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

ISBN-10: 3540403124

ISBN-13: 9783540403128

On the summer season tuition Saint Petersburg 2001, the most lecture classes bore on fresh development in asymptotic illustration conception: these written up for this quantity take care of the speculation of representations of limitless symmetric teams, and teams of countless matrices over finite fields; Riemann-Hilbert challenge strategies utilized to the research of spectra of random matrices and asymptotics of younger diagrams with Plancherel degree; the corresponding important 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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**Example text**

For m = 1, 2, 3, . . , EN t∈R (2) χN m →E χ(2) m as N → ∞. ) by Okounkov (1999), Borodin, Olshanski, Okounkov (1999) and Johansson (1999). These authors proved convergence in distribution. Convergence of the moments for the k rows was proved subsequently by Baik, Deift and Rains (2001). The results in Theorems 2 and 3 have generated a lot of activity in a variety of areas including the representation theory of large groups, polynuclear growth models, percolation models, random topologies on surfaces, digital boiling, amongst many others (see Adler, Baik, Borodin, Diaconis, Forrester, Johansson, van Moerbeke, Okounkov, Olshanski, Pr¨ ahoﬀer, Spohn, Tracy, Widom, .

Ai(n) ) = 0, whenever there exist 1 ≤ p, q ≤ n with i(p) = i(q). 62 R. 7 Remarks 1) An example of the vanishing of mixed cumulants is that for a, b free we have k3 (a, a, b) = 0, which, by the deﬁnition of k3 just means that ϕ(aab) − ϕ(a)ϕ(ab) − ϕ(aa)ϕ(b) − ϕ(ab)ϕ(a) + 2ϕ(a)ϕ(a)ϕ(b) = 0. This vanishing of mixed cumulants in free variables is of course just a reorganization of the information about joint moments of free variables – but in a form which is much more useful for many applications.

An ) = (a1 , . . , an ∈ A), kπ [a1 , . . , an ] π∈N C(n) where kπ denotes a product of cumulants according to the block structure of π: kπ [a1 , . . , an ] := kV1 [a1 , . . , an ] . . kVr [a1 , . . , an ] for π = {V1 , . . , Vr } ∈ N C(n) and kV [a1 , . . , an ] := k#V (av1 , . . , avl ) for V = (v1 , . . , vl ). 5 Remarks and Examples 1) Note: the above equations have the form ϕ(a1 . . an ) = kn (a1 , . . , an ) + smaller order terms and thus they can be resolved for the kn (a1 , .

### Asymptotic Combinatorics with Applications to Mathematical Physics by V.A. Malyshev, A.M. Vershik

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