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

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

On the summer time college Saint Petersburg 2001, the most lecture classes bore on fresh growth in asymptotic illustration concept: 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 thoughts utilized to the examine of spectra of random matrices and asymptotics of younger diagrams with Plancherel degree; the corresponding significant restrict theorems; the combinatorics of modular curves and random timber with software to QFT; unfastened likelihood and random matrices, and Hecke algebras.

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**Example text**

If N = 5 and π is the permutation (1 2 3 4 5) → (5 1 3 2 4), then 1 3 4 and 1 2 4 are both increasing subsequences of length 3 and are the longest increasing subsequences. Equip SN with the uniform distribution so that (1) qn,M ≡ Prob {lN (π) ≤ n} = fn,N N! where fn,N = # of π’s with lN (π) ≤ n. Our aim is the following: Determine the asymptotics of qn,N as N → ∞. Now why is lN of interest? There are connections to • Representation theory of SN (Young tableaux — more later). • Ulam’s metric on SN , d(π, σ) = N − lN (πσ −1 ) which is useful for a variety of statistical questions.

E. limN →∞ Prob (2) χN ≤ t = F (2) (t), Theorem 5. 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, .

Where fn,N = # of π’s with lN (π) ≤ n. Our aim is the following: Determine the asymptotics of qn,N as N → ∞. Now why is lN of interest? There are connections to • Representation theory of SN (Young tableaux — more later). • Ulam’s metric on SN , d(π, σ) = N − lN (πσ −1 ) which is useful for a variety of statistical questions. • A variety of statistical mechanical particle models — directed ﬁrst passage percolation questions, growth models, exclusion process models, “vicious walker” models, super-conductivity models • Random topologies on surfaces (Okounkov).