International Congress of Mathematicians

766 P. Bianewith classical cumulants associated with random matrices, and with characters ofsymmetric groups. Finally in section 5 we explain the asymptotic behaviour ofrepresentations of symmetric groups in terms of free probability concepts.2. Freeness and random matricesThe usual framework for free probability is a von Neumann algebra A, equippedwith a faithful, tracial, normal state r. To any self-adjoint element X £ A one canassociate its distribution, the probability measure on the real line, uniquely determinedby the identity r(X n ) = f- R x n p(dx) for all n > 1. This makes it natural tothink of the elements of A as noneommutative random variables, and of r as an expectationmap, and one usually calls noneommutative probability space such a pair(A,T). Although a great deal of the theory, especially the combinatorial side, canbe developped in a purely algebraic way, assuming only that A is a complex algebrawith unit, and r a complex linear functional, we shall stick to the von Neumannframework in the present exposition.Given (A, T), one considers a family {AJ; i £ 1} of von Neumann subalgebras.This family is called a free family if the following holds: for any k > 1 and fc-tuplecti,..., ak £ A such that• each aj belongs to some algebra A tj ,• T(OJ) = 0 for all j,with ii ^ i 2 ,i 2 ^ h, • • •, iu-i ^ ik,one has r(ai... ak) = 0.Moreover, a family of elements of A is called free if the von Neumann algebraseach of them generates form a free family. Freeness is a noneommutativenotion analogous to the independence of a-fields in probability theory, but whichincorporates also the notion of algebraic independence.Observe that if cti and a 2 are free elements in (A,T), and one defines thecentered elements ô, = a, — r(a,)l then one can conputer(aia 2 ) = r(âiâ 2 ) -V r(ai)r(a 2 ) = r(ai)r(a 2 )where the freeness condition has been used to get r(âiÔ2) = 0. Actually, if {Affi £1} is a free family, it is not dificult to see that one can compute the value of r onany product of the form cti... ak, where each aj belongs to some of the A t 's, interms of the quantities r(aj t ... a-j l ) where all the elements aj 1 ,..., a-j l belong tothe same subalgebra. This implies that the value of r on the algebra generatedby the family {Affi £ 1} is completely determined by the restrictions of r toeach of these subalgebras. However the problem of finding an explicit formula isnontrivial, and this is where combinatorics comes in. We shall describe Speicher'stheory of noncrossing cumulants, which solves this problem, in the next section, butbefore that we explain how free probability is relevant to understand large randommatrices.Consider n random N x N matrices X} ,..., X n , of the formXf ] = UjDf ] U* (2.1)