Edgeworth Expansion of the Largest Eigenvalue Distribution Function of GUE and LUE

Details Edgeworth Expansion of the Largest Eigenvalue Distribution Function of GUE and LUE Edgeworth Expansion of the Largest Eigenvalue Distribution Function of GUE and LUE

2006-03-28

arxiv.org/abs/math/0603639v2

IMRN Volume 2006, ID 61049, Pages 1-33.

Mathematics

Probability

Scientific paper

We derive expansions of the Hermite and Laguerre kernels at the edge of the spectrum of the finite n Gaussian Unitary Ensemble (GUEn) and the finite n Laguerre Unitary Ensem- ble (LUEn), respectively. Using these large n kernel expansions, we prove an Edgeworth type theorem for the largest eigenvalue distribution function of GUEn and LUEn. In our Edgeworth expansion, the correction terms are expressed in terms of the same Painleve II function appearing in the leading term, i.e. in the Tracy-Widom distribution. We conclude with a brief discussion of the universality of these results.

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