Critical phenomena in complex and real spectra

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

Project description
Non-Hermitian matrices have their eigenvalues in the complex plane. For random non-Hermitian matrices the typical behavior is that the complex eigenvalues behave like 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 doctoral project studies deformations which lead to more complicated droplets. The average characteristic polynomial in these models will be a polynomial with an orthogonality in the complex plane. The zeros of these polynomials typically accumulate along certain contours within the droplet, called a motherbody.

The aim of the project is to describe the topology of the droplets and their motherbodies, their evolution in terms of parameters and to study phase transitions. The goal is to analyse certain models in great detail with tools from integrable probability and asymptotic analysis.

The project will be complemented by The University of Melbourne based project and the collaboration will ensure a successful completion of the project

Supervision team:

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)