By Alexei Borodin (auth.), Anatoly M. Vershik, Yuri Yakubovich (eds.)

At the summer time tuition Saint Petersburg 2001, the most lecture classes bore on fresh growth in asymptotic illustration idea: these written up for this quantity care for the idea of representations of endless symmetric teams, and teams of countless matrices over finite fields; Riemann-Hilbert challenge options utilized to the research of spectra of random matrices and asymptotics of younger diagrams with Plancherel degree; the corresponding critical restrict theorems; the combinatorics of modular curves and random bushes with program to QFT; unfastened chance and random matrices, and Hecke algebras.

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**Extra resources for Asymptotic Combinatorics with Applications to Mathematical Physics: A European Mathematical Summer School held at the Euler Institute, St. Petersburg, Russia July 9–20, 2001**

**Example text**

We now return to the RHP (21). The calculations that follow are motivated by work which Zhou and I did together with Venakides (1994) on the zerodispersion problem for KdV where genuinely non-linear oscillations (sn(αx+β) as opposed to sin(αx + β), where sn denotes the Jacobi function) develop. Our ﬁrst task is to turn the RHP into a normalized RHP. To this end set l˜ T (z) = T (N ) (z) = eN 2 0 l˜ 0 e−N 2 e−N g(z) 0 S(z) 0 eN g(z) l˜ e−N 2 0 l˜ 0 eN 2 (29) Here ˜l is a constant, to be determined below, and g(z) is an as yet undetermined function with the following properties (i) g(z) is analytic in C \ R (ii) g(z) ∼ log z as z → ∞.

Four Lectures on Random Matrix Theory 29 c) {Mkj } , k ≤ j are independent so fkj (Mkj ). e. M = M ∗ and M = σ JM T J −1 where J = 0 σ .. 0 where σ = . 0 1 −1 0 . e. U ∗ U = I and U T JU = J. This invariance is most conveniently expressed by writing M as a N × N matrix with quaternion entries (see Mehta). c) The linearly independent components of M are also statistically independent as in Ic, Ic. Again, this is most conveniently expressed by writing M as a N × N matrix with quaternion entries. The analysis of GUE is the simplest and I am going to restrict myself to this case throughout these lectures.

AN and variables A1 , . . , Am (living in some abstract algebra A equipped with a state ϕ). We say that 56 R. Speicher (1) (m) (AN , . . , AN ) → (A1 , . . , Am ) if (i ) in distribution, (i ) lim trN ⊗ E[AN1 · · · ANk ] = ϕ(Ai1 · · · Aik ) N →∞ for all choices of k, 1 ≤ i1 , . . , ik ≤ m. 4 Remark The A1 , . . , Am arising in the limit of random matrices are a priori abstract elements in some algebra A, but it is good to know that in many cases they can also be concretely realized by some kind of creation and annihilation operators on a full Fock space.