Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges |
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| Institution: | 1. Department of Mathematics, University of Melbourne, Parkville, Victoria 3052, Australia;2. Department of Physics, Faculty of Science, Osaka University, Toyonaka, Osaka 560, Japan;1. Department of Mathematics, Southwestern University of Finance and Economics, Chengdu, 611130, PR China;2. Department of Mathematics, University of Central Florida, Orlando, FL 32816, United States;1. Department of Mathematics and Mechanics, St. Petersburg State University, St. Petersburg, Russia;2. National Research University Higher School of Economics, St. Petersburg, Russia;3. St. Petersburg State University, St. Petersburg, Russia;4. I2M, CNRS, Centrale Marseille, Aix-Marseille Université, 13453 Marseille, France;1. Department of Mathematics, Tongji University, Shanghai 200092, China;2. Graduate School of Mathematics, Nagoya University, Nagoya 464-8602, Japan;1. Department of Liberal Arts, Faculty of Science and Technology, Tokyo University of Science, 2641 Yamazaki, Noda-shi, Chiba-ken, 278-8510, Japan;2. Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87, 61-614 Poznań, Poland;3. Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan |
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| Abstract: | For the orthogonal-unitary and symplectic-unitary transitions in random matrix theory, the general parameter dependent distribution between two sets of eigenvalues with two different parameter values can be expressed as a quaternion determinant. For the parameter dependent Gaussian and Laguerre ensembles the matrix elements of the determinant are expressed in terms of corresponding skew-orthogonal polynomials, and their limiting value for infinite matrix dimension are computed in the vicinity of the soft and hard edges respectively. A connection formula relating the distributions at the hard and soft edge is obtained, and a universal asymptotic behaviour of the two point correlation is identified. |
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