Hahn TASEP and they generalize q-Whittaker symmetric functions (t=0 degeneration of Macdonald functions). I will describe how these functions originated from integrable probability and will explain connections to some other families of symmetric functions, namely, spin Hall-Littlewood functions and interpolation symmetric functions. Time permitting, I will also cover the connection of the model with the representation theory of quantum affine sl_2.

• THURSDAY, March 17. 4:00pm in 4-261

Mustafa Alper Gunes (Oxford): Moments of Characteristic Polynomials of Random Matrices, Painleve Equations and L-functions

Abstract: In this talk, we will consider various different types of joint moments of characteristic polynomials of random matrices that are sampled according to the Haar measure on classical compact groups. In each case, we will see how one can obtain the asymptotics of these quantities as the matrix size tends to infinity, and see the implications that these asymptotics have regarding moments of L-functions. Finally, we will see that these asymptotics have representations in terms of solutions of certain Painleve equations, giving us conjectures relating L-functions and solutions of these Painleve systems. Based on a joint work with Assiotis, Bedert and Soor, and some original results obtained as a part of my undergraduate thesis.

• THURSDAY, March 31. 4:00pm in 4-261

Promit Ghosal (MIT): Universality of multiplicative statistics of Hermitian random matrix ensembles

Abstract: Multiplicative statistics of random matrix ensembles and their scaling limits are important ingredients for many findings throughout the last few decades. In this talk, we focus on large matrix limit of multiplicative statistics of eigenvalues of unitarily invariant Hermitian random matrices. We show that for one-cut regular potential and a large class of multiplicative functional, the associated statistics converge to an universal limit which is described in terms of integro-differential Painleve II equation. This talk will be based on a joint work with Guilherme Silva.

• THURSDAY, April 7. 3:00pm in 4-261 (NOTE UNUSUAL TIME)

Guillaume Barraquand (ENS): Positive random walks in random environment

Abstract: We will consider random walks in random environment and discuss how their asymptotic behavior differs from that of simple random walks. I will first survey what is known about an exactly solvable model on Z, the Beta RWRE. Then I will show that there exists a variant of the model, where the walks are restricted to positive integers, which remains exactly solvable. The large scale behavior of this model present some similarities with the model on Z, but also some differences due to the presence of a boundary. This is a joint work with Mark Rychnovsky.

• THURSDAY, April 14. 4:00pm in 4-261

Jimmy He (MIT): Shift invariance of some half space integrable models

Abstract: I'll discuss work in progress on shift invariance in a half space setting. The starting point is the colored stochastic six vertex model in a half space, from which we obtain results on the asymmetric simple exclusion process, as well as for the beta polymer through a fusion procedure. An application to the asymptotics of a type B analogue of the oriented swap process is also given.

• THURSDAY, April 21. 4:00pm in 4-261

Marianna Russkikh (MIT): Everything you always wanted to know about t-embeddings*

Abstract: We introduce a concept of ‘t-embeddings’ of weighted bipartite planar graphs. We believe that these t-embeddings always exist and that they are good candidates to recover the complex structure of big bipartite planar graphs carrying a dimer model. We also develop a relevant theory of discrete holomorphic functions on t-embeddings; this theory unifies Kenyon’s holomorphic functions on T-graphs and s-holomorphic functions coming from the Ising model.

• THURSDAY, April 28. 4:00pm in 4-261

Terrence George (University of Michigan): Shufflings in the dimer model

Abstract: Domino-shuffling is a technique introduced by Elkies, Kuperberg, Larsen and Propp to enumerate and generate domino tilings of the Aztec diamond graph, and was used to give the first proof of the arctic circle theorem. Domino tilings are dual to the dimer model on the square lattice. There are generalizations of domino-shuffling for any biperiodic dimer model, and they form a group called the cluster modular group. This group was studied by Fock and Marshakov, who gave an explicit conjecture for its isomorphism type. We will discuss these generalized shufflings and describe the cluster modular group for any biperiodic dimer model. (joint work with Giovanni Inchiostro).

• THURSDAY, May 5. 4:00pm in 4-261

Alexander Moll (UMass Boston): Exact Results for the Quantum Benjamin-Ono Equation on the Torus

Abstract: In this talk, we use a fractional Gaussian field on the 1-dimensional real torus to quantize Benjamin-Ono waves and give a dynamical interpretation of a result of Stanley (1989). Precisely, the classical Benjamin-Ono equation on the torus is a non-linear and non-local model of dispersive waves. Starting from the standard Gaussian in the symplectic space of this equation, we construct the quantum Benjamin-Ono equation on the torus without any path integrals. Remarkably, a result of Stanley (1989) for Jack polynomials implies an exact description of the spectrum of the quantum Hamiltonian in this model. We prove that if one considers Bohr-Sommerfeld quantization of the classical Benjamin-Ono multi-phase solutions on the torus found by Satsuma-Ishimori (1979), then the resulting approximation of the quantum spectrum found by Stanley is exact after an explicit renormalization of the coefficient of dispersion predicted by Abanov-Wiegmann (2005).

