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 -