Olivia Auster

Limit Theorems for Non-Hermitian Ensembles

Academic research in Random Matrix Theory provides a means to analyse problems in high dimensions and perform related calculations analytically. Random matrix ensembles are mathematical concepts used to model and understand the dynamics of physical phenomena from the study of the distribution of the eigenvalues and eigenvalue moduli. However, the literature deals primarily with analytical methods to establish the nature of the limiting distribution of the spectral radius. Thus, little is devoted to the analysis of the distribution of its extreme counterpart, the minimum modulus.
The limiting distribution of the minimum modulus is the main topic that we will discuss for the complex Ginibre ensemble and its generalisation, the complex induced Ginibre ensemble, as the size of the matrices increases to infinity. The study of the left and right tail distributions of this extreme modulus is also of interest. Applying Andreief’s integration formula and the analytical approach established by B. Rider at the edge of the support of the spectrum, we will see that the minimum modulus has a Gumbel distribution for the complex induced Ginibre ensemble and follows a Half-Normal Rayleigh distribution for the complex Ginibre ensemble in the limit of large random matrices.
The independence of the spectral radius and the minimum modulus is a self-evident fact in this limit. I will provide proof of this for the complex induced Ginibre ensemble from the aforementioned formula and analytical framework. Then, I will explain how to use known results on gap probabilities to establish the independence of these extreme moduli for the complex Ginibre ensemble.
This research adds new insight into unexplored areas in the field, where the minimum modulus of certain non-Hermitian ensembles is the centre of attention.