# Critical phenomena in complex and real spectra

This joint PhD project will be based at The University of Melbourne with a minimum 12 month stay at KU Leuven

**Project description
**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. The long-standing problem of their relation shall 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 shall 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 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.

The ultimate goal of this project is to find novel universal correlations between eigenvalues and singular values. The student will learn techniques from harmonic analysis on matrix spaces, bi-orthogonal functions and asymptotic analysis.

The project will be complemented by the KU Leuven based project and the collaboration will ensure a successful completion of the project.

**Supervision team:**

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)

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