Integrable Probability Working Group

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

Join our mailing list for the regular announcements.

Talks in Spring 2024 (most recent at the top):

• THURSDAY, May 9. 3:00pm in 2-132
  
  Jimmy He (MIT): Symmetries of periodic measures on partitions
  
  Abstract: The periodic $q$-Whittaker measure is a probability measure on partitions defined in terms of $q$-Whittaker functions and an additional parameter $u$ known as the winding fugacity. I will explain a hidden distributional symmetry of this model which exchanges the $u$ and $q$ parameters, as well as related results on the periodic Hall-Littlewood measure. As a special case, we recover an identity of Imamura--Mucciconi--Sasamoto. This is joint work with Michael Wheeler.
  
  

• THURSDAY, May 2. 3:00pm in 2-132
  
  Chenyang Zhong (Columbia University): Large deviation principle for the Airy point process
  
  Abstract: The Airy point process is a universal determinantal point process and a central object in random matrix theory. It arises from eigenvalues near the soft edge of large random matrix ensembles and the largest parts of random partitions picked from the Plancherel measure. In this talk, I will present a large deviation principle for the Airy point process. Our large deviations result resolves a conjecture of Corwin and Ghosal, and is connected to lower tail large deviations of the KPZ equation. Our result also extends to point processes arising from the stochastic Airy operator.
  
  

• THURSDAY, April 11. 3:00pm in 2-132
  
  Nikolai Bobenko (University of Geneva): Dimers and M-Curves: Limit Shapes from Riemann Surfaces
  
  Abstract: We present a general scheme for the study of dimer models via integrable systems techniques. This results in dimer models with quasi periodic weights. Putting an M-curve at the center of the construction allows one to define weights and algebraic objects describing the behavior of the corresponding dimer model. We obtain explicit formulas for the limit shapes of these models for certain boundary conditions. Furthermore everything in this approach can be computed numerically and the results match simulation. This talk is based on joint work with A. I. Bobenko and Y. B. Suris.
  
  

• TUESDAY, April 9. 3:30pm in 2-255
  
  Panagiotis Zografos (Leipzig University): Infinitesimal free probability via Fourier transform
  
  Abstract: Infinitesimal free probability provides a framework for studying random matrices not only in the large N limit but in the 1/N correction as well. The 1/N correction provides information for outliers for finite-rank perturbations of random matrices. In this talk we will begin by discussing how the method of characteristic functions of random matrices is connected to this framework. We will also discuss how problems of asymptotic representation theory fit into this framework, using again the characteristic functions approach. For this direction we will introduce some probability measures on R which depend on a parameter -1 \leq q \leq 1. These measures encode signatures and are related to classical symmetric polynomials. They will play the role of the empirical spectral distribution. We will show how the infinitesimal limits of these measures lead to quantized analogues of infinitesimal free probability. Applications include outliers of random tilings.
  
  

• THURSDAY, April 4. 3:00pm in 2-132
  
  Shankar Balasubramanian (MIT): The word problem for groups and slow dynamics
  
  Abstract: A major area of study in the dynamics of classical and quantum systems seeks to understand how these systems reach equilibrium. One particularly interesting class of systems are kinetically constrained models, which appear in physically realistic settings. We construct a new class of kinetically constrained (classical or quantum) systems based on finitely generated groups. The relaxation times of these systems can be related to the geometric and complexity theoretic properties of the underlying group. In one class of groups, fluctuations of a conserved charge density takes an exponentially long time to relax. In a second class of groups, relaxation only occurs if the system is connected to a large bath whose size scales exponentially (or more rapidly) in the system size. When the constraint imposed by the group is violated in a weak sense, relaxation to the new equilibrium distribution can still take an exponentially long time, an observation which we relate to a conjecture in the theory of expander graphs.
  
