×

Asymptotics for characteristic polynomials of Wishart type products of complex Gaussian and truncated unitary random matrices. (English) Zbl 1351.30023

The paper studies the asymptotic behavior of the characteristic polynomials associated to Wishart-type random matrices that are formed as products consisting of independent standard complex Gaussians and a truncated Haar distributed unitary random matrix. The multivariate saddle point approach is used for the derivation of the results. The oscillatory behavior on the asymptotic interval of zeros by means of formulae of Plancherel-Rotach type is investigated which is applied to derive the limiting distribution of the rescaled zeros. It is proved that the asymptotic zero distribution lies in the class of Raney distributions.

MSC:

30E15 Asymptotic representations in the complex plane
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
15B52 Random matrices (algebraic aspects)
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Adhikari, K.; Reddy, N.; Reddy, T.; Saha, K., Determinantal point processes in the plane from products of random matrices, Ann. Inst. H. Poincaré Probab. Stat., 52, 16-46 (2016) · Zbl 1331.60014
[2] Akemann, G.; Burda, Z., Universal microscopic correlation functions for products of independent Ginibre matrices, J. Phys. A, 45, Article 465201 pp. (2012) · Zbl 1261.15041
[3] Akemann, G.; Ipsen, J.; Kieburg, M., Products of rectangular random matrices: singular values and progressive scattering, Phys. Rev. E, 88, 5, 052118 (2013)
[4] Burda, Z.; Janik, R.; Waclaw, B., Spectrum of the product of independent random Gaussian matrices, Phys. Rev. E, 81, Article 041132 pp. (2010)
[5] Fedoryuk, M., Saddle-point Method (1977), Nauka: Nauka Moscow, (Russian) · Zbl 0463.41020
[6] Forrester, P., Eigenvalue statistics for product complex Wishart matrices, J. Phys. A, 47, Article 345202 pp. (2014) · Zbl 1309.15053
[8] Forrester, P.; Liu, D., Raney distributions and random matrix theory, J. Stat. Phys., 158, 5, 1051-1082 (2015) · Zbl 1352.60024
[10] Hardy, A., Average characteristic polynomials of determinantal point processes, Ann. Inst. H. Poincaré Probab. Stat., 51, 283-303 (2015) · Zbl 1332.60023
[11] Kieburg, M.; Kuijlaars, A.; Stivigny, D., Singular value statistics of matrix products with truncated unitary matrices, Int. Math. Res. Not. IMRN (2015), in press
[12] Kuijlaars, A.; Stivigny, D., Singular values of products of random matrices and polynomial ensembles, Random Matrices: Theory Appl., 3, 1450011 (2014) · Zbl 1303.15045
[13] Kuijlaars, A.; Zhang, L., Singular values of products of ginibre random matrices, multiple orthogonal polynomials and hard edge scaling limits, Comm. Math. Phys., 332, 759-781 (2014) · Zbl 1303.15046
[14] Mlotkowski, W., Fuss-Catalan numbers in noncommutative probability, Doc. Math., 15, 939-955 (2010) · Zbl 1213.44004
[15] Mlotkowski, W.; Penson, K.; Zyczkowski, K., Densities of the raney distributions, Doc. Math., 18, 1573-1596 (2013) · Zbl 1291.44016
[16] Neuschel, T., Apéry polynomials and the multivariate saddle point method, Constr. Approx., 40, 487-507 (2014) · Zbl 1304.30050
[17] Neuschel, T., Plancherel-Rotach formulae for average characteristic polynomials of products of Ginibre random matrices and the Fuss-Catalan distribution, Random Matrices: Theory Appl., 3, 1450003 (2014) · Zbl 1288.30033
[19] Penson, K.; Zyczkowski, K., Product of Ginibre matrices: /Fuss-Catalan and Raney distributions, Phys. Rev. E, 83, 6, Article 061118 pp. (2011)
[20] Pólya, G.; Szegő, G., Problems and Theorems in Analysis, II (1976), Springer · Zbl 0311.00002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.