A unitary extension of virtual permutations
  Analogously to the space of virtual permutations, we define projective limits of isometries: these sequences of unitary operators are natural in the sense that they minimize the rank norm between successive matrices of increasing sizes. The space of virtual isometries we construct this way may be viewed as a natural extension of the space of virtual permutations of Kerov, Olshanski and Vershik as well as the space of virtual isometries of Neretin. We then derive with purely probabilistic methods an almost sure convergence for these random matrices under the Haar measure: for a coherent Haar measure on virtual isometries, the smallest normalized eigenangles converge almost surely to a point process whose correlation function is given by the sine kernel. This almost sure convergence actually holds for a larger class of measures as is proved by Borodin and Olshanski. We give a different proof, probabilist in the sense that it makes use of martingale arguments and shows how the eigenangles interlace when going from dimension n to n+1. Our method also proves that for some universal constant epsilon>0, the rate of convergence is almost surely dominated by n^{-epsilon} when the dimension n goes to infinity. 
P. Bourgade, J. Najnudel, A. Nikeghbali
math.PR