# Athanasios Tsareas

I am currently a PhD student working with Dr. Roger Tribe and Dr. Oleg Zaboronski on "real and complex Ginibre random matrix ensembles" and "determinantal point processes and rigidity".

We are currently finishing our work on studying the diagonal overlaps $O_{11}$ and the off-diagonal overlaps $O_{12}$ off the complex Ginibre. Given a matrix with eigenvalues $\lambda_i$ and their respective left and right eigenvectors $v_i,u_i$, those objects are defined to be

$$O_{11}(z)=<u_1,\bar u_1><v_1,\bar v_1>$$

and $$O_{12}(z_1,z_2)=<u_1,\bar u_2><v_1,\bar v_2>$$

The diagonal overlaps $O_{11}$ formula is a well know result by Chalker and Mehlig, whose work we are trying to expand.

We have focused on $O_{12}$ and $D^{(k)}_{12}=E(O_{12}|\lambda_1,\bar \lambda_1,...,\lambda_k,\bar \lambda_k)$ which have a determinantal structure with kernel $K(x,\bar y)$.

We have computed an exact formula for the kernel and have studied the asymptotics in the bulk and edge as the size of the matrix $n\rightarrow \infty$, and we are checking our results.

**Academic Background **

**Undergraduate degree** at the National and Kapodistrian University of Athens

**Postgraduate degree** (MSc) at the University of Warwick

Modules taken

- Stochastic Analysis
- Brownian Motion
- Fourier Analysis
- Ergodic Theory
- Advanced PDEs
- Advanced Real Analysis
- Complex Analysis

- Asymptotic Methods
- Reading module on Multiscale Methods

s.tsareas@warwick.ac.uk