Critical phenomena in complex and real spectra

4 minute read

Symbolic representation of maths

This research is comprised of two distinct, but related projects. KU Leuven is the home institution for one project and the University of Melbourne will host the second.

KU Leuven-based project:

Non-Hermitian matrices have their eigenvalues in the complex plane. For random non-Hermitian matrices, the typical behaviour is that the complex eigenvalues behave as mutually repelling charged particles, like electrons in a trap that accumulate on a region in the complex plane, known as the droplet. In the simplest cases, the droplet is a disk.

The research project studies deformations that lead to more complicated droplets. The average characteristic polynomial in these models will be a polynomial with orthogonality in the complex plane. The zeros of these polynomials typically accumulate along certain contours within the droplet, called a motherbody.

Project aims:

  • To describe the topology of the droplets and their motherbodies, their evolution in terms of parameters and to study phase transitions.
  • Analyse models in great detail with tools from integrable probability and asymptotic analysis

Graduate researcher: Sampad Lahiry

Supervision team (Leuven)

Principal Investigators (PIs):

Professor Dr Arno Kuijlaars (KU Leuven)

Dr Mario Kieburg (The University of Melbourne)

Co-Principal Investigators (Co-PIs):

Professor Peter Forrester (The University of Melbourne)

Melbourne-based project:

Non-Hermitan linear operators describe a multitude of systems like open scattering systems in physics or the signal transmission in wireless telecommunications. Such operators can be characterised either by their eigenvalues or their singular values.

In this project, the long-standing problem of their relation will be addressed for a specific class of random matrices with the goal to revealing universal correlations between these two sets of quantities.

We aim for the computation of the joint level densities of the eigenvalues and singular values at finite matrix dimension. A well-known relation, called the Haagerup-Larson theorem, for infinite dimensional matrices shall be re-derived and its corrections for finite matrix dimensions will be quantified. Furthermore, two simple deformations of the probability density of the random matrix will be investigated in this context, too.

The first deformation introduces holes in the complex spectrum and makes contact with the Leuven-based project. The second kind of deformation squeezes the originally isotropic spectrum to an elongated shape. Both kinds of deformations are encountered in quantum field theoretical applications.

Project aims:

  • The ultimate goal of this project is to find novel universal correlations between eigenvalues and singular values.
  • To achieve this, the graduate researcher will apply techniques from harmonic analysis on matrix spaces, bi-orthogonal functions and asymptotic analysis

Graduate researcher: Matthias Allard

Supervision team (Melbourne)

Principal Investigators (PIs):

Dr Mario Kieburg (The University of Melbourne)

Professor Dr Arno Kuijlaars (KU Leuven)

Co-Principal Investigators (Co-PIs):

Professor Peter Forrester (The University of Melbourne)

Other joint PhD projects