# Universal microscopic correlation functions for products of independent Ginibre matrices

Akemann G, Burda Z (2012)

J. Phys. A: Math. Theor. 45(46).

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Abstract

We consider the product of n complex non-Hermitian, independent randommatrices, each of size NxN with independent identically distributed Gaussianentries (Ginibre matrices). The joint probability distribution of the complexeigenvalues of the product matrix is found to be given by a determinantal pointprocess as in the case of a single Ginibre matrix, but with a more complicatedweight given by a Meijer G-function depending on n. Using the method oforthogonal polynomials we compute all eigenvalue density correlation functionsexactly for finite N and fixed n. They are given by the determinant of thecorresponding kernel which we construct explicitly. In the large-N limit atfixed n we first determine the microscopic correlation functions in the bulkand at the edge of the spectrum. After unfolding they are identical to that ofthe Ginibre ensemble with n=1 and thus universal. In contrast the microscopiccorrelations we find at the origin differ for each n and generalise the knownBessel-law in the complex plane for n=2 to a new hypergeometric kernel 0_F_n-1.

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Akemann G, Burda Z. Universal microscopic correlation functions for products of independent Ginibre matrices.

*J. Phys. A: Math. Theor.*2012;45(46).Akemann, G., & Burda, Z. (2012). Universal microscopic correlation functions for products of independent Ginibre matrices.

*J. Phys. A: Math. Theor.*,*45*(46).Akemann, G., and Burda, Z. (2012). Universal microscopic correlation functions for products of independent Ginibre matrices.

*J. Phys. A: Math. Theor.*45.Akemann, G., & Burda, Z., 2012. Universal microscopic correlation functions for products of independent Ginibre matrices.

*J. Phys. A: Math. Theor.*, 45(46).G. Akemann and Z. Burda, “Universal microscopic correlation functions for products of independent Ginibre matrices”,

*J. Phys. A: Math. Theor.*, vol. 45, 2012.Akemann, G., Burda, Z.: Universal microscopic correlation functions for products of independent Ginibre matrices. J. Phys. A: Math. Theor. 45, (2012).

Akemann, Gernot, and Burda, Zdzislaw. “Universal microscopic correlation functions for products of independent Ginibre matrices”.

*J. Phys. A: Math. Theor.*45.46 (2012).
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arXiv 1208.0187