It is known since the seminal paper [1] that correlators of the Gaussian Unitary Ensemble (GUE) are related to the counting problem of so called ribbon graphs; however, their explicit computation was given only a few years ago by Dubrovin and Yang [2]. Recently [3], correlators of the Laguerre Unitary Ensemble (LUE) have been linked to Hurwitz numbers.

In this seminar we will first review the connection between GUE correlators and ribbon graphs and LUE and Hurwitz numbers. We then show how a generating function for these objects, as well as for the LUE ones, can be obtained starting from the Riemann-Hilbert problem of the associated orthogonal polynomials.

This is a joint work with T.Grava and G.Ruzza.

[1] D. Bessis, C. Itzykson, J.B. Zuber. Quantum field theory techniques in graphical enumeration. Adv. Appl. Math. 1(2), 109–157 (1980)

[2] B. Dubrovin and D. Yang. Generating series for GUE correlators. Lett. Math. Phys. 107(11), 1971–2012, (2017)

[3] F.D. Cunden, A. Dahlqvist, N. O’Connell. Integer moments of complex Wishart matrices and Hurwitz numbers. arXiv:1809.10033