Abstract
We introduce and study a family of random processes with a discrete time related to products of random matrices. Such processes are formed by singular values of random matrix products, and the number of factors in a random matrix product plays a role of a discrete time. We consider in detail the case when the (squared) singular values of the initial random matrix form a polynomial ensemble, and the initial random matrix is multiplied by standard complex Gaussian matrices. In this case, we show that the random process is a discrete-time determinantal point process. For three special cases (the case when the initial random matrix is a standard complex Gaussian matrix, the case when it is a truncated unitary matrix, or the case when it is a standard complex Gaussian matrix with a source) we compute the dynamical correlation functions explicitly, and find the hard edge scaling limits of the correlation kernels. The proofs rely on the Eynard-Mehta theorem, and on contour integral representations for the correlation kernels suitable for an asymptotic analysis.
Original language | American English |
---|---|
Article number | 1550020 |
Journal | Random Matrices: Theory and Application |
Volume | 4 |
Issue number | 4 |
DOIs | |
State | Published - 1 Oct 2015 |
Bibliographical note
Publisher Copyright:© 2015 World Scientific Publishing Company.
Keywords
- Products of random matrices
- determinantal point processes