# Interpolation between Airy and Poisson statistics for unitary chiral non-Hermitian random matrix ensembles

Akemann G, Bender M (2010) *J. Math. Phys.* 51(10): 103524.

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Akemann, Gernot

^{UniBi}; Bender, MartinAbstract

We consider a family of chiral non-Hermitian Gaussian random matrices in theunitarily invariant symmetry class. The eigenvalue distribution in this modelis expressed in terms of Laguerre polynomials in the complex plane. These areorthogonal with respect to a non-Gaussian weight including a modified Besselfunction of the second kind, and we give an elementary proof for this. In thelarge $n$ limit, the eigenvalue statistics at the spectral edge close to thereal axis are described by the same family of kernels interpolating betweenAiry and Poisson that was recently found by one of the authors for the ellipticGinibre ensemble. We conclude that this scaling limit is universal, appearingfor two different non-Hermitian random matrix ensembles with unitary symmetry.As a second result we give an equivalent form for the interpolating Airy kernelin terms of a single real integral, similar to representations for theasymptotic kernel in the bulk and at the hard edge of the spectrum. This makesits structure as a one-parameter deformation of the Airy kernel moretransparent.

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Akemann G, Bender M. Interpolation between Airy and Poisson statistics for unitary chiral non-Hermitian random matrix ensembles.

*J. Math. Phys.*2010;51(10):103524.Akemann, G., & Bender, M. (2010). Interpolation between Airy and Poisson statistics for unitary chiral non-Hermitian random matrix ensembles.

*J. Math. Phys.*,*51*(10), 103524. doi:10.1063/1.3496899Akemann, G., and Bender, M. (2010). Interpolation between Airy and Poisson statistics for unitary chiral non-Hermitian random matrix ensembles.

*J. Math. Phys.*51, 103524.Akemann, G., & Bender, M., 2010. Interpolation between Airy and Poisson statistics for unitary chiral non-Hermitian random matrix ensembles.

*J. Math. Phys.*, 51(10), p 103524. G. Akemann and M. Bender, “Interpolation between Airy and Poisson statistics for unitary chiral non-Hermitian random matrix ensembles”,

*J. Math. Phys.*, vol. 51, 2010, pp. 103524. Akemann, G., Bender, M.: Interpolation between Airy and Poisson statistics for unitary chiral non-Hermitian random matrix ensembles. J. Math. Phys. 51, 103524 (2010).

Akemann, Gernot, and Bender, Martin. “Interpolation between Airy and Poisson statistics for unitary chiral non-Hermitian random matrix ensembles”.

*J. Math. Phys.*51.10 (2010): 103524.
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