Journal article

Integrable structure of products of finite complex Ginibre random matrices

Vladimir V Mangazeev, Peter J Forrester



We consider the squared singular values of the product of [Formula presented] standard complex Gaussian matrices. Since the squared singular values form a determinantal point process with a particular Meijer G-function kernel, the gap probabilities are given by a Fredholm determinant based on this kernel. It was shown by Strahov (2014) that a hard edge scaling limit of the gap probabilities is described by Hamiltonian differential equations which can be formulated as an isomonodromic deformation system similar to the theory of the Kyoto school. We generalize this result to the case of finite matrices by first finding a representation of the finite kernel in integrable form. As a result we ob..

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University of Melbourne Researchers


Awarded by Australian Research Council

Awarded by ARC Centre of Excellence for Mathematical and Statistical Frontiers

Funding Acknowledgements

We would like to thank V. Bazhanov and J.R. Ipsen for useful discussions and N. Witte for careful reading of the manuscript and his comments. We acknowledge support by the Australian Research Council through grant DP140102613 (PJF, VVM) and the ARC Centre of Excellence for Mathematical and Statistical Frontiers through grant CE140100049 (PJF).