Complex Hermite polynomials: from the semi-circular law to the circular law | Zendy

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Complex Hermite polynomials: from the semi-circular law to the circular law

Author(s) -

Michel Ledoux

Publication year - 2008

Publication title -

communications on stochastic analysis

Language(s) - English

Resource type - Journals

eISSN - 2688-6669

pISSN - 0973-9599

DOI - 10.31390/cosa.2.1.03

Subject(s) - mathematics , hermite polynomials , hermitian matrix , law , orthogonal polynomials , circular law , hermite interpolation , mathematical analysis , polynomial , gaussian , differential equation , hermitian function , random variable , pure mathematics , physics , quantum mechanics , multivariate random variable , political science , statistics , sum of normally distributed random variables

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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