• THURSDAY, May 12. 4:00pm in 4-261

Mirjana Vuletić (UMass Boston/MIT): Free boundary Schur process

Abstract: In this talk I will present our work on the free boundary Schur process, which is a generalization of the original Schur process of Okounkov and Reshetikhin. This model after some shift-mixing is a Pfaffian process. I will introduce combinatorial random corner growth/LPP models corresponding to the free boundary Schur process and present some asymptotic results. The limiting behavior for these models is given by some new deformations of universal distributions of Schur processes. This is a joint work with D. Betea, J. Bouttier, and P. Nejjar.

Talks in Spring 2019:

• TUESDAY, April 2. 3.00pm in 2-255

Ryosuke Sato (Nagoya University): Asymptotic representation theory of inductive systems of compact quantum groups

• WEDNESDAY, April 10. 3.00pm in 2-255

Claus Koestler (University College Cork)Characters of the infinite symmetric group from the viewpoint of distributional symmetries

• TUESDAY, May 14. 3.00pm in 2-255

Elia Bisi (Dublin): Transition between characters of classical groups, decomposition of Gelfand-Tsetlin patterns, and last passage percolation

Talks in Fall 2018:

• TUESDAY, Sep 18. 3.00pm in 2-255

Cesar Cuenca (MIT): A partial q-scheme of symmetric functions

• TUESDAY, Oct 2. 3.00pm in 2-255.

Zhongyang Li (UConn): Limit shape of perfect matchings on square-hexagon lattices

• THURSDAY, Oct 25. 3.00pm in 2-136

Benjamin Brubaker (Minnesota): Whittaker functions from solvable lattice models and vertex operators

• THURSDAY, Nov 1. 3.00pm in 2-136

Pavel Galashin (MIT): Ising Model and Total Positivity

•  Nov 13 - Nov 15, Tuesday-Thursday: FRG Integrable probability meeting.

Wednesday lunch sign-up form (deadline Nov 9)

Tuesday, Nov 13, 3.00pm in 2-255

Jiaoyang Huang (Harvard University): The law of large numbers and central limit theorem via Jack generating functions.

Wednesday, Nov 14, 11.00am in 2-361

Alisa Knizel (Columbia University): Gap probability function for discrete log-gases

Thursday, Nov 15, 2.00pm in 2-361

Promit Ghosal (Columbia University):  Tails of the KPZ equation

• TUESDAY, Nov 27. 3.30pm in 2-255.

Maurice Duits (KTH): Doubly periodic tilings and matrix-valued orthogonal polynomials.

Talks in Spring 2018:

• TUESDAY, March 6. 3.00pm in 2-255.

Peter Nejjar(Vienna): Symmetric Last Passage Percolation and Schur Processes

• TUESDAY, April 3. 3.00pm in 2-255.

Alejandro Morales (Amherst):  Hook formulas for skew shapes: border strips and product formulas

• THURSDAY, April 26. 1.00pm in 2-139.

Cesar Cuenca (MIT): Quantized Vershik-Kerov Theory and Quantized Central Measures on Branching Graphs

• THURSDAY, April 26. 3.00pm in 2-136. CANCELLED

• TUESDAY, May 1. 1.00pm in 2-139.

Konstantin Matveev (Brandeis): Boundary of the Young graph with Macdonald multiplicities and Kerov’s conjecture

• THURSDAY, May 3. 1.00pm in 2-139.

Konstantin Matveev (Brandeis):
Boundary of the Young graph with Macdonald multiplicities and Kerov’s conjecture

• TUESDAY, May 8. 1.00pm in 2-139.

Leonid Petrov (Virginia): Cauchy identities, Yang-Baxter equation, and their randomization

• THURSDAY, May 10. 1.00pm in 2-139.

Nicolai Reshetikhin (Berkeley): The 6-vertex model in statistical mechanics.

• May 14 - May 18: Workshop IntProb-2018, Boston

Talks in Fall 2017:

• TUESDAY, September 19. 3.00pm in 2-255.

Alexander Moll (Bonn): Integrability and Fractional Gaussian Fields in Geometric Quantization

• THURSDAY, September 21. 3.00pm in 2-142.

Marianna Russkikh (Geneva):  Playing dominos in different domains

• THURSDAY, September 28. 3.00pm in 2-142.

Herbert Spohn (Munich): KPZ growing interfaces: How flat is flat?

• TUESDAY, October 3. 3.00pm in 2-255.

Jiaoyang Huang (Harvard): Discrete beta ensembles: dynamics and universality.

• TUESDAY, Oct 17. 3.00pm in 2-255.

Axel Saenz (University of Virginia): Transition Probabilities for ASEP on the Ring

• TUESDAY, October 24. 3.00pm in 2-255.

