Spectra of large unitary matrices
by Thierry Lévy (Université Pierre et Marie Currie Paris 6, France)
The study of large random matrices can be approached from many branches of mathematics and physics. For instance, Euclidean gauge theories produce physically relevant models of random matrices and it has been recognized since the work of 't Hooft in 1974 that their "large N limit", when the size of the matrices tends to infinity, is both simple and important.
Moreover, the flavour of this study depends a great deal on the technical apparatus with which it is undertaken. In this talk, I will focus on what physicists would call a one-matrix model and try to illustrate on several classical and less classical examples how a combinatorial approach to the problem of the repartition of the eigenvalues of a large unitary (or, in a first time, Hermitian) matrix becomes simpler in the large N limit, to the point that it becomes solvable.
The prototype of the theorems that I will discuss is the celebrated result of Wigner which asserts that the eigenvalues of a large Hermitian matrix with independent Gaussian entries are distributed according to a semi-circle law. I will discuss analogues of this theorem for matrices picked in the unitary group under the Haar measure and under the heat kernel measure at a given time. If time allows, I will discuss more recent results relating to fluctuations and possible extensions to other matrix groups.
The core of our combinatorial approach will be Schur-Weyl duality, which operates as a dictionary translating questions about unitary groups into the realm of symmetric groups. In this translation, the computation of expected values of traces of powers of random matrices is turned into the enumeration of certain paths in the symmetric group, and in the large N limit, only those paths which are geodesic in a natural sense contribute.