Семинары под руководством М.Э.Казаряна и С.К.Ландо : А.Ю.Орлов

Hurwitz numbers and matrix integrals labeled with chord diagrams

We consider products of complex random matrices from independent complex Ginibre ensembles. The products include complex random matrices Z_i, Z _i ^†, i = 1, . . ., n, and 2n sources (these are the complex matrices C_i, C _i^∗, i = 1, . . ., n, which play the role of parameters). We consider collections of products X₁, . . ., X_F , constrained by the property, that each of the matrices of the set {Z_iC_i, Z _i ^† C _i^∗, i = 1, . . ., n} is included only once on the product X = X₁ · · · X_F . It can be represented graphically as a collection of F polygons with a total number of edges 2n, and the polygon with number a encodes the order of the matrices in X_a. The matrices Z_i and Z _i ^† are distributed along the edges of this collection of polygons, and the sources are distributed at their vertices. The calculation of the expected values involves pairing the matrices Z_i and Z _i ^†. There is a standard procedure for constructing a 2D surface by pairwise gluing edges of polygons, this procedure results to a ribbon graph embedded in the surface Σ_e∗ of some Euler characteristic e∗ (this graph also known as embedded graph or fatgraph). We propose a matrix model that generates spectral correlation functions for matrices X_a, a = 1, . . ., F in the Ginibre ensembles, which we call the matrix integral, labeled network chord diagram. We show that the spectral correlation functions generate Hurwitz numbers H_e∗ that enumerate nonequivalent branched coverings of Σ_e∗. The role of sources is the generation of ramification profiles in branch points which are assigned to the vertices of the ribbon graph drawn on the base surface Σ_e∗. The role of coupling constants of our model is to generate ramification profiles in F additional branch points assigned to the faces of the ribbon graph (the faces of the ’triangulated’ Σ_e∗). The Hurwitz numbers for Klein surfaces can also be obtained by a small modification of the model. To do this, we pair any of the source matrices (in that case presenting a hole on Σ_e∗) with the tau function, which we call Mobius one. The presented matrix models generate Hurwitz numbers for any given Euler characteristic of the base surface e∗ and for any given set of ramification profiles.