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

Journal Article · Fri Oct 15 00:00:00 EDT 2010 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.3496899· OSTI ID:21483628

Akemann, G ^[1]; Bender, M ^[2]

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1. Department of Mathematical Sciences and BURSt Research Centre, Brunel University West London, Uxbridge UB8 3PH (United Kingdom)
2. Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, 3001 Leuven (Belgium)

We consider a family of chiral non-Hermitian Gaussian random matrices in the unitarily invariant symmetry class. The eigenvalue distribution in this model is expressed in terms of Laguerre polynomials in the complex plane. These are orthogonal with respect to a non-Gaussian weight including a modified Bessel function of the second kind, and we give an elementary proof for this. In the large n limit, the eigenvalue statistics at the spectral edge close to the real axis are described by the same family of kernels interpolating between Airy and Poisson that was recently found by one of the authors for the elliptic Ginibre ensemble. We conclude that this scaling limit is universal, appearing for two different non-Hermitian random matrix ensembles with unitary symmetry. As a second result we give an equivalent form for the interpolating Airy kernel in terms of a single real integral, similar to representations for the asymptotic kernel in the bulk and at the hard edge of the spectrum. This makes its structure as a one-parameter deformation of the Airy kernel more transparent.

• OSTI ID: 21483628
• Journal Information: Journal of Mathematical Physics, Vol. 51, Issue 10; Other Information: DOI: 10.1063/1.3496899; (c) 2010 American Institute of Physics; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ASYMPTOTIC SOLUTIONS
BESSEL FUNCTIONS
CHIRAL SYMMETRY
CHIRALITY
EIGENFUNCTIONS
EIGENVALUES
INTERPOLATION
KERNELS
LAGUERRE POLYNOMIALS
MATRICES
RANDOMNESS
STATISTICS
UNITARY SYMMETRY
FUNCTIONS
MATHEMATICAL SOLUTIONS
MATHEMATICS
NUMERICAL SOLUTION
PARTICLE PROPERTIES
POLYNOMIALS
SYMMETRY