Abstract
We study asymptotics of orthogonal polynomial measures of the form |HN| 2 d∞ where HN are real or complex Hermite polynomials with re- spect to the Gaussian measure ∞. By means of dierential equations on Laplace transforms, interpolation between the (real) arcsine law and the (complex) uniform distribution on the circle is emphasized. Suitable aver- ages by an independent uniform law give rise to the limiting semi-circular and circular laws of Hermitian and non-Hermitian Gaussian random matrix models. The intermediate regime between strong and weak non-Hermiticity is clearly identified on the limiting dierential equation by means of an addi- tional normal variable in the vertical direction.
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