Researcher Title Abstract Date

Transcription

1 FROM RANDOM MATRIX THEORY TO TOEPLITZ Understanding the eigenvalues of random matrices in the classical compact ensembles led (in joint work with Shahshahani) to showing that the joint distribution of the traces of powers have Gaussian limits (to amazing accuracy). This turns out to be equivalent to some theorems about Toeplitz operators---the strong Szegö limit theorems (work with Dan Bump). The equivalence is quite mysterious (at least to me) and linked to strange identities (cf work of Tracy-Widom-deHaye). I hope that the assembled brain power at our Diaconis OPERATORS conference can shed some light on it. 6/26/2017 Forrester Baik Chatterjee Speicher Borot Applications of decomposition of measure in random matrix theory to number theory and integral geometry Fluctuations of the free energy of spherical Sherrington-Kirkpatrick model A general method for lower bounds on fluctuations of random variables Fundamental to random matrix theory is various factorisations of Lebesgue product measure implied by matrix change of variables. In number theory, factorisation of Siegel's invariant measure for SL${}_N(\mathbb R)$ is an ingredient in Duke, Rudnik and Sarnak's asymptotic computation of the number of matrices in SL${}_N(\mathbb Z)$, with a bounded norm. It allows to for calculations in the space of integral lattices SL${}_N(\mathbb R)/{\rm SL}_N(\mathbb Z)$ and generalisations such as SL${}_N(\mathbb C)/{\rm SL}_N(\mathbb Z[i])$ Factorisation of measure is also fundamental to integral geometry, with one of the most important results due to Blaschke and Petkantschin. Following recent work of Moghadasi, we show how the latter is related to matrix polar decomposition, and can be applied to the calculation of the moments of the volume content of the convex hull of random points in higher dimensional spaces. 6/26/2017 Spherical Sherrington-Kirkpatrick (SSK) model is an example of disordered systems, called spin glasses. The free energy of 2-spin SSK model is a finite temperature version of the largest eigenvalue of a random symmetric matrix. We obtain the limiting distribution of the free energy using the results from random matrix theory. A special interest is how the limiting distribution changes depending on the temperature. We also consider a SSK plus a ferromagnetic Hamiltonian which is related to spiked random matrices. This is a joint work with Ji Oon Li. 6/27/2017 There are numerous ways of establishing upper bounds on fluctuations of random variables, but there is no systematic approach for lower bounds. As a result, lower bounds are unknown in many important problems. I will talk about a general method for lower bounds on fluctuations. The method gives new results for the stochastic traveling salesman problem, the stochastic minimal matching problem, the random assignment problem, the Sherrington--Kirkpatrick model of spin glasses, first-passage percolation and random matrices. I will discuss some of these examples. 6/27/2017 The free probability perspective on random matrices is that the large size limit of random matrices is given by some (usually interesting) operators on Hilbert spaces and corresponding operator algebras. The prototypical example for this is that independent GUE random matrices converge to free semicircular operators, which generate the free group von Neumann algebra. The usual convergence in distribution has been strengthened in recent years to a strong convergence, also taking operator norms into account. All this Distributions of Random Matrices and Their Limits is on the level of polynomials. In my talk I will recall this and then go over from polynomials to rational functions (in non-commuting variables). Unbounded operators will also play a role. 6/27/2017 Discrete matrix models, Nekrasov equations and asymptotic expansions 6/28/2017

2 Extremal statistics in the classical We consider a one-dimensional classical Coulomb gas of N like-charges in a harmonic potential --~also known as the one-dimensional one-component plasma (1dOCP). We compute analytically the probability distribution of the position x_{max} of the rightmost charge in the limit of large N. We show that the typical fluctuations of x_{max} around its mean are described by a non-trivial scaling function, with asymmetric tails. This distribution is different from the Tracy-Widom distribution of x_{max} for the Dyson's log-gas. We also compute the large deviation functions of x_{max} explicitly and show that the system exhibits a third-order phase transition, as in the log-gas. I'll also discuss