Asymptotics of Harish-Chandra-Itzykson-Zuber integrals and free probability theory

We discuss an asymptotic exponent N-1 log EA[exp(β tr X* AX/2)] in the limit N → ∞, where β = 1 or 2, where A is an N-by-N real symmetric (β = 1) or complex Hermitian (β = 2) random matrix, and where X is an N-by-M real (β = 1) or complex (β = 2) matrix, with Mbeing kept finite while taking the limit N → ∞. Relation of the problem to the so-called Harish-Chandra or Itzykson-Zuber integrals are also discussed. Assuming that the result is given in terms of limiting eigenvalues of Q = N-1 X*X, we show that the exponent is given by a sum of integrated R-transforms of the limiting eigenvalue distribution of A. We also provide some examples for which the same result holds without the above assumption