Asymptotics of Harish-Chandra-Itzykson-Zuber integrals and of Schur polynomials

Abstract

Let Y^N,? be the random matrix D^N + X ^N,? with a deterministic diagonal matrix D ^N and X ^N,? a Gaussian Wigner matrix.We now show how the deviations of the law of the spectral measure of Y ^N,? are related to the asymptotics of the Harish–Chandra–Itzykson–Zuber (or spherical) integrals

$$I_N (A,B) = \int {e^{N{\rm Tr}(AUBU*)} } dm_N^\beta (U)$$

where $m_N^\beta (U)$ is the Haar measure on U(N) when ? = 2 and O(N) when ? = 1. m _N will stand for $m_N^2$ to simplify the notations. Here, IN(A,B) makes sense for any A,B ? M _n (C), but we shall consider asymptotics only when $A,B \in H_N^{(\beta )} (C)$ (the extension of our results to non-self-adjoint matrices is still open). To this end, we shall make the following hypothesis:

Assumption 1. 1. There exists d_max ? R⁺ such that for any integer number $N,L_{D_N } (\{ |x| \ge d_{\max } \} ) = 0$. Moreover, LDN converges weakly to ?_D ? P(R).

2. $L_{D_N }$ converges to ?E ? P(R) while $L_{D_N }$ (x²) stays uniformly bounded.