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Singular values of products of Ginibre random matrices, multiple orthogonal polynomials and hard edge scaling limits
- Source :
- Communications in Mathematical Physics 332 (2014), 759--781
- Publication Year :
- 2013
-
Abstract
- Akemann, Ipsen and Kieburg recently showed that the squared singular values of products of M rectangular random matrices with independent complex Gaussian entries are distributed according to a determinantal point process with a correlation kernel that can be expressed in terms of Meijer G-functions. We show that this point process can be interpreted as a multiple orthogonal polynomial ensemble. We give integral representations for the relevant multiple orthogonal polynomials and a new double contour integral for the correlation kernel, which allows us to find its scaling limits at the origin (hard edge). The limiting kernels generalize the classical Bessel kernels. For M=2 they coincide with the scaling limits found by Bertola, Gekhtman, and Szmigielski in the Cauchy-Laguerre two-matrix model, which indicates that these kernels represent a new universality class in random matrix theory.<br />Comment: 30 pages, 1 figure. Revised version with the first three paragraphs completely rewritten, typos corrected, references updated. To appear in Communications in Mathematical Physics
Details
- Database :
- arXiv
- Journal :
- Communications in Mathematical Physics 332 (2014), 759--781
- Publication Type :
- Report
- Accession number :
- edsarx.1308.1003
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1007/s00220-014-2064-3