Exact 2-point function in Hermitian matrix model

TL;DR

This paper derives an exact, elementary formula for 2-point correlators in the Gaussian Hermitian matrix model using Toda integrability, extending previous 1-point results and exploring higher-point generalizations.

Contribution

It provides the first exact formula for 2-point correlators in the model, revealing their elementary arctangent form and connecting to resolvents, with initial steps toward higher-point functions.

Findings

Exact 2-point correlator is an arctangent function.

Generalization of 1-point to 2-point correlators achieved.

Connections to resolvents and potential higher-point extensions.

Abstract

J. Harer and D. Zagier have found a strikingly simple generating function for exact (all-genera) 1-point correlators in the Gaussian Hermitian matrix model. In this paper we generalize their result to 2-point correlators, using Toda integrability of the model. Remarkably, this exact 2-point correlation function turns out to be an elementary function - arctangent. Relation to the standard 2-point resolvents is pointed out. Some attempts of generalization to 3-point and higher functions are described.