some results on the distribution of the index, i.e., the Majumdar one dimensional Coulomb gas number of charges on the positive semi-axis. 6/29/2017 The purpose of this talk is to advertize a new stochastic Loewner evolution that was developed in my student Vivian Healey's 2017 thesis. This work was motivated by the desire to better understand the Brownian map by constructing explicit conformal maps that embed random trees, especially Aldous' continuum random tree, into the upper half plane. The main results are (i) conformal maps that embed finite Galton-Watson trees into the upper half plane via a stochastic Loewner evolution that includes branching; and (ii) a description of the scaling limit. Part (i) is completely rigorous, while part (ii) includes both formal calculations and rigorous results. Menon Schnelli Gorin Stochastic Loewner evolution with branching and the Dyson superprocess Free addition of random matrices and the local single ring theorem. Local limits of Random Sorting Networks At least formally, the scaling limit is a stochastic Loewner evolution driven by a measure valued diffusion that is the scaling limit of Dyson Brownian motion plus branching. We call this diffusion the Dyson superprocess. It can be seen as the free analogue of the Dawson-Watanabe superprocess and it has several interesting connections with stochastic PDE. 6/29/2017 In the first part of this talk, I will discuss some recent results on local laws and rigidity of eigenvalues for additive random matrix models. In the second part, I will explain how these results can be used to derive the optimal convergence rate of the empirical eigenvalue distribution in the Single Ring Theorem. 6/29/2017 A sorting network is a shortest path between 12..n and n..21 in the Caley graph of the symmetric group spanned by swaps of adjacent letters. We will discuss the bulk local limit of the swap process of uniformly random sorting networks and encounter universal distributions of the random matrix theory. 6/30/2017 Liechty Free fermions at finite temperature and the MNS matrix model The Moshe Neuberbger Shapiro (MNS) random matrix model is equivalent to one-dimensional fermions at finite temperature trapped in a quadratic well and was introduced by those authors in the mid-1990 s as an interpolant between random matrix-type statistics and Poissonian statistics. In the 2000's Johansson studied this crossover rigorously for a grand-canonical version of the ensemble, which is determinantal, and obtained kernels describing the limiting local behavior of the eigenvalues in the crossover regime both in the bulk and at the edge. I will discuss the analysis of the canonical MNS model, in which the number of particles is deterministic. This model is not determinantal, but correlation functions and gap probabilities can be written explicitly in a form amenable to asymptotic analysis. This is joint work with Dong Wang. 6/30/2017 According to the well known conjecture, there is a crossover in the local eigenvalue statistics behavior of 1d band $n\times n$-matrices with a band width $W$. For $W<<n^{1/2}$ it is expected that the local eigenvalue statistics is Poisson, while for $W>>n^{1/2}$ the behavior is expected to be the GUE type. We will discuss the application of the transfer matrix approach to the proof of the conjecture for some classes of band and Wegner matrices. 7/3/2017 Transfer matrix approach to 1d Shcherbina random band matrices. Yau 7/3/2017

3 Periodic and free boundary Schur The Schur process is a probability measure over sequences of integer partitions, where the weight of a sequence is given by a product of skew Schur functions. It can be thought as a discrete random matrix/orthogonal polynomial ensemble, and encompasses several well-known models such as plane partitions, domino tilings of the Aztec diamond and last-passage percolation with geometric weights. In the original version considered by Okounkov and Reshetikhin, the first and last partitions in the sequence are both constrained to be the empty partition, and the process was shown to have determinantal correlation functions. A Pfaffian variant was considered by Sasamoto-Imamura and Borodin-Rains, and consists in having, say, the first partition in the sequence unconstrained. Another variant is the periodic Schur process (in which the first and last partitions are constrained to be equal) considered by Borodin, and shown to have a slightly more involved structure for correlations. In this talk we will revisit the Pfaffian and periodic Schur process, and consider also the case where both endpoints can be arbitrary (possibly different) partitions. In all cases we compute the correlation functions using the free fermionic approach as used originally by Okounkov and Reshetikhin, adapted to the case of different boundary conditions thanks to suitable variants of Wick's lemma. This talk is based on works done Bouttier processes via fermions in collaboration with Dan Betea, Peter Nejjar and Mirjana Vuletić. 