The Harer--Zagier recursion for an irregular spectral curve

TL;DR

This paper derives a recursion formula for the one-loop mean of an irregular spectral curve, expressing it in polynomial form and generalizing it to Laguerre polynomials, advancing spectral curve analysis.

Contribution

It introduces a new derivation of the Do and Norbury recursion for irregular spectral curves and extends it to Laguerre polynomials.

Findings

Recursion formula for irregular spectral curves derived

Expression of recursion in polynomial form for all genus terms

Generalization to Laguerre polynomial case

Abstract

We derive the Do and Norbury recursion formula for the one-loop mean of an irregular spectral curve from a variant of replica method by Brez\'in and Hikami. We express this recursion in special times in which all terms $W_1^{(g)}$ of the genus expansion of the one-loop mean are polynomials. We find a generalization of this recursion to the generalized Laguerre polynomial case.