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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-classicalskew-orthogonal polynomials, in analogue to the approach of Nadal and Majumdar(NM) for the Gaussian unitary ensemble (GUE). Generalizing the techniques ofAdler, Forrester, Nagao and van Moerbeke, and using "overlapping Pfaffian"identities due to Knuth, we explicitly construct these semi-classicalskew-orthogonal polynomials in terms of the semi-classical orthogonalpolynomials studied by NM in the case of the GUE. With these polynomials weobtain expressions for the cumulative distribution functions of the largesteigenvalue in the GOE and the GSE. Further, by performing asymptotic analysisof these skew-orthogonal polynomials in the limit of large matrix size, weobtain an alternative derivation of the Tracy-Widom distributions for GOE andGSE. This asymptotic analysis relies on a certain Pfaffian identity, the proofof which employs the characterization of Pfaffians in terms of perfectmatchings and link diagrams.

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