International Congress of Mathematicians

Free Probability and Combinatorics 771This was first observed by P. Zinn-Justin [25], a proof using representation theoryhasbeen found by B. Collins [6].We have related noncrossing cumulants to usual cumulants via random matrixtheory, we shall see that that noncrossing cumulants are also useful in evaluatingcharacters of symmetric groups. The precise relation however is not obvious at firstsight.Let us recall a few facts about irreducible representations of symmetric groups.It is well known that they can be parametrized by Young diagrams. In the followingit will be convenient to represent a Young diagram by a function u : R —¥ R suchthat (jj(x) = \x\ for |x| large enough, and a; is a piecewise affine function, with slopes±1, see the following picture which shows the Young diagram corresponding to thepartition 8 = 3 + 2 + 2+1.Xl J/1 X2 y 2 x 3 t/3 XiAlternatively we can encode the Young diagram using the local minima andlocal maxima of the function OJ, denoted by xi,..., Xk and 2/1, • • •, 2/fc-i respectively,which form two interlacing sequences of integers. These are (-3,-1,2,4) and (-2,1,3)respectively in the above picture. Associated with the Young diagram there is aunique probability measure m w on the real line, such thatR1,(dx)for all z £ C \ R.This probability measure is supported by the set {xi,...,Xk} and is called thetransition measure of the diagram, see [8]. Let a denote the conjugacy class in £„of a permutation with fc 2 cycles of length 2, fc 3 of length 3, etc.. Here fc 2 ,fc3,...are fixed while we let n —¥ 00. Denote by Xw the normalized character of £„associated with the Young diagram OJ, then the following asymptotic evaluationholds uniformly on the set of A-balanced Young diagrams, i.e. those whose longestrow and longest column are less that Ay/n (where A is some constant > 0),xAo-) = f[n- jk >Rf +1 (ou) + 0(r -i-\*\ß\ (4.1)Note that Rk is scaled by A* if we scale the diagram u by a factor A, therefore thefirst term in the right hand side is of order 0(n^- , i^+1 ' kj ' 2 ~^- , ' : ' kj ) = 0(n~^^2),