Patrik Ferrari (Bonn): Limit law of a second class particle in TASEP with non-random initial condition

• THURSDAY, Nov 2. 3.00pm in 2-142

Cesar Cuenca (MIT): Interpolation Macdonald operators at infinity

• THURSDAY, Nov 9. 11.30am in 2-361

Jeremy Quastel (Toronto):  From TASEP to KPZ fixed point

IMPORTANT: This is a lunch seminar. You can sign up here before Nov. 1 to reserve a sandwich. Otherwise, please, bring your own food

• TUESDAY, Nov 14. 3.00pm in 2-255

Andrew Ahn (MIT): Asymptotics of Periodically Weighted Random Plane Partitions

• TUESDAY, Dec 5. 3.00pm in 2-255

Drazen Petrovic (IUPUI): Pfaffian Sign Theorem for the Dimer Model on a Triangular Lattice

Talks in Fall 2016:

• THURSDAY, October 13. 3.00pm in 2-146

Mark Adler (Brandeis University): Tilings of non-convex polygons and limiting processes.

• TUESDAY, November 1. 3.00pm in 2-146

Boris Hanin (MIT): Jack polynomials and 2D beta ensembles

• THURSDAY, November 10. 3.00pm in 2-146

Yuchen Pei (Harvard): A qRSK and its q-polymer

• TUESDAY, November 15. 3.00pm in 2-146

Cesar Cuenca (MIT): Asymptotics of Macdonald and Jack polynomials

• TUESDAY, November 29. 3.00pm in 2-146

Lingfu Zhang (MIT): Interlacing adjacent levels of β-Jacobi corners processes

• TUESDAY, December 6. 3.00pm in 2-146

Asad Lodhia (MIT): Marchenko-Pastur Law for Kendall's Tau

Talks in Spring 2016:

• THURSDAY,  February 4. 3.00pm in 2-136.

Alexei Borodin (MIT):  Stochastic spin models, Lecture 1.

• THURSDAY,  February 11. 3.00pm in 2-136.

Alexei Borodin (MIT):  Stochastic spin models, Lecture 2.

• THURSDAY,  February 18.

NO SEMINAR

• THURSDAY,  February 25. 3.00pm in 2-136.

Yi Sun (MIT):  Laguerre and Jacobi analogues of the Warren process

• THURSDAY,  March 3. 3.00pm in 2-136.

Alisa Knizel (MIT): Asymptotics of random domino tilings of rectangular Aztec diamonds

• THURSDAY,  March 10.

NO SEMINAR

• THURSDAY,  March 17. 3.00pm in 2-136.

Alexei Borodin (MIT):  Stochastic spin models, Lecture 3.

• TUESDAY, March 22. 3.30pm in 2-361

Michael Wheeler (University of Melbourne): The interplay between quantum integrability and symmetric functions

• THURSDAY,  March 31. 3.00pm in 2-136.

Alexei Borodin (MIT):  Stochastic spin models, Lecture 4.

• THURSDAY,  April 7. 3.00pm in 2-136.

Vadim Gorin (MIT):  Discrete loop equations and their use in statistical mechanics.

• THURSDAY,  April 14. 3.00pm in 2-136.

Amol Aggarwal (Harvard):  On Spin 1/2 Models with (Half) Stationary Initial Data.

• THURSDAY,  April 21. 3.00pm in 2-136.

Alex Moll (MIT):  Limit Shapes: Correspondence Principle as Large Deviations Principle

• THURSDAY,  April 28. 3.00pm in 2-136.

Alisa Knizel (MIT):  Gaussian asymptotics of q-Racah ensemble

• THURSDAY,  May 5. 3.00pm in 2-136.

Alex Moll (MIT):  Limit Shapes: Correspondence Principle as Large Deviations Principle - Lecture 2

Talks in Fall 2015:

• TUESDAY, Sep 15. 3.00pm in E18-466B.

Jeffrey Kuan (Columbia)
Two time distribution in Brownian directed percolation

• THURSDAY, Sep 17. 3.00pm in E17-128.

Organizational meeting:   Please, bring with you the articles and topics which you are interested in and which you would like to be discussed during the semester.

• THURSDAY, Sep 24. 3.00pm in E17-128.

Asad Lodhia (MIT)Coulomb Gasses and Renormalized Energy

• THURSDAY, Oct 1.

NO SEMINAR
due to  Charles River Lectures on Probability and Related Topics   on Friday, Oct 2 at MSR

• THURSDAY, Oct 8. 3.00pm in E17-128

Vincent Genest (MIT)ASEP with open boundaries and Koornwinder polynomials

• THURSDAY, Oct 15. 3.00pm in E17-128

Konstantin Matveev (Harvard)Efficient quantum tomography

• TUESDAY, Nov 3, 3.00pm in E18-358

Michael La Croix (MIT):   The Singular Values of the GOE

• TUESDAY, Nov 10, 3.00pm in E18-358

Evgeni Dimitrov (MIT):   Asymptotic results for a Hall-Littlewood process on plane partitions.

• TUESDAY, Nov 24, 3.00pm in E18-358

Guillaume Barraquand (Columbia):     First particle in exclusion processes and the Pfaffian Schur process.