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Strong convergence of eigenangles and eigenvectors for the circular unitary ensemble. (English) Zbl 1471.60070

The limiting spectral behavior of a large class of random hermitian and unitary ensembles, including the Gaussian Unitary Ensemble (with independent, complex Gaussians above the diagonal), and the Circular Unitary Ensemble (corresponding to the Haar measure on the unitary group of a given dimension), involves a random point process, called the determinantal sine-kernel process. From an observation of Montgomery in 1972, it has been conjectured that the limiting short-scale behavior of the imaginary parts of the zeros of the Riemann zeta function is also described by the same point process. In 1999, Katz and Sarnak [N. M. Katz and P. Sarnak, Random matrices, Frobenius eigenvalues, and monodromy. Providence, RI: American Mathematical Society (1999; Zbl 0958.11004)] proved the Montgomery conjecture in the function field case, which gave rise to several other questions in this field. “In particular, Katz and Sarnak asked whether it is possible to give a meaning to strong convergence (i.e., almost sure convergence) for the eigenvalues of random unitary matrices to the determinantal sine kernel point process.” This problem was first solved by Borodin and Olshanski [A. Borodin and G. Olshanski, Commun. Math. Phys. 223, No. 1, 87–123 (2001; Zbl 0987.60020)] and then Bourgade-Najnudel-Nikeghbali in [P. Bourgade et al., Int. Math. Res. Not. 2013, No. 18, 4101–4134 (2013; Zbl 1314.60023)] proposed an alternative solution which was probabilistic and quantitative (quantified the rate of convergence to the sine kernel point process) using the so called virtual isometries.
In the words of the authors, “The goal of this paper is twofold:
to improve on our estimates in the rate of convergence to the sine-kernel point process by refining several other estimates;
to give a complete panorama of the spectral analysis of virtual isometries by establishing quantitative strong convergence for the eigenvectors as well.”

They believe that these refined estimates “can be very useful in tackling other problems at the interface of random matrix theory and analytic number theory”.
A unitary matrix can be decomposed into a product of complex reflections, one for each dimension, and these reflections are independent and uniformly distributed on the space where they live if the initial matrix is Haar-distributed. If one takes an infinite sequence of such reflections, and consider their successive products, then one gets an infinite sequence of unitary matrices of increasing dimension, all of them following the circular unitary ensemble. In this coupling, the authors show that “the eigenvalues of the matrices converge almost surely to the eigenvalues of the flow, which are distributed according to a sine-kernel point process, and get some estimates of the rate of convergence”. Moreover, they prove that “the eigenvectors of the matrices converge almost surely to vectors which are distributed as Gaussian random fields on a countable set.”
First, the authors “consider the sequence of eigenvalues of the virtual isometry and prove that it converges almost surely, and then, conditioning on the eigenvalues of every matrix in the virtual isometry, consider the sequence of eigenvectors and show that they also converge in a suitable sense”.

MSC:

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60B20 Random matrices (probabilistic aspects)
60F15 Strong limit theorems
11M26 Nonreal zeros of \(\zeta (s)\) and \(L(s, \chi)\); Riemann and other hypotheses
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References:

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