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 |
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 |
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. |