Aug 23, 2021
Monday

09:10 AM  09:30 AM


Introductory Remarks

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video


 Abstract
 
 Supplements



09:30 AM  10:30 AM


Schubert polynomials, the inhomogeneous TASEP, and evilavoiding permutations
Lauren Williams (Harvard University)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
The totally asymmetric simple exclusion process (TASEP) was introduced around 1970 as a model for translation in protein synthesis and traffic flow. The inhomogeneous TASEP is a Markov chain of weighted particles hopping on a lattice, in which the hopping rate depends on the weight of the particles being interchanged. We will consider the case where the lattice is a ring, and each particle has a distinct weight, so that we can think of this model as a Markov chain on permutations. We will see that in many cases, and in particular for w an "evilavoiding" permutation, the steady state probability of w can be expressed in terms of Schubert polynomials. Based on joint work with Donghyun Kim.
 Supplements


10:45 AM  11:45 AM


On eigenvectors of perturbed Toeplitz matrices
Ofer Zeitouni (Weizmann Institute of Science)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
We consider Toeplitz matrices perturbed by vanishing noise, i.e. noise of the form $N^{\gamma} G$ with $\gamma>1/2$ and $G$ having iid zero mean entries of variance $1$; the resulting spectrum concentrates around spectral curves determined by the symbol. For $\gamma>1$, we prove localization of eigenvectors and show they are close to certain pseudomodes. I will also discuss the case $\gamma\in (1/2,1)$, where a different behavior emerges. Joint work with Anirban Basak and Martin Vogel.
 Supplements



Aug 24, 2021
Tuesday

09:30 AM  10:30 AM


Moments of Characteristic Polynomials and Integrability
Jon Keating (University of Oxford)

 Location
 SLMath: Online/Virtual
 Video

 Abstract
I will discuss the moments of characteristic polynomials and their derivatives in the context of integrability. Specifically, I will discuss formulae for the moments of the characteristic polynomials of GUE random matrices, and formulae for the joint moments of the characteristic polynomials of CUE random matrices and their derivatives.
 Supplements


10:45 AM  11:45 AM


Unearthing random matrix theory in the statistics of Lfunctions: the story of Beauty and the Beast
Nina Snaith (University of Bristol)

 Location
 SLMath: Online/Virtual
 Video

 Abstract
There has been very convincing numerical evidence since the 1970s that the positions of zeros of the Riemann zeta function and other Lfunctions show the same statistical distribution (in the appropriate limit) as eigenvalues of random matrices. Proving this connection, even in restricted cases, is difficult, but if one accepts the connection then random matrix theory can provide unique insight into longstanding questions in number theory. I will give a history of the attempt to prove the connection, as well as propose that the way forward may be to forgo the enticing beauty of the determinantal formulae available in random matrix theory in favour of something a little less elegant.
 Supplements


12:00 PM  01:00 PM


On the circle, GMC = CBE
Reda Chhaibi (Université de Toulouse III (Paul Sabatier))

 Location
 SLMath: Online/Virtual
 Video

 Abstract
In this talk, I would like to advertise the strict equality between two objects from very different areas of mathematical physics:
 Kahane's Gaussian Multiplicative Chaos (GMC), which uses a logcorrelated field as input and plays an important role in certain conformal field theories
 A reference model in random matrices called the Circular Beta Ensemble (CBE)
The goal is to give a precise theorem whose loose form is GMC = CBE. Although it was known that random matrices exhibit logcorrelated features, such an exact correspondence is quite a surprise.
 Supplements



Aug 25, 2021
Wednesday

09:30 AM  10:30 AM


Cointegration, S&P, and random matrices
Vadim Gorin (University of WisconsinMadison)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Cointegration is a property of Ndimensional time series, which says that each individual component is non stationary (growing like a random walk), but there exists a stationary linear combination. Testing procedures for the presence of cointegration has been extensively studied in statistics and economics, but most results are restricted to the case when N is much smaller than the length of the time series. I will discuss the recently discovered mathematical structures, which make the large N case accessible. On the applied side we will see a remarkable match between predictions of random matrix theory and behavior of S&P 100 stocks. On the theoretical side we will see how ideas from statistical mechanics and asymptotic representation theory play a crucial role in the analysis. (Based on joint work with Anna Bykhovskaya.)
 Supplements

Slides
1.06 MB application/pdf


10:45 AM  11:45 AM


Dynamical Loop Equations
Jiaoyang Huang (New York University, Courant Institute)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Loop (or DysonSchwinger) equation is an important tool to study the global fluctuations of one dimensional loggas type interacting particle systems. In this talk I will present a dynamical version of loop equations for large families of two dimensional interacting particle systems. Some examples include Dyson’s Brownian motion, Nonintersecting Bernoulli/Poisson random walks, corner process, measures on GelfandTsetlin patterns and Macdonald process. Then I will explain how to use dynamical loop equations to understand global fluctuations of these systems. This is a joint work with Vadim Gorin.
 Supplements


12:00 PM  01:00 PM


Transfer matrix approach to random band matrices
Tetiana Shcherbyna (University of WisconsinMadison)

