TY - JOUR

T1 - Dynamical correlation functions for products of random matrices

AU - Strahov, Eugene

N1 - Publisher Copyright:
© 2015 World Scientific Publishing Company.

PY - 2015/10/1

Y1 - 2015/10/1

N2 - 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.

AB - 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.

KW - Products of random matrices

KW - determinantal point processes

UR - http://www.scopus.com/inward/record.url?scp=85030167846&partnerID=8YFLogxK

U2 - 10.1142/S2010326315500203

DO - 10.1142/S2010326315500203

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AN - SCOPUS:85030167846

SN - 2010-3263

VL - 4

JO - Random Matrices: Theory and Application

JF - Random Matrices: Theory and Application

IS - 4

M1 - 1550020

ER -