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Mathematics Institute

Pre-workshop talks


Date Time Room Speaker Title Abstract

Tue,

8/3/16

13:00-13:50 B3.02 Mario Kieburg

Products of random matrices:

what is new?

Products of random matrices were first studied in the late 60's, early 70's to analyze the stability of time evolutions in complex systems. Since then other applications as in transport theory, quantum
information, quantum chromodynamics, etc. were found. Especially for the newer applications new challenges arose for this kind of random matrix ensembles. In the past years a new radical progress was achieved in the derivation of the spectral statistics of product matrices at finite matrix dimensions. I will review this development for a certain class of random matrices multiplied from the point of view of harmonic analysis. For this purpose I will recall the class of polynomial ensembles and a distinguished subclass therein. Moreover I will show how the kernels for the spectral statistics of the eigenvalues and of the singular values change when multiplying the corresponding random matrices.

Thu,

10/3/16

16:00-17:00 D1.07 Christopher Joyner GSE statistics without spin

Energy level statistics following the Gaussian Symplectic Ensemble (GSE) of Random Matrix Theory have been predicted theoretically and observed numerically in numerous quantum chaotic systems. However, in all these systems there has been one unifying feature: the com-bina- tion of half-integer spin and time-reversal invariance. Here we provide an alternative mechanism for obtaining GSE statistics that is derived from geometric symmetries of a quantum system which alleviates the need for spin. As an example, we construct a quantum graph with a discrete symme- try given by the quaternion group Q8 and observe GSE statistics within one of its subspectra. We then show how to isolate this subspectrum and construct a quantum graph with a scalar valued wave function and a pure GSE spectrum. This is work together with M. Sieber and S. Muller.

Mon,

14/3/16

16:00-17:00 B3.01 Elliot Paquette

The Brownian Carousel description of

local eigenvalue statistics

 This talk will give an introduction and overview of Valkó and Virág's Brownian carousel description of the local eigenvalue statistics for beta-ensembles. This talk will start from the tridiagonal description of the Gaussian beta-ensembles, build Valkó and Virág's discrete Sturm-Liouville theory, and roughly derive the scaling limit of the discrete process to the limiting Carousel.

Tue,

15/3/16

13:00-13:50 B3.02 Gernot Akemann

Products of random matrices:

dropping the independence

 The subject of products of independent random matrices has seen considerable progress recently. This is due to the observation that its complex eigenvalues and its singular values form determinantal point
processes that are integrable. In this talk I will drop the independence and present results for the singular values of the product of two linearly coupled random matrices as a first step. The resulting ensemble is no longer polynomial, but can be solved explicitly in terms of an integrable kernel of biorthogonal functions. The process interpolates between that of two independent and a single Gaussian random matrix. For genuine coupling in the local scaling regime at the origin we find back the Meijer G-kernel of Kuijlaars and Zhang which is universal. Sending the coupling to zero at a rate 1/N, where N is the matrix size, we find a one-parameter family of kernels that interpolates between the Bessel- and Meijer G-kernel for two independent matrices. In this limit the two matrices in the product become strongly coupled and almost adjoint to each other. This is joint work with
Eugene Strahov.

Tue,

15/3/16

16:00-17:00 B1.01 Anna Maltsev

Local laws for Wigner

random matrices

The spectral measure of Hermitian matrices with centered independent identically distributed entries (Wigner matrices) tends to the semicircle law weakly in the limit of large dimension. This has been
proven by Wigner in the 50's. We ask to what extent this convergence continues to hold on small intervals, i.e. when the interval size tends to 0 with dimension. I will give an overview of the field,
mention some recent results, and outline some of the methods.

Thu,

17/3/16

16:00-17:00 MS.03 Mikhail Poplavskyi

Reaction diffusion point processes

and the real Ginibre ensemble

We discuss the family of reaction-diffusion stochastic particle systems and their connections to the real Ginibre ensemble of random matrices. We show that for a wide class of initial distributions and mixed annihilating/coalescing interaction of particles they form a Pfaffian Point Process with the kernel satisfying a differential equation. Two types of homogeneous initial conditions are considered and corresponding kernels in the case of pure annihilating interaction are calculated. We show how to correct the answer obtained by A. Borodin and R. Sinclair and prove that rescaled kernel for the real Ginibre ensemble coincides with the ones obtained for particle systems. The talk is based on a joint paper with B. Garrod, R. Tribe. O.Zaboronski (preprint: http://arxiv.org/abs/1507.01843).

Fri,

18/3/16

16:00-17:00 B3.02 Gernot Akemann

Random matrices - an overview and

recent developments (departmental

colloquium)

 Why are random matrices as popular and exciting after more than 50 years of research? In this talk I will try to answer this question. First, the most basic random matrix models will be introduced, solved and the typical questions explained that are raised, including that of universality. One of the popular tools, the theory of orthogonal polynomials, will be used as an example that has benefitted a lot from the
solution of more sophisticated and realistic random matrix models in recent years. Examples for various applications from physics to finance will be used throughout as illustrations.