Toby Shepherd

Random Matrix Ensembles With an Edge Spectrum Singularity

The study of the distribution of the largest eigenvalue in random matrix ensembles begun with the seminal work of Tracy-Widom who derived the relation between a Painlevé-II equation and the distribution of the largest eigenvalue in the GUE. This work was extended by Forrester-Witte who derived an ensemble that related to the general Painlevé-II equation. In my first paper with Thomas, we showed that this behaviour is universal in a wider class of RMT ensembles. In ongoing work, I am now looking at generalising the work of Forrester-Witte to an ensemble with the same weight, instead sampling over complex matrices (i.e. Ginibre Ensemble with an edge singularity) in an attempt to derive a similar law that generalises the Gumbel distribution. In this talk, I will briefly review the results of our first paper, and present the methods I am now using in this extension and explain how they differ from the Hermitian case. Time permitting, I will mention how this problem can be motivated in electrostatics!