Transition between Hermitian and non-Hermitian Gaussian ensembles

O. Bohigas 1 and M. P. Pato 2

J. Phys. A: Math. Theor. 46 (2013) 115001 (11pp)

The transition between Hermitian and non-Hermitian matrices of the Gaussian unitary ensemble is revisited. An expression for the kernel of the rescaled Hermite polynomials is derived which expresses the sum in terms of the highest order polynomials. From this Christoffel–Darboux-like formula some results are derived including an extension to the complex plane of the Airy kernel.

  • 1 CNRS, Universite Paris-Sud, UMR8626, LPTMS, Orsay Cedex, F-91405, France
  • 2 Instıtuto de F ́ısica, Universidade de Sao Paulo, Caixa Postal 66318, 05314-970 Sao Paulo, S.P., Brazil