7/5/2017 We discuss limit shape theorems for random tilings and a recent extension to a "non-determinantal" setting, Kenyon Zeitouni Cook Ben Arous Keating Limit shapes beyond dimers a special case of the 6-vertex model. This is joint work with Jan de Gier and Sam Watson. 7/5/2017 Noise (in)stability for non-normal I will describe several approaches that allow one to describe limiting eigenvalue distributions for certain matrices sparse non-normal matrices matrices perturbed by polynomially vanishing independent noise. 7/5/2017 The celebrated Circular Law for i.i.d. matrices roughly states that for an $n\times n$ matrix with i.i.d. centered entries of unit variance, in the large $n$ limit the spectrum becomes uniformly distributed over the disk of radius $\sqrt{n}$ in the complex plane. I will discuss some recent extensions of this result to matrices with dependent or non-identically distributed entries. In particular, I will first discuss how the Circular Law holds for adjacency matrices of random regular directed graphs with slowly growing degrees, which are challenging to analyze due to their global dependence structure and sparsity. Then I will describe generalizations of the Circular Law for matrices with independent, centered entries having non-identical variances. A key challenge for all of these results is to establish control on the pseudospectrum of the matrix, which we do with the help of tools from graph regularity theory. Based on joint work with Anirban Basak, Ofer Circular laws for non-i.i.d. matrices Zeitouni, Walid Hachem, Jamal Najim and David Renfrew. 7/6/2017 Gaussian Random Matrices and stationary random functions 7/6/2017 extreme values of characteristic polynomials I will review recent developments concerning the extreme value statistics of the characteristic polynomials of random matrices. I will also briefly discuss connections with the extreme values taken by the Riemann zeta function on its critical line. 7/6/2017

4 Random matrices with slow The resolvent of a large dimensional self-adjoint random matrix approximately satisfies the matrix Dyson equation (MDE) up to a random error. We show that for random matrices with arbitrary expectation and slow decay of correlation among its entries this error matrix converges to zero both in an isotropic and averaged sense with optimal rates of convergence as the dimension tends to infinity. This result requires a delicate cancellation (self-energy renormalization) which is seen through a diagrammatic cumulant expansion that automatically exploits the cancellation to all orders. Furthermore, we provide a comprehensive isotropic stability analysis of the MDE down to the length scale of the eigenvalue spacing. This analysis is then used to show convergence of the resolvent to the non-random solution of the MDE and to prove that the local eigenvalue statistics are universal, i.e. they do not depend on the distribution of the entries of the random matrix under consideration (Wigner-Dyson-Mehta spectral universality). [Joint work with Oskari Ajanki & Laszlo Erdös & Dominik Kruger correlation decay Schröder] 7/7/2017 Maltsev Le Doussal Paquette Petrov Spectral properties of random fermionic models Two-time persistent correlation in Kardar Parisi Zhang growth via the replica Bethe ansatz, and its comparison to experiments Tridiagonal matrix models for beta- Dyson Brownian motion TASEP in continuous inhomogeneous space I will discuss our work on fermionic XY models with interactions that are taken to be i.i.d. Gaussian random variables. We focus on the ground state energy gap, density of states, and the local eigenvalue statistics in the case of all-to-all interactions. We also examine what happens as the number of interacting neighbors changes. This is based on a joint work with F. Mezzadri and F. Cunden. 