  

• THURSDAY, March 14. 3:00pm in 2-132
  
  Cesar Cuenca (Ohio State): The Symplectic Schur Process
  
  Abstract: We introduce a new symmetric function that plays the role of skew Schur function for symplectic groups. Many combinatorial identities for this function are developed and employed to construct the Symplectic Schur Process. This new probability ensemble shares many desirable properties with the classical Schur process of Okounkov-Reshetikhin and we will explain some of them in this talk, including the property of being a determinantal point process, the connection to the Berele insertion algorithm and, finally, we discuss some applications. This talk is based on joint work with Matteo Mucciconi.
  
  

• THURSDAY, March 7. 3:00pm in 2-132
  
  Jiaming Xu (University of Wisconsin): Edge universality of Beta-additions through Dunkl operators
  
  Abstract: It is well known that the edge limit of Gaussian/Laguerre Beta ensembles is given by Airy(\beta) point process. We prove an universality result that this also holds for a general class of additions of Gaussian and Laguerre ensembles. In order to make sense of Beta-addition, we introduce type A Bessel function as the characteristic function of our matrix ensemble, then extract its moment information through the action of Dunkl operators, a class of differential operators originated from special function theory. Joint work in preparation with David Keating.
  
  

• THURSDAY, February 29. 3:00pm in 2-132
  
  Yuhan Jiang (Harvard): The doubly asymmetric simple exclusion process
  
  Abstract: The multispecies ASEP (mASEP) is a Markov chain in which particles of different species hop on a one-dimensional lattice. The doubly ASEP (DASEP) is like the mASEP, but it additionally allows spontaneous change of species. We will introduce two new Markov chains that the DASEP lumps to, which give relations between sums of steady state probabilities. We also give explicit formulas for the stationary distribution of a particular infinite family.
  
  

• THURSDAY, February 15. 3:00pm in 2-132
  
  Vidya Venkateswaran: Quasi-polynomial generalizations of Macdonald polynomials
  
  Abstract: In joint work with Siddhartha Sahi and Jasper Stokman, we constructed (quasi-)polynomial generalizations of nonsymmetric Macdonald polynomials for arbitrary reduced root systems. We will introduce these objects and discuss how they arise from a family of cyclic $Y$-parabolically induced DAHA modules where the $T_i$ operators act via ``truncated'' divided difference operators. We will also connect a limiting case of these polynomials to metaplectic Whittaker functions, and discuss recent duality results between (anti-)symmetric quasi-polynomials and partially (anti-)symmetric polynomials.
  
  

Talks in Fall 2023 (most recent at the top):

• THURSDAY, November 30. 3:00pm in 2-132
  
  Jimmy He (MIT): Random growth models with half space geometry
  
  Abstract: Random growth models in 1+1 dimension capture the behavior of interfaces evolving in the presence of noise. These models are expected to exhibit universal behavior, but we are still far from proving such results even in relatively simple models. A key development which has led to recent progress is the discovery of exact formulas for certain models with a rich algebraic structure, leading to asymptotic results. I will survey some of these results, with a focus on models where a single boundary wall is present, as well as applications to rates of convergence for a Markov chain.
  
  

• THURSDAY, November 16. 3:00pm in 2-132
  
  Sveta Gavrilova (MIT): Grothendieck measures on partitions
  
  Abstract: We study probability measures on partitions based on symmetric Grothendieck polynomials, which are certain deformations of Schur polynomials. Despite the similarity of determinantal formulas for the probability weights of Schur and Grothendieck measures, we demonstrate that Grothendieck measures are not determinantal point processes. This question is related to the principal minor assignment problem in algebraic geometry. We identify Grothendieck random partitions with a cross-section of a Schur process, which allows us to derive limit shape result. The limit shape curve is not particularly explicit as it arises as a cross-section of the limit shape surface for the Schur process. Also, by extending our approach, we introduce a new framework of tilted biorthogonal ensembles.
  
