On asymptotics of large Haar distributed unitary matrices

The set U(n) of n × n unitary matrices forms a compact topological group with respect to the matrix multiplication and the usual topology, therefore there exists a unique (up to the scalar multiplication) translation invariant measure on U(n), the so-called Haar measure. We will consider a random variable Un which maps from a probability space to U(n), and take its values uniformly from U(n), i.e. if H ⊂ U(n), then Prob (Un ∈ H) = γ(H), where γ is the normalized Haar measure on U(n). We call this random variable a Haar unitary random variable, or shortly Haar unitary.