# Universal microscopic correlation functions for products of truncated unitary matrices

Akemann G, Burda Z, Kieburg M, Nagao T (2014)

Journal of Physics: A Mathematical and Theoretical 47(25): 255202.

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We investigate the spectral properties of the product of $M$ complexnon-Hermitian random matrices that are obtained by removing $L$ rows andcolumns of larger unitary random matrices uniformly distributed on the group${\rm U}(N+L)$. Such matrices are called truncated unitary matrices or randomcontractions. We first derive the joint probability distribution for theeigenvalues of the product matrix for fixed $N,\ L$, and $M$, given by astandard determinantal point process in the complex plane. The weight howeveris non-standard and can be expressed in terms of the Meijer G-function. Theexplicit knowledge of all eigenvalue correlation functions and thecorresponding kernel allows us to take various large $N$ (and $L$) limits atfixed $M$. At strong non-unitarity, with $L/N$ finite, the eigenvalues condenseon a domain inside the unit circle. At the edge and in the bulk we find thesame universal microscopic kernel as for a single complex non-Hermitian matrixfrom the Ginibre ensemble. At the origin we find the same new universalityclasses labelled by $M$ as for the product of $M$ matrices from the Ginibreensemble. Keeping a fixed size of truncation, $L$, when $N$ goes to infinityleads to weak non-unitarity, with most eigenvalues on the unit circle as forunitary matrices. Here we find a new microscopic edge kernel that generalizesthe known results for M=1. We briefly comment on the case when each productmatrix results from a truncation of different size $L_j$.

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Akemann G, Burda Z, Kieburg M, Nagao T. Universal microscopic correlation functions for products of truncated unitary matrices.

*Journal of Physics: A Mathematical and Theoretical*. 2014;47(25):255202.Akemann, G., Burda, Z., Kieburg, M., & Nagao, T. (2014). Universal microscopic correlation functions for products of truncated unitary matrices.

*Journal of Physics: A Mathematical and Theoretical*,*47*(25), 255202. doi:10.1088/1751-8113/47/25/255202Akemann, G., Burda, Z., Kieburg, M., and Nagao, T. (2014). Universal microscopic correlation functions for products of truncated unitary matrices.

*Journal of Physics: A Mathematical and Theoretical*47, 255202.Akemann, G., et al., 2014. Universal microscopic correlation functions for products of truncated unitary matrices.

*Journal of Physics: A Mathematical and Theoretical*, 47(25), p 255202. G. Akemann, et al., “Universal microscopic correlation functions for products of truncated unitary matrices”,

*Journal of Physics: A Mathematical and Theoretical*, vol. 47, 2014, pp. 255202. Akemann, G., Burda, Z., Kieburg, M., Nagao, T.: Universal microscopic correlation functions for products of truncated unitary matrices. Journal of Physics: A Mathematical and Theoretical. 47, 255202 (2014).

Akemann, Gernot, Burda, Zdzislaw, Kieburg, Mario, and Nagao, Taro. “Universal microscopic correlation functions for products of truncated unitary matrices”.

*Journal of Physics: A Mathematical and Theoretical*47.25 (2014): 255202.
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arXiv 1310.6395