Pierre Bousseyroux

The Stieltjes transform as a random object

The Stieltjes transform is a powerful tool for studying the distribution of eigenvalues of large random matrices because it builds a bridge between algebra and probability. The idea is simple: for Hermitian matrices, you introduce a function, defined outside the real axis, whose poles are exactly the eigenvalues, and you hope that this function converges to a deterministic limit as the matrix size goes to infinity. Knowing this limiting function then allows you to recover the continuous distribution of eigenvalues. This method is very effective. But what happens if you stay on the real axis and still let the matrix size go to infinity? In that case, you obtain a random object whose tails carry a significant amount of local information. While this phenomenon is well understood in the Hermitian case, I will present a new result describing it explicitly for the Ginibre ensemble, and then move on to more general conjectures.