Download PDF by Alexei Borodin (auth.), Anatoly M. Vershik, Yuri Yakubovich: Asymptotic Combinatorics with Applications to Mathematical

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

ISBN-10: 3540403124

ISBN-13: 9783540403128

ISBN-10: 354044890X

ISBN-13: 9783540448907

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

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Additional 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

Sample text

What converges and survives in the limit are the moments of the random matrices. Research supported by a grant of NSERC, Canada A. M. ), pp. 53–73, 2003. c Springer-Verlag Berlin Heidelberg 2003 54 R. Speicher To talk about moments we need in addition to the random matrices also a state. e. for A = (aij )N i,j=1 we have trN (A) := 1 N N aii . e. for A = (aij (ω))N i,j=1 (where the entries aij are random variables on some probability space Ω equipped with a probability measure P ) we have trN ⊗ E(A) := 1 N N aii (ω)dP (ω).

The critical region is around n ∼ 2 λ ∼ 2 N . Poissonization helps because of the following wonderful (1) fact: there is an exact formula for φn (λ) −λ φ(1) Dn−1 e2 n (λ) = e √ λ cos θ = e−λ Dn−1 (λ) Dn−1 is the n × n Toeplitz determinant with weight function f (eiθ ) = √ 2 λ cos θ , e π Dn−1 (f ) = det ckj = ck −j = e−i(k−j)θ f (eiθ ) −π dθ 2π 0≤k,j≤n−1 This formula was first found by Gessel (1990), but has since been discovered independently by many authors: Johansson, Diaconis–Shahshahani, Gessel– Weinstein–Wilf, Odylyzsko, Poonen, Widom, Wilf, Rains, Baik, Deift and Johansson (1999) also give a new proof.

If the shuffled deck is in a “permutation state” π, we let pN (π) denote the number of piles one obtains by playing patience sorting starting from π and using the greedy strategy. For example, suppose N = 6 cards are in the order 4 1 3 5 6 2 (corresponding to the permutation π(1) = 4, π(2) = 1, . . ). Then the game proceeds as follows: 11 1 1 12 4 4 43 435 4356 4356 and p6 (π) = 4. Question: Putting uniform distribution on the set of shuffles {π ∈ SN }, SN = symmetric group, how does pN (π) behave as N → ∞?

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Asymptotic Combinatorics with Applications to Mathematical Physics: A European Mathematical Summer School held at the Euler Institute, St. Petersburg, Russia July 9–20, 2001 by Alexei Borodin (auth.), Anatoly M. Vershik, Yuri Yakubovich (eds.)


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