 Location
 SLMath: Eisenbud Auditorium, Online/Virtual
 Video

 Abstract
Random band matrices (RBM) are natural intermediate models to study eigenvalue statistics and quantum propagation in disordered systems, since they interpolate between meanfield type Wigner matrices and random Schrodinger operators. In particular, RBM can be used to model the Anderson metalinsulator phase transition (crossover) even in 1d. In this talk we will discuss some recent progress in application of the supersymmetric method (SUSY) and transfer matrix approach to the analysis of local spectral characteristics of some specific types of 1d RBM.
 Supplements



Aug 26, 2021
Thursday

09:30 AM  10:30 AM


Fluctuations of the Stieltjes transform of the empirical spectral distribution of selfadjoint polynomials in Wigner and deterministic diagonal matrices
Mireille Capitaine (Centre National de la Recherche Scientifique (CNRS))

 Location
 SLMath: Online/Virtual
 Video

 Abstract
We will present the following result. When the dimension goes to infinity, the recentered analytic process on nonreal complex numbers of nonnormalized traces of resolvents of any selfadjoint noncommutative polynomial in a Wigner matrix and a deterministic diagonal matrix converges to a centred complex Gaussian process whose covariance is expressed in terms of operatorvalued subordination functions in free probability theory. This is a joint work with Serban Belinschi, Sandrine Dallaporta and Maxime Février.
 Supplements


10:45 AM  11:45 AM


Eigenstate thermalisation hypothesis and Gaussian fluctuations for Wigner matrices
Laszlo Erdos (Institute of Science and Technology Austria)

 Location
 SLMath: Online/Virtual
 Video

 Abstract
We prove that any deterministic matrix is approximately the identity in the eigenbasis of a large random Wigner matrix W with an optimal error inversely proportional to the square root of the dimension. This verifies a strong form of Quantum Unique Ergodicity with an optimal convergence rate and we also prove Gaussian fluctuations around this convergence. The key technical tool is a new multiresolvent local law for Wigner ensemble and the Dyson Brownian motion for eigenvector overlaps.
 Supplements

Slides
4.08 MB application/pdf


12:00 PM  01:00 PM


Delocalization of random band matrices in high dimensions
Jun Yin (University of California, Los Angeles)

 Location
 SLMath: Online/Virtual
 Video

 Abstract
One famous conjecture in quantum chaos and random matrix theory is the socalled phase transition conjecture of random band matrices. It predicts that the eigenvectors' localizationdelocalization transition occurs at some critical bandwidth $\mathrm{Wc}(\mathrm{d})$, which depends on the dimension $\mathrm{d}$. The wellknown Anderson model and Anderson conjecture have a similar phenomenon. It is widely believed that $\mathrm{Wc}(\mathrm{d})$ matches $1 / \lambda \mathrm{c}(\mathrm{d})$ in the Anderson conjecture, where $\lambda \mathrm{c}(\mathrm{d})$ is the critical coupling constant. Furthermore, this random matrix eigenvector phase transition coincides with the local eigenvalue statistics phase transition, which matches the BohigasGiannoniSchmit conjecture in quantum chaos theory. We proved the eigenvector's delocalization property for most of the general $d>=8$ random band matrix as long as the size of this random matrix does not grow faster than its bandwidth polynomially. In other words, as long as bandwidth $W$ is larger than $L^ \epislon$ for some $epislon $>0$. It is joint work with H.T. Yau (Harvard) and F. Yang (Upenn).
 Supplements



Aug 27, 2021
Friday

09:30 AM  10:30 AM


Deformed Polynuclear Growth in (1+1) Dimensions
Alexei Borodin (Massachusetts Institute of Technology)

 Location
 SLMath: Online/Virtual
 Video

 Abstract
We introduce and study a one parameter deformation of the polynuclear growth (PNG) in (1+1)dimensions, which we call the tPNG model. It is defined by requiring that, when two expanding islands merge, with probability t they sprout another island on top of the merging location. At t=0, this becomes the standard (nondeformed) PNG model that, in the droplet geometry, can be reformulated through longest increasing subsequences of uniformly random permutations or through an algorithm known as patience sorting. In terms of the latter, the tPNG model allows errors to occur in the sorting algorithm with probability t. We prove that the tPNG model exhibits onepoint TracyWidom GUE asymptotics at large times for any fixed t∈[0,1), and onepoint convergence to the narrow wedge solution of the KardarParisiZhang (KPZ) equation as t tends to 1. We further construct distributions for an external source that are likely to induce BaikBen ArousPeche type phase transitions. The proofs are based on solvable stochastic vertex models and their connection to the determinantal point processes arising from Schur measures on partitions.
Joint work with Amol Aggarwal and Michael Wheeler.
 Supplements



10:45 AM  11:45 AM


Towards KPZ universality
Jeremy Quastel (University of Toronto)

 Location
 SLMath: Online/Virtual
 Video

 Abstract
This will be a survey talk on asymptotic fluctuations in the KPZ universality class. These are governed by an invariant fixed point whose transition probabilities are given by determinants and satisfy integrable partial differential equations. Convergence to the fixed point of some nonintegrable models like the KPZ equation and partially asymmetric exclusions is obtained by comparison with the integrable case using energy estimates.
 Supplements


