Combinatorics of Even-Valent Graphs on Riemann Surfaces

Abstract: We present a new method for determining the topological expansion of the recurrence coefficients of orthogonal polynomials with weights of the form ( e^{-N\mathscr{V}(z)} ), where ( \mathscr{V}(z) = \frac{z^2}{2} + \frac{u z^{2\nu}}{2\nu} ) with ( u > 0 ) and ( \nu \in \mathbb{N} ). A key feature of our approach is that it is non-recursive in ( \nu ). For fixed choices of vertices, $j$, and genus, $g$, our expansion yields closed-form formulae in $\nu$ which in turn count the number of connected ( 2\nu )-valent labeled graphs (and their 2-legged counterparts) with ( j ) vertices on a compact Riemann surface of genus ( g ). We also outline the steps required to extend the methodology of \cite{BGM} to hexic weights, where ( \mathscr{V}(z) = \frac{z^2}{2} + \frac{u z^6}{6} ). This enables us to derive closed-form formulae in $j$ for the number of connected 6-valent labeled graphs with ( j ) vertices on a compact Riemann surface of genus ( g ), for ( g = 0, 1, ) and ( 2 ). Note the difference to our new method which instead holds for general $\nu$ and fixed $j$.