7/7/2017 We recall the mapping of the KPZ equation, which describes the stochastic growth of an interface, to the integrable delta Bose gas model. We also recall the calculation of the probability distribution (PD) of the height of the interface at a single time, using the replica Bethe ansatz (RBA). We then study the the two-time problem, which is still outstanding: the aim is to obtain the joint PD of heights at time t and t', in the limit of large times with fixed ratio t'/t>1. We use the RBA to conjecture the exact form of this joint PD in its partial tail, i.e. when the height h at the earlier time is large (for arbitrary value of the height h' at t') for droplet initial conditions. Comparison with experiments and numerics shows a very nice agreement in a broad window of values of h. Further using Airy processes, we derive the exact form of the persistent correlations which quantify the memory effect in the time evolution. Most is joint work with J. de Nardis and K. Takeuchi. 7/10/2017 We characterize the tridiagonal matrix diffusions A(t), the eigenvalues of which evolve according to beta- Dyson Brownian motion. This characterization is given in terms of the orthogonal polynomials associated to this tridiagonal matrix. We highlight one particular choice of diffusion in which the spectral weights are frozen in time. This gives a dynamics on tridiagonal matrices that is reversible with respect to the Dumitriu- Edelman measure. For this particular choice, we show a scaling limit of the bounded order principal submatrices of A(t) - A(0). This is joint work with Diane Holcomb. 7/10/2017 We discuss a version of TASEP in continuous time and space, in which the space can be equipped with arbitrary inhomogeneity. This system is integrable owing to a new connection with Schur measures. This leads to limit shape and KPZ-type fluctuation results. We also discuss a q-deformation of this system, and its mapping to a corner growth like model. 7/10/2017

5 We will discuss a new approach to the analysis of the global behavior of stochastic discrete particle systems. The approach links the asymptotics of these systems with properties of certain observables related to the Schur symmetric functions. This provides a natural analog of classical characteristic functions for discrete probability measures. Bufetov Colins Seppalainen Characteristic functions for random tilings Free probability for purely discrete eigenvalues of random matrices Random walk in random environment and the Kardar-Parisi- Zhang class In this talk, our main focus will be on applications of this method to the Law of Large Numbers and the Central Limit Theorem for models of random lozenge and domino tilings in various domains. In particular, for some non-simply connected domains we will prove the convergence of fluctuations of the height function to the Gaussian Free Field. Based on joint works with V. Gorin and A. Knizel. 7/11/2017 Let $A_n$ be the n-dimensional truncation of a compact operator, and $B_n$ be a unitarily invariant random matrix that admits a joint distribution as $n\to\infty$ in the sense of Voiculescu. Take a non-commutative polynomial P in formal non-commuting variables a, b such that specializing to a=0 gives the zero polynomial. We show that almost surely, as n goes to infinity, the l-th largest singular value of $P(A_n, B_n)$ converges and we explain how to compute it in some cases. I will also discuss variants, e.g. the fact that the result actually holds true in the more general context where $A_n$ and $B_n$ are replaced by k-tuples of matrices with an appropriate joint distribution. We compare this result with previous and also subsequent results by Shlyakhtenko, and Belinschi, Bercovici and Capitaine. Joint work with Takahiro Hasebe and Noriyoshi Sakuma. 7/11/2017 This talk describes 1+1 dimensional directed random walks in correlated random environments that obey the Kardar-Parisi-Zhang 2/3 fluctuation exponent. This is in marked contrast with the IID environment case where the walk satisfies a quenched central limit theorem with the standard diffusive scaling. We discuss walks in correlated environments that arise from two sources: (i) from limits of quenched polymer measures when the length of the polymer path is taken to infinity, and (ii) from limits of RWRE in an IID environment conditioned on an atypical velocity. In both cases results currently exist only for exactly solvable models. In case (i) the exactly solvable case is the log-gamma polymer. In case (ii) the exactly solvable model is the RWRE in an IID beta environment, whose exact solvability was discovered by Barraquand and Corwin. This talk is based on joint papers with M\'arton Bal\'azs (Bristol), Nicos Georgiou (Sussex), Firas Rassoul-Agha (Utah), and Atilla Y{\i}lmaz (Ko\c{c}). 