Quantum Lunch: Marchenko--Pastur distributions and hypergeometric Hurwitz numbers

Speaker: Guest Professor Leonid Chekov from NBI

Marchenko--Pastur distributions and hypergeometric Hurwitz numbers

(based on joint papers with Jan Ambjorn, arXiv:1404.4240 and 1409.3553)

I will show how generalisations of the Marchenko--Pastur distribution (the one corresponding to the Gaussian measure on complex rectangular matrices of size n x m) naturally arise in problems of counting hypergeometric Hurwitz numbers enumerating homotopy types of maps $CP^1\to \Sigma_g$ (coverings of the complex plane by genus-g Riemann surfaces) ramified over $r\ge 3$ points on $CP^1$ (the case $r=3$ corresponds to Grothendieck's dessins d'enfant). Generating functions for these numbers are given in general by chains of $(r-2)$-Hermitian matrices with a new interaction type between neighbour matrices in the chain. We were able to obtain the spectral curve of such a model (for $g=0$), which is a basic ingredient of the topological recursion method enabling constructing the corresponding numbers for all $g$. I will shortly mention possible relations to quantum information theory.