Pfaffian Expressions for Random Matrix Correlation Functions

TL;DR

This paper reviews Pfaffian formulas for eigenvalue correlations in real and quaternion random matrices, highlighting their derivation, relation to determinant formulas, and applications in random matrix theory.

Contribution

It provides a comprehensive overview of Pfaffian expressions, connecting them to skew orthogonal polynomials and classical orthogonal polynomial cases, and discusses their applications.

Findings

Pfaffian formulas effectively describe eigenvalue correlations.

Relation established between Pfaffian and determinant formulas.

Applications extend to various fields beyond random matrix theory.

Abstract

It is well known that Pfaffian formulas for eigenvalue correlations are useful in the analysis of real and quaternion random matrices. Moreover the parametric correlations in the crossover to complex random matrices are evaluated in the forms of Pfaffians. In this article, we review the formulations and applications of Pfaffian formulas. For that purpose, we first present the general Pfaffian expressions in terms of the corresponding skew orthogonal polynomials. Then we clarify the relation to Eynard and Mehta's determinant formula for hermitian matrix models and explain how the evaluation is simplified in the cases related to the classical orthogonal polynomials. Applications of Pfaffian formulas to random matrix theory and other fields are also mentioned.