Oct 18, 2021
Monday
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08:30 AM - 09:00 AM
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Welcome Tea
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09:00 AM - 09:10 AM
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Introduction
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
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09:10 AM - 10:00 AM
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Strong Szego Theorem on a Jordan Curve
Kurt Johansson (Royal Institute of Technology (KTH))
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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- Abstract
I will discuss certain determinants with respect to a sufficiently regular Jordan curve in the complex plane that generalize Toeplitz determinants which are obtained when the curve is the circle. This also corresponds to studying a planar Coulomb gas on the curve at inverse temperature beta =2. Under suitable assumptions on the curve we prove a strong Szego type asymptotic formula as the size of the determinant grows. The resulting formula involves the Grunsky operator built from the Grunsky coefficients of the exterior mapping function for the curve. As a consequence of our formula we obtain the asymptotics of the partition function on the curve. Interestingly, this formula involves the Fredholm determinant of the absolute value squared of the Grunsky operator which equals, up to a multiplicative constant, the so called Loewner energy of the curve.
- Supplements
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10:20 AM - 11:10 AM
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Large Deviations for Generalized Gibbs Ensembles of the Classical Toda Chain
Alice Guionnet (École Normale Supérieure de Lyon)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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Large deviations principles for the distribution of the empirical measure of the equilibrium measure for the Generalized Gibbs ensembles of the classical Toda chain introduced by H. Spohn. We deduce its almost sure convergence and characterize its limit in terms of the limiting measure of Beta-ensembles. Our results apply to general smooth potentials. This is joint work with Ronan Memin.
Article discussed: https://arxiv.org/abs/2103.04858
- Supplements
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11:30 AM - 12:20 PM
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Exact Solution of TASEP and Generalizations
Daniel Remenik (Universidad de Chile)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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I will present a general result which allows to express the multipoint distribution of the particle locations in the totally asymmetric exclusion process (TASEP) and several related processes, for general initial conditions, in terms of the Fredholm determinant of certain kernels involving the hitting time of a random walk to a curve defined by the initial data. This scheme generalizes an earlier result for the particular case of continuous time TASEP, which has been used to prove convergence of TASEP to the KPZ fixed point. The result covers processes in continuous and discrete time, with push and block dynamics, and with sequential and parallel update, as well as some extensions to processes with memory length larger than 1. Based on joint work with Konstantin Matetski.
- Supplements
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12:20 PM - 01:30 PM
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Lunch
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01:30 PM - 02:20 PM
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Gibbsian Line Ensembles and Beta-Corners Processes
Evgeni Dimitrov (Columbia University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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Gibbs measures are ubiquitous in statistical mechanics and probability theory. In this talk I will discuss two types of classes of Gibbs measures – random line ensembles and triangular particle arrays, which have received considerable attention due, in part, to their occurrence in integrable probability.
Gibbsian line ensembles can be thought of as collections of finite or countably infinite independent random walkers whose distribution is reweighed by the sum of local interactions between the walkers. I will discuss some recent progress in the asymptotic study of Gibbsian line ensembles, summarizing some joint works with Barraquand, Corwin, Matetski, Wu and others.
Beta-corners processes are Gibbs measures on triangular arrays of interacting particles and can be thought of as analogues/extensions of multi-level spectral measures of random matrices. I will discuss some recent progress on establishing the global asymptotic behavior of beta-corners processes, summarizing some joint works with Das and Knizel.
- Supplements
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03:00 PM - 03:30 PM
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Afternoon Tea
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Oct 19, 2021
Tuesday
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09:10 AM - 10:00 AM
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On Finite-Rank Non-Hermitian Deformations of Random Matrix Ensembles
Yan Fyodorov (King's College London)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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I will present results, both old and new, on complex eigenvalues and left/right eigenvectors of finite-rank non-Hermitian deformations of classical ensembles (GUE/GOE) and - time permitting - Random Band Matrices.
- Supplements
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10:20 AM - 11:10 AM
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Interacting Diffusions on Positive Definite Matrices
Neil O'Connell (University College Dublin)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
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We consider systems of Brownian particles in the space of positive definite matrices, which evolve independently apart from some simple interactions. We give examples of such processes which have an integrable structure. These are related to K-Bessel functions of matrix argument and multivariate generalisations of these functions. The latter are eigenfunctions of a particular quantisation of the non-Abelian Toda lattice.
