A~matrix model for hypergeometric Hurwitz numbers

J. Ambjørn ; L. O. Chekhov

Teoretičeskaâ i matematičeskaâ fizika, Tome 181 (2014) no. 3, pp. 421-435

Résumé

We present multimatrix models that are generating functions for the numbers of branched covers of the complex projective line ramified over $n$ fixed points $z_i$, $i=1,\dots,n$ (generalized Grothendieck's dessins d'enfants) of fixed genus, degree, and ramification profiles at two points $z_1$ and $z_n$. We sum over all possible ramifications at the other $n-2$ points with a fixed length of the profile at $z_2$ and with a fixed total length of profiles at the remaining $n-3$ points. All these models belong to a class of hypergeometric Hurwitz models and are therefore tau functions of the Kadomtsev–Petviashvili hierarchy. In this case, we can represent the obtained model as a chain of matrices with a (nonstandard) nearest-neighbor interaction of the type $\operatorname{tr} M_iM_{i+1}^{-1}$. We describe the technique for evaluating spectral curves of such models, which opens the way for obtaining $1/N^2$-expansions of these models using the topological recursion method. These spectral curves turn out to be algebraic.

Détail

Keywords: Hurwitz number, random complex matrix, Kadomtsev–Petviashvili hierarchy, spectral curve.
Mots-clés : matrix chain, bipartite graph