International Congress of Mathematicians

Free Probability and Combinatorics 769allowing the computation of the moments of Xi + X 2 , hence its distribution, interms of the distributions of Xi and X 2 . It remains to give a compact form to therelation between moments and noncrossing cumulants. For any self-adjoint elementX with distribution p, letbe its Cauchy transform, and let1 °° r 1G x (z)=- + Yz- k - 1 T(X k )= / p(dx)zitiK(z) = - +be the inverse series for composition.Theorem 3. [19]Y R kZ kk=0Jnz-xOne has R k = R (k) (X,..., X) for all fc.The operation which associates to the two distributions of Xi and X 2 thedistribution of their sum is called the free convolution of measures on the real line,and was introduced by D. Voiculescu, who first considered the coefficients Rk andproved the formula for the free convolution of two measures, using very differentmethods [22].Combining theorems 1 and 2, given two large random matrices of known spectraone can predict the spectral distribution of their sum, with a good accuracyand probability close to 1. It is illuminating to look at the following example. Thehistogram below is made of the 800 eigenvalues of a random matrix of the formHi + n 2 where Hi and n 2 are two orthogonal projections onto some random subpacesof dimension 400 in C 800 , chosen independently. The curve y = —, 40T\JX(2-x)which corresponds to the large N limit predicted by free probability has been drawn.0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2