  

• THURSDAY, October 26. 3:00pm in 2-132
  
  David Keating (University of Wisconsin): Coupled tilings, double dimers, and LLT polynomials
  
  Abstract: In this talk will begin by constructing coupled tilings of the Aztec diamond. These can be viewed as a pair of a tilings, superimposed one on top of the other, with a certain local interaction between the tilings. We will show how to use the underlying integrable structure to compute the generating function of these tilings. Then we will discuss some ways to combinatorially understand the multi-colored nature of the model. We will then introduce the closely related LLT symmetric polynomials, defining them as colored, interacting copies of a five vertex model. Again, our goal will be to give some combinatorial understanding of interplay between the colors.
  
  

• THURSDAY, October 19. 2:30pm in 2-132
  
  Leonid Petrov (University of Virginia): Stationarity for Colored Interacting Particle Systems on the Ring
  
  Abstract: Recently, there has been much progress in understanding stationary measures for colored (also called multi-species or multi-type) interacting particle systems motivated by asymptotic phenomena and rich underlying algebraic and combinatorial structures (such as nonsymmetric Macdonald polynomials). I will describe a unified approach to constructing stationary measures for colored ASEP, q-Boson, and q-PushTASEP systems based on integrable stochastic vertex models and the Yang-Baxter equation. Stationary measures become partition functions of new "queue vertex models" on the cylinder, and stationarity is a direct consequence of the Yang-Baxter equation. Our construction recovers and generalizes known stationary measures constructed using multiline queues and the Matrix Product Ansatz. In the quadrant, Yang-Baxter implies a colored version of Burke's theorem, which produces stationary measures for particle systems on the line. We also compute the colored particle currents in stationarity. Joint work with Amol Aggarwal and Matthew Nicoletti.
  
  

• THURSDAY, September 28. 1:00pm in 2-429
  
  Patrik Ferrari (Bonn): TASEP with a moving wall
  
  Abstract: We study TASEP on Z with the step initial condition, under the additional restriction that the first particle cannot cross a deterministically moving wall. We prove that such a wall may induce asymptotic fluctuation distributions of particle positions equal to the probability that the Airy2 process is below a barrier function g. This is the same class of distributions that arises as one-point asymptotic fluctuations of TASEPs with arbitrary initial conditions.
  
  

• THURSDAY, September 21. 2:30pm in 2-361
  
  Catherine Wolfram (MIT): Large deviations for the 3D dimer model
  
  Abstract: In 2000, Cohn, Kenyon, and Propp showed that 2D dimer tilings satisfy a large deviations principle. In joint work with Nishant Chandgotia and Scott Sheffield, we prove an analogous large deviations principle for dimers in 3D. A lot of the results for dimers in two dimensions use tools and exact formulas (e.g. the height function representation of a tiling or the Kasteleyn determinant formula) that are specific to dimension 2. I will explain how to formulate the large deviations principle in 3D, show simulations, try to give some intuition for why three dimensions is different from two, and explain some of the ways that we use a smaller set of tools (e.g. Hall’s matching theorem or a double dimer swapping operation) in our arguments.
  
  

• THURSDAY, September 14. 3:00pm in 2-132
  
  Lingfu Zhang (UC Berkeley): Random Lozenge Tiling at Cusps and the Pearcey Process
  
  Abstract: It has been known since Cohn-Kenyon-Propp (2000) that uniformly random tiling by lozenges exhibits frozen and disordered regions, which are separated by the 'arctic curve'. For a generic simply connected polygonal domain, the microscopic statistics are widely predicted to be universal, being one of (1) discrete sine process inside the disordered region (2) Airy line ensemble around a smooth point of the curve (3) Pearcey process around a cusp of the curve (4) GUE corner process around a tangent point of the curve. These statistics were proved years ago for special domains, using exact formulas; as for universality, much progress was made more recently. In this talk, I will present proof of the universality of (3), the remaining open case. Our approach is via a refined comparison between tiling and non-intersecting random walks, for which a new universality result of the Pearcey process is also proved. This is joint work with Jiaoyang Huang and Fan Yang.
  
