By V.A. Malyshev, A.M. Vershik

On the summer season tuition Saint Petersburg 2001, the most lecture classes bore on contemporary growth in asymptotic illustration idea: these written up for this quantity care for the speculation of representations of countless symmetric teams, and teams of limitless matrices over finite fields; Riemann-Hilbert challenge concepts utilized to the research of spectra of random matrices and asymptotics of younger diagrams with Plancherel degree; the corresponding principal restrict theorems; the combinatorics of modular curves and random bushes with software to QFT; loose likelihood and random matrices, and Hecke algebras.

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It puts mass 1/N on each of the eigenvalues of A (counted with multiplicity): µA = 1 N N δλi , i=1 where λ1 , . . , λN are the eigenvalues of A. In the same way, for a random matrix A, µA is given by the averaged eigenvalue distribution of A. Thus, moments of random matrices with respect to the averaged trace trN ⊗ E contain exactly that type of information in which one is usually interested when dealing with random matrices. 2 Example Let us consider the basic example of random matrix theory, expressed in our new language.

It remains to make the connection between moments and cumulants as explicit as possible. On a combinatorial level, our deﬁnition specializes in the one-dimensional case to the following relation. 3 Proposition Let (mn )n≥1 and (kn )n≥1 be the moments and free cumulants, respectively, of some random variable. The connection between these two sequences of numbers is given by mn = kπ , π∈N C(n) where kπ := k#V1 · · · k#Vr for π = {V1 , . . , Vr }. Example. For n = 3 we have m3 = k +k +k = k3 + 3k1 k2 + +k +k k13 .

Let χ(1) be a random variable whose distribution is F (1) . Then as N → ∞ √ lN − 2 N →χ χN = 1 N6 in distribution. That is, lim Prob (χN ≤ t) = F (1) (t), t ∈ R. N →∞ We also have convergence of the moments. Theorem 3. For any m = 1, 2, 3, . . we have m lim EN (χm N ) = E (χ ) N →∞ where E denotes expectation with respect to F (1) . 771 which agrees pretty well with the Mont´e-Carlo values c0 and c1 above. Now it turns out that there is a very interesting connection between the above results and random matrix theory, in particular with GUE.