7/11/2017 Remenik Extreme statistics of nonintersecting Brownian paths and LOE In this talk I will discuss results about the maximal height and its location for the top path of N nonintersecting Brownian bridges on the line and on the half-line (with either reflecting or absorbing boundary conditions). A result of K. Johansson implies that, suitably rescaled, the max converges as N goes to infinity to the Tracy- Widom GOE distribution. I will present a discrete version of this result which states that, in the case of Brownian bridges on the line, and for fixed N, the squared maximal height of the top path in this system is distributed as the top eigenvalue of a (finite) random matrix drawn from the Laguerre Orthogonal Ensemble. More generally, I will present explicit formulas for the joint distribution of the max and argmax for the three models and show directly that they converge to the analog distribution for the Airy2 process minus a parabola. In the case of Brownian bridges on the line I will also discuss a small deviation inequality for the argmax which matches the tail behavior of the limiting polymer endpoint distribution. (Joint work with Gia Bao Nguyen). 7/12/2017

6 Barraquand Liu, Zhipeng ASEP and KPZ equation in a halfspace with open boundary Multi-time distribution of periodic TASEP We consider the asymmetric simple exclusion process (ASEP) on the positive integers with an open boundary at the origin, and discuss two asymptotic results. (1) For a certain boundary condition, the fluctuations of the height function at the origin weakly converge to the Tracy-Widom GOE distribution. (2) The statistics of the KPZ equation in a half-space, which is a limit of weakly asymmetric ASEP, are characterized by a remarkably simple functional of the eigenvalues of the GOE (in the edge scaling limit). We will also discuss predictions for other boundary conditions and compare these results with analogous ones for ASEP on the whole line. Joint work with Alexei Borodin, Ivan Corwin and Michael Wheeler. 7/13/2017 The scaling limit of the height function for the models in the KPZ class is expected to be universal. The spatial process at an equal time is known to be given by the Airy process or its variations. However, the temporal process, or more generally the two-dimensional space-time field, is less well understood. We consider this question for the periodic TASEP (totally asymmetric simple exclusion process). For a special initial condition, we evaluate the joint distribution of the height function at multiple locations and times under the KPZ scale in the so-called relaxation time scale. The limit is given by multiple integrals involving a Fredholm determinant. This result is obtained by finding the multi-time, multi-location joint distribution explicitly. This is a joint work with Jinho Baik. 7/13/2017 This talk concerns two results on the algebraic structure of multilevel densities for sample covariance matrices. In the first half of the talk, we study the static multilevel structure of certain generalized Wishart and Jacobi ensembles. In the real Wishart case, which corresponds to the spiked covariance model, we identify the process with one defined via branching of multivariate Bessel functions, generalizing a result of Dieker-Warren. In the real and complex Jacobi case, we identify the process as a principally specialized Heckman-Opdam measure, resolving a conjecture of Borodin-Gorin. Sun Duits Algebraic structures for multilevel eigenvalue densities Asymptotic analysis of the two periodic Aztec diamond using Riemann-Hilbert methods In the second half of the talk, we define Laguerre and Jacobi analogues of the Warren process. That is, we construct local dynamics on a triangular array of particles so that (1) the projections to each level recover the Laguerre and Jacobi eigenvalue processes of König-O'Connell and Doumerc and (2) the fixed time distributions recover the joint distribution of eigenvalues in multilevel Laguerre and Jacobi random matrix ensembles. Our techniques extend and generalize the framework of intertwining diffusions developed by Pal- Shkolnikov. 7/13/2017 In this talk, I will report on joint work in progress with Arno Kuijlaars on a connection between random tilings of certain planar domains and polynomials that satisfy an orthogonality relation in the complex plane. For the classical examples of the Aztec diamond and lozenge tilings of a regular hexagon, these polynomials are Jacobi polynomials with non-standard parameters. The benefit of this point of view is that there is good hope that the asymptotic analysis of the random tiling models can be carried using the Riemann-Hilbert toolkit. A key example in the talk will be the asymptotic analysis of the two periodic Aztec diamond. 7/14/2017