To participate in this seminar, please register HERE.

This talk will describe the interplay between methods and results from algebraic combinatorics and the study of lozenge tilings. This includes the Schur generating function to prove the GUE corners process and limit surfaces, as well as the Naruse Hook-length formula and factorial Schur functions towards weighted lozenge tilings and asymptotics of skew Standard Young Tableaux. Some of the further applications and questions concern the asymptotics of certain representation theoretic multiplicities, structure constants.

To register for this course, go to: https://www.msri.org/seminars/26228

This class aims at understanding some important classes of smooth random functions of very many variables.

What can be said about the complexity of the topology of the landscapes they define?

How efficient are the natural exploration or optimization algorithms in these landscapes?

The toolbox of Random Matrix Theory will be used for both questions.

We will concentrate on two wide classes of interesting smooth random functions of many variables.

A first source of such functions is to be found in statistical mechanics of disordered systems, i.e. the Hamiltonians of disordered models, like spin-glasses. There the randomness is assumed to model quenched disorder in the medium.

Another rich class of such functions comes from Data Science and studies the random landscapes of inference problems in high-dimensional statistical estimation. Here the randomness of these landscapes is the randomness inherent in sampling.

To participate in this seminar, please register HERE.

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.

To participate in this seminar, please register HERE.

Recently we discovered a direct connection between solvable models in the Kardar-Parisi-Zhang (KPZ) universality class and free fermionic models at positive temperature. In this talk we will explain this. 

  In 2000, Johansson showed that the current of the totally asymmetric simple exclusion process (TASEP) corresponds to a marginal of a free fermionic system at zero temperature described by the Schur measure. This was achieved using the Robinson-Schensted-Knuth (RSK) bijection and an immediate consequence is a  Fredholm determinant formula for the current distribution which admits further asymptotic analysis.      

  A similar Fredholm determinant formula was found for KPZ equation in 2010 by a few different groups and also for its various discretized models in the following years. However they are usually found by calculations through non-free fermionic methods such as Bethe ansatz and Macdonald operators. 

  In this talk, we will present a direct connection between discretized models of the KPZ equation and free fermions at finite temperature. The key ingredient in our theory is a new identity between marginals of the q-Whittaker measure and the periodic Schur measure. The latter measure was first introduced by Borodin (2007) and studied recently in connection with free fermions by Betea and Buttier (2018). The identity is proved in a bijective fashion by generalizing  and developing  substantially the RSK algorithm, and studying its properties leveraging the theory of affine crystal. 

   We also discuss applications of this approach to a few KPZ models.  In particular we plan to report on a symmetrized variant of the identity and its applications to the KPZ models in half spaces with Pfaffian point process. 

 The talk is based on collaborations with Takashi Imamura and Matteo Mucciconi.

To participate in this seminar, please register HERE.

We consider eigenvector statistics of large symmetric random matrices. When the matrix entries are sampled from independent Gaussian random variables, eigenvectors are uniformly distributed on the sphere and numerous properties can be computed exactly. In particular, we can bound their extremal coordinates with high probability. There has been an extensive amount of work on generalizing such a result, known as delocalization, to more general entry distributions. After giving a brief overview of the previous results going in this direction, we present an optimal delocalization result for matrices with sub-exponential entries for all eigenvectors. The proof is based on the dynamical method introduced by Erdos-Yau, an analysis of high moments of eigenvectors as well as new level repulsion estimates which will be presented during the talk. This is based on a joint work with P. Lopatto.

To register for this course, go to: https://www.msri.org/seminars/26228

This class aims at understanding some important classes of smooth random functions of very many variables.

What can be said about the complexity of the topology of the landscapes they define?

How efficient are the natural exploration or optimization algorithms in these landscapes?

The toolbox of Random Matrix Theory will be used for both questions.

We will concentrate on two wide classes of interesting smooth random functions of many variables.

A first source of such functions is to be found in statistical mechanics of disordered systems, i.e. the Hamiltonians of disordered models, like spin-glasses. There the randomness is assumed to model quenched disorder in the medium.

Another rich class of such functions comes from Data Science and studies the random landscapes of inference problems in high-dimensional statistical estimation. Here the randomness of these landscapes is the randomness inherent in sampling.

To participate in this seminar, please register HERE.

Using the notion of duality, or bispectrality, in Macdonal theory, can derive the Pieri operators from the eigenvalue equation for Macdonald-Koornwinder operators. These commuting difference  operators, in the q-Whittaker limit, can be regarded as q-difference relativistic Toda Hamiltonians for some root systems, and their eigenfunctions as q-Whittaker functions. One can use the SL_2(Z)-action on the spherical DAHA, acting on the Macdonald or Koornwinder operators, to obtain a set of operators which, in the limit, act as raising operators for q-Whittaker functions. [Joint work with P. Di Francesco]

To participate in this seminar, please register HERE.

Consider the continuous time totally asymmetric simple exclusion process (TASEP) on the line with the step initial condition. The infinite time scaling limit of this model belongs to the Kardar-Parisi-Zhang (KPZ) universality class. Denote its random space-time height function as h(t,y). The one point distribution as the function of three variables is the tau function of the Kadomtsev-Petviashvili (KPII) equation. The multitime distribution has the representation as the multiple contour integral of the integrable Fredholm determinant. We study the differential equations associated with it.

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.

We study the large matrix limit of a family of multiplicative statistics for eigenvalues of hermitian matrix models, showing that they universally connect with the integro-differential Painlevé II equation and, in turn, with the KPZ equation. But the connection does not stop there, and we will also explain how the norming constants of the associated orthogonal polynomials and the underlying correlation kernel are asymptotically described in terms of the same solution to the integro-differential PII. Although we work under the assumption of a regular one-cut potential and a family of multiplicative statistics satisfying certain regularity conditions, we also plan to discuss how our approach indicates that other classes of potentials may give rise to new families of integrable systems.

Based on ongoing work with Promit Ghosal (MIT).

To participate in this seminar, please register HERE.

To participate in this seminar, please register HERE.