Tracy-Widom distributions for the Gaussian orthogonal and symplectic ensembles revisited: a skew-orthogonal polynomials approach. (English) Zbl 1469.60034
Summary: 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 analogy with the approach of C. Nadal and S. N. Majumdar [J. Stat. Mech. Theory Exp. 2011, No. 4, Paper No. P04001, 29 p. (2011; Zbl 1456.60025)] (NM) for the Gaussian unitary ensemble (GUE). Generalizing the techniques of M. Adler et al. [J. Stat. Phys. 99, No. 1–2, 141–170 (2000; Zbl 0989.82020)], and using “overlapping Pfaffian” or “compound Pfaffian” identities, 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.
MSC:
| 60B20 | Random matrices (probabilistic aspects) |
| 15B52 | Random matrices (algebraic aspects) |
| 60G70 | Extreme value theory; extremal stochastic processes |
| 33C47 | Other special orthogonal polynomials and functions |
Keywords:
random matrices; extreme value statistics; Tracy-Widom distributions; skew-orthogonal polynomialsReferences:
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