Extreme value statistics of random matrices and log-correlated fields

I plan to study the characteristic polynomial of random matrices from the compact classical groups. It has drawn a lot of interest lately thanks to the Fyodorov-Hiary-Keating conjecture.

Very recently the scaled asymptotic distribution of its maximum has been shown to equal a sum of a Gumbel and another independent random variable.

A (difficult) question is if we can obtain the convergence to a sum of two independent Gumbels using its Fourier decomposition. This would prove the entire Fyodorov-Hiary-Keating conjecture related to extreme values of random matrix theory.

The second project is to obtain the first and second order asymptotics, but for a random matrix model evolving in time, called the Dyson Brownian Motion.

The last project is to study the orthogonal Dyson Brownian Motion, and obtain convergence to a two-dimensional Gaussian Multiplicative Chaos measure.These projects will require a good understanding of the compact classical groups, especially the trace of the powers, and of techniques of proof which are specific to these matrices (e.g. the Basor-Ehrhardt identities) which I have acquired while working on my first two papers.

They will also rest on fundamental techniques from extreme value theory and dynamical models which I have encountered in various papers and workshops during my Ph.D. studies, and which are a field of expertise of my intended supervisor, Professor Paul Bourgade.