  

Talks in Spring 2023:

• THURSDAY, March 16. 2:30pm in 2-147
  
  Jimmy He (MIT): Boundary current fluctuations in the half space ASEP
  
  Abstract: The half space asymmetric simple exclusion process (ASEP) is an interacting particle system on the half line, with particles allowed to enter/exit at the boundary. I will discuss recent work on understanding fluctuations for the number of particles in the half space ASEP started with no particles, which exhibits the Baik-Rains phase transition between GSE, GOE, and Gaussian fluctuations as the boundary rates vary. As part of the proof, we find new distributional identities relating this system to two other models, the half space Hall-Littlewood process, and the free boundary Schur process, which allows exact formulas to be computed.
  
  
• THURSDAY, April 20. 2:30pm in 2-147
  
  Matthew Lerner-Brecher (MIT): Edge Limits of the Laguerre Corners Process at Zero Temperature
  
  Abstract: In this talk, we will discuss the hard edge limit of a multilevel extension of the Laguerre $\beta$-ensemble at $\beta=\infty$. In particular, we will see that this limit can be captured by a discrete time Gaussian process defineable in two different ways: (1) in terms of its covariance function and (2) in terms of the partition functions of certain additive polymers arising from Bessel functions. Our work here builds off Gorin-Kleptsyn (2020), which studied the soft edge limit of the corresponding Gaussian case, and we will spend some time expounding on interesting connections that arose between our results and theirs.
  
  
• THURSDAY, April 27. 2:30pm in 2-147
  
  Ron Nissim (MIT): Asymptotics of the One-point Distribution for the KPZ Fixed Point Conditioned on a Large Height at an Earlier Point
  
  Abstract: After a review of the KPZ universality class and the KPZ fixed point, we will discuss formulas for the multi-time distribution of the KPZ fixed point with step and flat initial conditions obtained by Zhipeng Liu (2022). Via these formulas, I will address the asymptotics for one point conditioned on a large height at an earlier point. This is based on joint work with Ruixuan Zhang. Lastly, we will compare this to another asymptotic regime studied by Zhipeng Liu and Yizao Wang (2022).
  
  
• THURSDAY, May 4. 2:30pm in 2-147
  
  Matthew Nicoletti (MIT): Six vertex model and random matrices
  
  Abstract: In this expository talk, we will introduce the six vertex model, a well studied quantum integrable lattice model of statistical mechanics, and on the other hand we will also introduce several universal distributions originating from the study of eigenvalues of various random matrix ensembles. In one special case, via the Izergin-Korepin determinant formula, we provide a self contained proof of the appearance of one such universal eigenvalue distribution in a random six vertex configuration in the scaling limit.
  
  
• THURSDAY, May 11. 2:30pm in 2-147
  
  Gopal Goel (MIT): A QUANTIZED ANALOGUE OF THE MARKOV-KREIN CORRESPONDENCE
  
  Abstract: We study a family of measures originating from the signatures of the irreducible components of representations of the unitary group, as the size of the group goes to infinity. Given a random signature $\lambda$ of length $N$ with counting measure $\mathbf{m}$, we obtain a random signature $\mu$ of length $N-1$ through projection onto a unitary group of lower dimension. The signature $\mu$ interlaces with the signature $\lambda$, and we record the data of $\mu,\lambda$ in a random rectangular Young diagram $w$. We show that under a certain set of conditions on $\lambda$, both $\mathbf{m}$ and $w$ converge as $N\to\infty$. We provide an explicit moment generating function relationship between the limiting objects. We further show that the moment generating function relationship induces a bijection between bounded measures and certain continual Young diagrams, which can be viewed as a quantized analogue of the Markov-Krein correspondence.
  
