Félix Parraud: Free probability and random matrices: the asymptotic behaviour of polynomials in independent random matrices
Time: Tue 2021-09-28 14.15
Location: Zoom, meeting ID: 698 3346 0369
Lecturer: Félix Parraud (KTH)
I will first introduce basic notions of free probability and in particular the notion of freeness. Indeed as we will see, independence of random matrices usually translates into freeness when studying their asymptotic behaviour, that is when the size of the matrices considered goes to infinity. More precisely as Voiculescu proved, the limit of the renormalized trace of a polynomial in GUE or Haar unitary matrices can be described with the help of free probability. One can however prove much sharper results, notably Haagerup and Thorbjørnsen considered smooth functions and proved that the difference with the limit was of order 1/N^2. This in turn has consequences on the spectrum of the random matrices considered. In this seminar I will show that we can actually compute a Taylor expansion of the expectation of the trace around the dimension of our random matrices. In order to do so we adapt to the free probability setting a well-known method in classical probability. Indeed by interpolating Wigner and GUE matrices with an Ornstein–Uhlenbeck process one can extend properties of GUE random matrices to Wigner matrices. Thus the method that we used was to interpolate GUE matrices and semicircular variables with a free Ornstein–Uhlenbeck process.