- Supplements
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11:30 AM - 12:20 PM
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Lecture Hall Tableaux, Non Intersecting Paths and Tilings
Sylvie Corteel (University of California, Berkeley)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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Lecture Hall tableaux were introduced by Corteel and Kim in 2018 to study the combinatorics of the multivariate Little Jacobi polynomials. They are in bijection with non intersecting paths on a lattice that is not translation invariant and with lozenge tilings of a "spiral" surface. I will present the combinatorics of these objects and asymptotic results.
- Supplements
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12:20 PM - 01:30 PM
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Lunch
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01:30 PM - 02:20 PM
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Random Melting Skew Young Diagram
Zhipeng Liu (University of Kansas)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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We consider a model of random melting skew Young diagram whose northwest and southeast corners melt independently at two rates $\gamma_1$ and $\gamma_2$ respectively. We find an exact formula for the joint distribution of the location of the last melting box and the melting time for an arbitrary initial skew Young diagram. This formula is suitable for asymptotic analysis for some special initial skew Young diagrams. As applications, we show how this result is related to the argmax of the sum of two independent Airy-type processes, such as two parabolic Airy2 processes, or a parabolic Airy2 process and an Airy1 process.
- Supplements
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03:00 PM - 03:30 PM
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Afternoon Tea
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Oct 20, 2021
Wednesday
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09:10 AM - 10:00 AM
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The Two-Periodic Aztec Diamond and Matrix Valued Orthogonality
Arno Kuijlaars (Katholieke Universiteit Leuven)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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I will discuss how polynomials with a non-hermitian orthogonality on a contour in the complex plane arise in certain random tiling problems. In the case of periodic weightings the orthogonality is matrixvalued. In work with Maurice Duits (KTH Stockholm) the Riemann-Hilbert problem for matrix valued orthogonal polynomials was used to obtain asymptotics for domino tilings of the two-periodic Aztec diamond. This model is remarkable since it gives rise to a gaseous phase, in addition to the more common solid and liquid phases.
- Supplements
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10:20 AM - 11:10 AM
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Local Universality of the Time-Time Covariance and of the Geodesic Tree for Last Passage Percolation
Patrik Ferrari (Rheinische Friedrich-Wilhelms-Universität Bonn)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
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We consider time correlation for KPZ growth in 1+1 dimensions in a neighborhood of a characteristics. We prove convergence and local universality of the covariance with droplet, flat and some random initial profiles. Furthermore, we show that also the geodesic tree is locally universal. These are joint works with Alessandra Occelli and Ofer Busani.
- Supplements
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11:30 AM - 12:20 PM
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Lozenge Tilings and the Gaussian Free Field on a Cylinder
Marianna Russkikh (Massachusetts Institute of Technology)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
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We discuss new results on lozenge tilings on an infinite cylinder, which may be analyzed using the periodic Schur process introduced by Borodin. Under one variant of the $q^{vol}$ measure, corresponding to random cylindric partitions, the height function converges to a deterministic limit shape and fluctuations around it are given by the Gaussian free field in the conformal structure predicted by the Kenyon-Okounkov conjecture. Under another variant, corresponding to an unrestricted tiling model on the cylinder, the fluctuations are given by the same Gaussian free field with an additional discrete Gaussian shift component. Fluctuations of the latter type have been previously conjectured for tiling models on planar domains with holes.
- Supplements
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12:20 PM - 01:30 PM
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Lunch
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03:00 PM - 03:30 PM
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Afternoon Tea
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Oct 21, 2021
Thursday
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09:10 AM - 10:00 AM
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Hankel Composition Structures in Random Matrix Theory and Beyond
Thomas Bothner (University of Bristol)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
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This talk will highlight a novel way of characterizing Fredholm determinants of Hankel composition operators via Riemann-Hilbert problems. Based on work in progress by the speaker.
- Supplements
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10:20 AM - 11:10 AM
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Properties of the chGUE at the Hard Edge: Spacing Distributions and Universality with External Field
Gernot Akemann (Universität Bielefeld)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
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The chiral Gaussian unitary ensemble also called Laguerre or Wishart unitary ensemble is probably one of the most studied random matrix ensembles. We will investigate its further properties at the hard edge of the spectrum at finite- and large-N. Due to applications to the Dirac operator in Quantum Chromodynamics, we will add a finite number $N_f$ of characteristic polynomials to the Gaussian distribution of matrix elements. In the first part we will focus on the spacing distribution between the smallest singular values, where we give exact determinantal formulae. In contrast to the k-point correlation functions, the spacing is almost immediately as close to the bulk spacing as the Wigner surmise. In the second part we will show that the k-point functions are universal at the hard edge under the addition of an external, deterministic field $A$ with full rank, as long as a hard edge is present. Using recent progress in polynomial ensembles we can show that previous results from supersymmetry with and orthogonal polynomials without external field agree and can be extended to a fixed number of zero modes. In particular we show that determinantal formulae of different sizes for the k-point functions are equivalent.
