# Hole probabilities and overcrowding estimates for products of complex Gaussian matrices

Akemann G, Strahov E (2013)

Journal of Statistical Physics 151(6): 987-1003.

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Akemann, Gernot

^{UniBi}; Strahov, EugeneAbstract

We consider eigenvalues of a product of n non-Hermitian, independent randommatrices. Each matrix in this product is of size N\times N with independentstandard complex Gaussian variables. The eigenvalues of such a product form adeterminantal point process on the complex plane (Akemann and Burda J. Phys A:Math. Theor. 45 (2012) 465201), which can be understood as a generalization ofthe finite Ginibre ensemble. As N\rightarrow\infty, a generalized infiniteGinibre ensemble arises. We show that the set of absolute values of the pointsof this determinantal process has the same distribution as{R_1^{(n)},R_2^{(n)},...}, where R_k^{(n)} are independent, and (R_k^{(n)})^2is distributed as the product of n independent Gamma variables Gamma(k,1). Thisenables us to find the asymptotics for the hole probabilities, i.e. for theprobabilities of the events that there are no points of the process in a discof radius r with its center at 0, as r\rightarrow\infty. In addition, we solvethe relevant overcrowding problem: we derive an asymptotic formula for theprobability that there are more than m points of the process in a fixed disk ofradius r with its center at 0, as m\rightarrow\infty.

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Akemann G, Strahov E. Hole probabilities and overcrowding estimates for products of complex Gaussian matrices.

*Journal of Statistical Physics*. 2013;151(6):987-1003.Akemann, G., & Strahov, E. (2013). Hole probabilities and overcrowding estimates for products of complex Gaussian matrices.

*Journal of Statistical Physics*,*151*(6), 987-1003. doi:10.1007/s10955-013-0750-8Akemann, G., and Strahov, E. (2013). Hole probabilities and overcrowding estimates for products of complex Gaussian matrices.

*Journal of Statistical Physics*151, 987-1003.Akemann, G., & Strahov, E., 2013. Hole probabilities and overcrowding estimates for products of complex Gaussian matrices.

*Journal of Statistical Physics*, 151(6), p 987-1003.G. Akemann and E. Strahov, “Hole probabilities and overcrowding estimates for products of complex Gaussian matrices”,

*Journal of Statistical Physics*, vol. 151, 2013, pp. 987-1003.Akemann, G., Strahov, E.: Hole probabilities and overcrowding estimates for products of complex Gaussian matrices. Journal of Statistical Physics. 151, 987-1003 (2013).

Akemann, Gernot, and Strahov, Eugene. “Hole probabilities and overcrowding estimates for products of complex Gaussian matrices”.

*Journal of Statistical Physics*151.6 (2013): 987-1003.
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arXiv 1211.1576