International Congress of Mathematicians

Free Probability and Combinatorics 767where D- ';j = l,...,n are diagonal, hermitian, nonrandom matrices and Uj areindependent unitary random matrices, each distributed with the Haar measure onthe unitary group YJ(N). In other words we have fixed the spectra of the X>but their eigenvectors are chosen at random. The n-tuple Xf ,..., X n canbe recovered, up to a global unitary conjugation X> H> UX t 'U*, (where Udoes not depend on i), from its mixed moments, i.e. the set of complex numbersjjTr(X h .. .X ik ) where ii,...ik are arbitrary sequences of indices in {l,...,n}.In particular the spectrum of any noneommutative polynomial of the X> can berecovered from these data. A most remarkable fact is that if we assume that theindividual moments j^Tr((Xl ) k ) converge as N tends to infinity, then the mixedmoments -^Tr(X> i .. .X ik ) converge in probability, and their limit is obtainedby the prescriptions of free probability.Theorem 1. Let (A,T) be a noneommutative probability space with free selfadjointelements Xi,...,X n , satisfying r(Xf) = limjv-s.00 ^Tr((X t ) k ), for alli and k, then, in probability, j^Tr(X ii ' ...X ik ') ^JV-S-OO T(X ì1 ...X ik ), for allii,--.,ik-This striking result was first proved by D. Voiculescu [23], and has lead to theresolution of many open problems about von Neumann algebras, upon which weshall not touch here.3. Noncrossing partitions and cumulantsA partition of the set {1,... ,n} is said to have a crossing if there exists aquadruple (i,j, k,l), with 1 < i < j < k < I < n, such that i and k belong to someclass of the partition and j and I belong to another class. If a partition has nocrossing, it is called noncrossing. The set of all noncrossing partitions of {1,... ,n}is denoted by NC(n). It is a lattice for the refinement order, which seems to havebeen first systematically investigated in [10].Let (A,T) be a non-commutative probability space, then we shall define afamily RS") of n-multilinear forms on A, for n > 1, by the following formulaHere, for n £ NC(n), one has definedr(ai...a n )= ^ R[n](ai,.. .,a n ). (3.1)TreNC(n)R[n](ai,...,a n )= JJ R,U v V(a v )where ay = (a,j 1 ,. • • ,aj k ) if V = {ji,-- -,jk} is a elass of the partition n, withji < J2 < • • • < jk and |F| = fc is the number of elements of V. In particularR[l n ] = R^ if l n is the partition with only one class. Thus one has, for n = 3,T(aia 2 a 3 ) = R^ (oi ,a 2 ,a 3 ) + R^ (oi, a 2 )R^ (o 3 ) + R {2) (ai,a 3 )R^ (a 2 )+RW (a 2 , a 3 )R^ (oi) + R^ (ai)R^ (oa)^1)(a 3 ).V67T