## 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 |
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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