Hole probabilities and overcrowding estimates for products of complex Gaussian matrices
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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987-1003
987-1003
Springer Science + Business Media