Tracy-Widom distributions for the Gaussian orthogonal and symplectic ensembles revisited: a skew-orthogonal polynomials approach

TL;DR

This paper revisits the Tracy-Widom distributions for GOE and GSE ensembles in random matrix theory, using skew-orthogonal polynomials to derive the distributions of the largest eigenvalue.

Contribution

It introduces a skew-orthogonal polynomial approach to derive Tracy-Widom distributions for GOE and GSE, extending methods used for GUE.

Findings

Explicit construction of semi-classical skew-orthogonal polynomials for GOE and GSE.

Derivation of the cumulative distribution functions for the largest eigenvalue.

Asymptotic analysis confirming Tracy-Widom distributions for large matrices.

Abstract

We study the distribution of the largest eigenvalue in the "Pfaffian" classical ensembles of random matrix theory, namely in the Gaussian orthogonal (GOE) and Gaussian symplectic (GSE) ensembles, using semi-classical skew-orthogonal polynomials, in analogue to the approach of Nadal and Majumdar (NM) for the Gaussian unitary ensemble (GUE). Generalizing the techniques of Adler, Forrester, Nagao and van Moerbeke, and using "overlapping Pfaffian" identities due to Knuth, we explicitly construct these semi-classical skew-orthogonal polynomials in terms of the semi-classical orthogonal polynomials studied by NM in the case of the GUE. With these polynomials we obtain expressions for the cumulative distribution functions of the largest eigenvalue in the GOE and the GSE. Further, by performing asymptotic analysis of these skew-orthogonal polynomials in the limit of large matrix size, we obtain an alternative derivation of the Tracy-Widom distributions for GOE and GSE. This asymptotic analysis relies on a certain Pfaffian identity, the proof of which employs the characterization of Pfaffians in terms of perfect matchings and link diagrams.