Content

The collective behavior of many independent degrees of freedom often gives rise to universal patterns. In probability theory the law of large numbers and the central limit theorem are prime examples. The celebrated insight that large disordered quantum systems exhibit a similar phenomenon on the level of their spectral information goes back at least to Eugene Wigner [18] and is known as the universality conjecture. It asserts that, in analogy to the emergence of the Gaussian distribution in the central limit theorem, the statistics of energy levels in these systems follow a universal distribution [16]. Despite the overwhelming physical evidence, this conjecture remains open for first principle physics models, such as the Anderson model that describes the motion of a single electron in a crystal with random impurities [1, 13]. However, in recent years significant progress has been made in the understanding of this non-commutative universality phenomenon for matrices with random entries [9]. These random matrices can be viewed as a toy model for chaotic quantum systems [6].

The most robust approach to establishing spectral universality relies on following the dynamics of a 1-dimensional particle system with logarithmic interaction and driven by stochastic noise, called Dyson Brownian motion [11]. In essence, the eigenvalues of a matrix with sufficiently many random degrees of freedom can be interpreted as a 1-dimensional system of such particles that has locally reached its thermodynamic equilibrium [8]. In this picture the distribution underlying spectral universality is interpreted in analogy to the Gaussian distribution that describes the velocities of gas particles in a 3-dimensional volume in thermodynamic equilibrium.

In this graduate seminar we will study the system of the N coupled stochastic differential equations that describe the Dyson Brownian motion of N particles at inverse temperature β. We will establish a connection between this system and the spectral universality of random matrices, derive an equation for the hydrodynamic limit of the particle density [7, 17] and study the local universality of its equilibrium state, the β-ensemble [3, 4, 5]. Depending on the level of the participants we may even try to understand how the strongly interacting particles reach their local equilibrium after very short times [12, 14, 15], providing a rigorous interpretation of the universality conjecture for random matrices [2, 9, 10].

Instructions

If you want to participate in the seminar, please send an email to torben.krueger(at)uni-bonn.de. In case you would like to give a talk and already have a preference, inform me about your preferred topic and/or about your background (so that I can assign an appropriate one). There are still slots available! If you need a certificate you also have to register in BASIS.

Schedule

The seminar takes place Mondays 10AM (c.t.) in SR N0.007. Starting date is April 23. The schedule will still be updated.

A summary of our first meeting with a comprehensive list of the possible topics and corresponding literature can be found here.

Prerequisites

The participants are expected to have a basic knowledge of linear algebra, functional analysis, probability theory and stochastic calculus.

Participation

Every participant who needs a certificate is expected to prepare a talk of approximately 90 minutes (projector or blackboard) that reflects a good understanding of the underlying literature source and a written summary of the chosen topic that is to be provided to the audience. The talks can be based on chapters of books or research articles. The reference list below provides ample examples but based on the preferences of the participants many other topics are possible.

Literature

[1] M. Aizenman and S. Warzel. Random Operators. American Mathematical Soc., 2015.

[2] O. H. Ajanki, L. Erdős, and T. Krüger. Stability of the matrix dyson equation and random matrices with correlations. Probability Theory and Related Fields, 2018.

[3] F. Bekerman, A. Figalli, and A. Guionnet. Transport maps for Beta-matrix models and Universality. Comm. Math. Phys., 338:589–619, 2015.

[4] P. Bourgade, L. Erdös, and H.-T. Yau. Edge universality of beta ensembles. Communications in Mathematical Physics, 332(1):261–353, 2014.

[5] P. Bourgade, L. Erdős, and H.-T. Yau. Universality of general β-ensembles. Duke Math. J., 163(6):1127–1190, 2014.

[6] P. Bourgade and J. P. Keating. Quantum Chaos, Random Matrix Theory, and the Riemann ζ-function, pages 125–168. Springer Basel, Basel, 2013.

[7] T. Chan. The wigner semi-circle law and eigenvalues of matrix-valued diffusions. Probability Theory and Related Fields, 93(2):249–272, 1992.

[8] F. J. Dyson. A Brownian-Motion Model for the Eigenvalues of a Random Matrix. J. Math. Phys., 3:1191, 1962.

[9] L. Erdős, B. Schlein, and H.-T. Yau. Universality of random matrices and local relaxation flow. Invent. Math., 185:75–119, 2011.

[10] L. Erdős and K. Schnelli. Universality for random matrix flows with time-dependent density. Ann. Inst. Henri Poincaré Probab. Stat., 53(4):1606–1656, 2017.

[11] P. Forrester. Log-Gases and Random Matrices (LMS-34). Princeton University Press, 2010.

[12] J. Huang and B. Landon. Local law and mesoscopic fluctuations of Dyson Brownian motion for general β and potential. arXiv:1612.06306, 2016.

[13] D. Hundertmark. A short introduction to anderson localization, 2007.

[14] B. Landon, P. Sosoe, and H.-T. Yau. Fixed energy universality for Dyson Brownian motion. arXiv:1609.09011, 2016.

[15] B. Landon and H.-T. Yau. Edge statistics of Dyson Brownian motion. arXiv:1712.03881, 2017.

[16] M. L. Mehta. Random matrices and the statistical theory of energy levels. Academic Press, New York-London, 1967.

[17] L. C. G. Rogers and Z. Shi. Interacting brownian particles and the wigner law. Probability Theory and Related Fields, 95(4):555–570, 1993.

[18] E. P. Wigner. Characteristic vectors of bordered matrices with infinite dimensions. Ann. of Math. (2), 62:548–564, 1955.