  
















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
  
  In the 1992 paper “Schur functions: theme and variations” Macdonald listed nine variations of Schur functions. The 6th variation is the factorial Schur functions. The 9th variation unifies variations 4–8. Recently Foley and King defined factorial versions of the irreducible characters of the symplectic and orthogonal groups. They proved combinatorial formulas and certain flagged versions of the Jacobi–Trudi identity for their factorial characters. We will discuss how to prove the unflagged Jacobi–Trudi identities. This allows us to define symplectic and orthogonal versions of Macdonald’s 9th variation. Along the proof we will introduce another generalization of the symplectic and orthogonal characters, which provides the most general framework for the symplectic and orthogonal versions of the Cauchy identity to hold. We will also carry over this discussion to the case of non-polynomial representations of GL(n).
  
  
• THURSDAY, October 27. 3:00pm in 4-153
  
  Patrick Lopatto (Brown): The Mobility Edge of Lévy Matrices
  
  Abstract: Lévy matrices are symmetric random matrices whose entry distributions lie in the domain of attraction of an alpha-stable law; such distributions have infinite variance when alpha is less than 2. Due to the ubiquity of heavy-tailed randomness, these models have been broadly applied in physics, finance, and statistics. When the entries have infinite mean, Lévy matrices are predicted to exhibit a phase transition separating a region of delocalized eigenvectors from one with localized eigenvectors. We will discuss the physical context for this conjecture, and describe a result establishing it for values of alpha close to one. This is joint work with Amol Aggarwal and Charles Bordenave.
  
  
• 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): Orthogonal polynomial dualities of multi-species ASEP and related processes via the $^*$--bialgebra structure of quantum groups
  
  Abstract: In this talk, I present a general method to produce orthogonal polynomial dualities from the $^*$--bialgebra structure of Drinfeld--Jimbo quantum groups. In the case of the quantum group $\mathcal{U}_q(\mathfrak{gl}_{n+1})$, the result is a nested multivariate $q$--Krawtchouk duality for the $n$--species ASEP$(q,\boldsymbol{\theta})$. The method also applies to other quantized simple Lie algebras and to stochastic vertex models. As a probabilistic application of the duality relation found, we provide an explicit contour integral formula of the $q-$shifted factorial moments for the two--species $q$--TAZRP.
  
  
• THURSDAY, December 1. 3:00pm in 4-153
  
  Roger Van Peski (MIT): Random groups, random matrices, and universality
  
  Abstract: In this talk I will discuss some recent joint work (https://arxiv.org/abs/2209.14957v1) with Hoi Nguyen, which concerns a universal distribution on sequences of abelian p-groups arising from products of random integer matrices. I will explain the context for the result from number theory, random graphs, and complex random matrix products, and also give a general introduction to the ideas behind the `moment method for random groups' used to prove this and many similar results in the literature.
  
  
• THURSDAY, December 8. 3:00pm in 4-153 (last talk of the semester)
  
  Mackenzie Simper (Broad Institute): Double coset Markov chains
  
  Abstract: Let G be a finite group. Any two subgroups H, K define equivalence classes of G called the `double cosets'. The sizes of double cosets can vary greatly (unlike single cosets), and the set of double cosets can be in bijection with interesting combinatorial objects. A Markov chain on the group induces a random process on the double coset space — I will present a sufficient condition for the induced random process to also be a Markov chain, and then discuss several examples. In particular, if G is the group of invertible matrices over a finite field, and H = K the subgroup of upper-triangular matrices, then the double cosets are indexed by permutations. The uniform distribution on G induces the Mallows measure on permutations, and multiplication by random transvections defines an interesting Markov chain on permutations which can be described using the Hecke algebra of double cosets. Based on joint work with Persi Diaconis and Arun Ram.
  
  













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*
  *but were afraid to ask
  
  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
  
  
  
• MONDAY, November 21. 3.00pm in 2-361
  
  Jon Novak (UCSD): Semiclassical Asymptotics of GL_N(ℂ) Tensor Products via Quantum Random Matrices
  
  
  
• 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.

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