From Brezin-Hikami to Harer-Zagier formulas for Gaussian correlators

TL;DR

This paper derives explicit formulas for Harer-Zagier multidensities in Gaussian correlators using Brezin-Hikami contour integrals, connecting classical matrix models with modern algebraic geometry and integrable systems.

Contribution

It introduces a method to explicitly compute all-genera Gaussian correlators via Brezin-Hikami representation, extending known results to higher-point functions.

Findings

Rederived the 1-point Harer-Zagier function.

Rederived the 2-point arctangent function.

Presented an explicit, though unproven, expression for the 3-point function.

Abstract

Brezin-Hikami contour-integral representation of exponential multidensities in finite N Hermitian matrix model is a remarkable implication of the old Hermitian-Kontsevich duality. It is also a simplified version of Okounkov's formulas for the same multidensities in the cubic Kontsevich model and of Nekrasov calculus for LMNS integrals, a central piece of the modern studies of AGT relations. In this paper we use Brezin-Hikami representation to derive explicit expressions for the Harer-Zagier multidensities (from arXiv:0906.0036): the only known exhaustive generating functions of all-genera Gaussian correlators which are fully calculable and expressed in terms of elementary functions. Using the Brezin-Hikami contour integrals, we rederive the 1-point function of Harer and Zagier and the 2-point arctangent function of arXiv:0906.0036. We also present (without a proof) the explicit expression for the 3-point function in terms of arctangents. Derivation of the 3-point and higher Harer-Zagier functions remains a challenging problem.