# 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 finished 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 $n\rightarrow \infty$.

Finished results in:"On the determinantal structure of conditional overlap for the complex Ginibre ensemble"

Currently working on conditioning on determinantal point processes. We know that by conditioning on having a particle on a certain point, the resulting process is also determinantal. For the discrete case I have an exact formula for the kernel of the new point process we get from conditioning on the original one having (or not having) particles on any finite number of points. Now I'm trying to extend this in the continuous case by considering Palm measures, hoping I will be able to also get some results on Rigidity.

**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