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Hole Probabilities and Overcrowding Estimates for Products of Complex Gaussian Matrices

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Abstract

We consider eigenvalues of a product of n non-Hermitian, independent random matrices. Each matrix in this product is of size N×N with independent standard complex Gaussian variables. The eigenvalues of such a product form a determinantal point process on the complex plane (Akemann and Burda in J. Phys. A, Math. Theor. 45:465201, 2011), which can be understood as a generalization of the finite Ginibre ensemble. As N→∞, a generalized infinite Ginibre ensemble arises. We show that the set of absolute values of the points of this determinantal process has the same distribution as \(\{R_{1}^{(n)},R_{2}^{(n)},\ldots\}\), where \(R_{k}^{(n)}\) are independent, and \((R_{k}^{(n)} )^{2}\) is distributed as the product of n independent Gamma variables Gamma(k,1). This enables us to find the asymptotics for the hole probabilities, i.e. for the probabilities of the events that there are no points of the process in a disc of radius r with its center at 0, as r→∞. In addition, we solve the relevant overcrowding problem: we derive an asymptotic formula for the probability that there are more than m points of the process in a fixed disk of radius r with its center at 0, as m→∞.

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Notes

  1. For a background on determinantal point processes we refer the reader to survey articles by Borodin [4], and by Hough, Krishnapur, Peres, and Virág [21].

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Acknowledgements

Part of this research was conducted during ZIF research program “Stochastic Dynamics: Mathematical Theory and Applications”. It is our pleasure to thank the Center for Interdisciplinary Research (ZIF) of Bielefeld University for hospitality, and the organizers of the ZIF Research Group 2012 “Stochastic Dynamics: Mathematical Theory and Applications” for the stimulating and encouraging environment they created at the program.

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Correspondence to Eugene Strahov.

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The first author (G.A.) is partly supported by the SFB|TR12 “Symmetries and Universality in Mesoscopic Systems” of the German research council DFG. The second author (E.S.) is supported in part by the US-Israel Binational Science Foundation (BSF) Grant No. 2006333, and by the Israel Science Foundation (ISF) Grant No. 1441/08.

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Akemann, G., Strahov, E. Hole Probabilities and Overcrowding Estimates for Products of Complex Gaussian Matrices. J Stat Phys 151, 987–1003 (2013). https://doi.org/10.1007/s10955-013-0750-8

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