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On unitarily invariant ensembles of infinite Hermitian matrices

The unitarily invariant probability measures on infinite Hermitian matrices have been classified by Olshanski and Vershik. In their work, and another work by Borodin and Olshanski, the authors prove the almost sure convergence of the renormalized extreme eigenvalues of the minors when the dimension goes to infinity. In this talk, I present these results, and another convergence result on eigenvectors I have recently proven. These results apply in particular to the image of the Circular Unitary Ensemble by the Cayley transform which maps the unit circle to the real line.