- Supplements
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11:30 AM - 12:20 PM
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Stationary Half-Space Last Passage Percolation
Alessandra Occelli (Instituto Superior Técnico)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
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We present our result on stationary last passage percolation in half-space geometry. We determine the limiting distribution of the last passage time in a critical window close to the origin. The result is a new two-parameter family of distributions: one parameter for the strength of the diagonal bounding the half-space and the other for the distance of the point of observation from the origin. It should be compared with the one-parameter family giving the Baik--Rains distributions for full-space geometry. Joint work with D. Betea and P. Ferrari.
- Supplements
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12:20 PM - 01:30 PM
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Lunch
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01:30 PM - 02:20 PM
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Fractal Geometry of the KPZ Equation
Promit Ghosal (Brandeis University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
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The Kardar-Parisi-Zhang (KPZ) equation is a fundamental stochastic PDE related to modeling random growth processes, Burgers turbulence, interacting particle systems, random polymers etc. In this talk, we focus on how the tall peaks and deep valleys of the KPZ height function grow as time increases. In particular, we will ask what are the appropriate scaling of the peaks and valleys of the KPZ equation and whether they converge to any limit under those scaling. These questions will be answered via the law of iterated logarithms and fractal dimensions of the level sets.
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03:00 PM - 03:30 PM
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Afternoon Tea
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Oct 22, 2021
Friday
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09:10 AM - 10:00 AM
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The Riemann Hilbert Problem in Higher Genus and Some Applications
Marco Bertola (Concordia University and SISSA; Concordia University and International School for Advanced Studies (SISSA))
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
- Video
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The role of (bi/multi/matrix) orthogonal polynomials in random matrices, integrable systems and combinatorics is well known. Our goal is to report on recent progress in the definition of suitable extensions of the notion of orthogonality where the polynomials are replaced by sections of appropriate line bundles on Riemann surfaces. We discuss their definition in the spirit of various generalizations of the Padé problem and the formulation of appropriate matrix Riemann Hilbert problems that allow to characterize them as well as control their asymptotic behaviour. Applications to Matrix Orthogonal Polynomials and the KP hierarchy will also be discussed.
- Supplements
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10:20 AM - 11:10 AM
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Marked and Conditional Determinantal Point Processes
Tom Claeys (Université Catholique de Louvain)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
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I will introduce a family of marked determinantal point processes and associated conditional ensembles, in which information about a random part of a point configuration is encoded. Special cases of these conditional ensembles appear in the Its-Izergin-Korepin-Slavnov method and in the study of number rigidity. I will discuss general properties of these ensembles, and show how they lead to a strengthened notion of number rigidity for determinantal point processes induced by a certain class of orthogonal projections, including the sine, Airy, and Bessel point processes. The talk will be based on joint work with Gabriel Glesner.
- Supplements
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11:30 AM - 12:20 PM
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Asymptotic Analysis of the Interaction Between a Soliton and a Regular Gas of Solitons
Ken McLaughlin (Colorado State University)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
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We provide a detailed asymptotic description of this interaction, for the case of the modified KdV equation. Kinetic velocity equations are derived, and they will be shown to be ubiquitous in singular limits of a number of integrable systems, and in random matrix theory as well.
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12:20 PM - 01:30 PM
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Lunch
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01:30 PM - 02:20 PM
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The Edge Scaling Limit of the Characteristic Polynomial of the Gaussian β-Ensembles
Gaultier Lambert (Universität Zürich)
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- Location
- SLMath: Eisenbud Auditorium, Online/Virtual
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In this talk, I report on the asymptotics of the characteristic polynomials of the Gaussian β-ensembles for general β > 0. Based on the Dumitriu-Edelman matrix models for the Gaussian β-ensemble, I will present a probabilistic coupling between the characteristic polynomial, a Gaussian analytic function and a new object called the stochastic Airy function. This random entire function arise as the scaling limit of the characteristic polynomial at the spectral edge and its zero set is exactly the Airy-β point process. This is joint work with Elliot Paquette and our results are based on the study of the transfer matrix recurrence satisfied by the characteristic polynomials.
- Supplements
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03:00 PM - 03:30 PM
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Afternoon Tea
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