Products of Random Matrices: from New Results to Applications

• Applicant: Professor Dr. Gernot Akemann
• Subject Area: Nuclear and Elementary Particle Physics, Quantum Mechanics, Relativity, Fields ; Mathematics
• Term: from 2015 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 281545347

Random matrices offer a very efficient description for the spectral statistics of operators where noise or fluctuations matter. They have led to deep insights in their applications to physics and mathematics, e.g. into the nature of the sign problem in Quantum Chromodynamics or into the distribution of zeros of the Riemann zeta-function and generalisations thereof. Typically the precise distribution of matrix elements is not important and can be chosen as Gaussian. This fact goes under the name of universality. Powerful tools have been developed to establish rigorous universality proofs for the spectral statistics of single random matrices. Products of several random matrices play an important role for example in the characterisation of chaotic dynamics or in modelling transport in physics. In mathematics they enjoy for instance a profound relation to generalisations of the Catalan number in combinatorics. The exact solvability of arbitrary many products of independent random matrices has been established only very recently in a series of papers with my collaborators. This has opened up the possibility to prove universality for such products, and to analyse the detailed distribution of Lyapunov exponents. However, in applications the matrices that are multiplied are not necessarily independent. In the above-mentioned example of Quantum Chromodynamics the coupling of the two matrices from the chiral block structure through chemical potential has been crucial. In this proposal we thus want to study products of coupled random matrices. Our first main goal will be to establish their exact solvability in terms of determinantal or Pfaffian point processes. For the singular values this will find applications to the microscopic Dirac operator spectrum of quantum field theories with high isospin chemical. At the same time these findings will lead to parameter dependent deformations of the classical Laguerre polynomials and to new universal distributions. Furthermore the exact solution for complex eigenvalues will lead to an enlarged class of non-trivial orthogonal polynomials in the complex plane.

• DFG Programme: Research Grants
• International Connection: Israel, Japan
• Co-Investigators: Professor Dr. Taro Nagao; Professor Dr. Eugene Strahov