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33
Asymptotics of Plancherel measures for symmetric groups
 J. Amer. Math. Soc
, 2000
"... 1.1. Plancherel measures. Given a finite group G, by the corresponding Plancherel measure we mean the probability measure on the set G ∧ of irreducible representations of G which assigns to a representation π ∈ G ∧ the weight (dim π) 2 /G. For the symmetric group S(n), the set S(n) ∧ is the set o ..."
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Cited by 134 (33 self)
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1.1. Plancherel measures. Given a finite group G, by the corresponding Plancherel measure we mean the probability measure on the set G ∧ of irreducible representations of G which assigns to a representation π ∈ G ∧ the weight (dim π) 2 /G. For the symmetric group S(n), the set S(n) ∧ is the set of partitions λ of the number
Seibergwitten theory and random partitions
"... We study N = 2 supersymmetric four dimensional gauge theories, in a certain N = 2 supergravity background, called Ωbackground. The partition function of the theory in the Ωbackground can be calculated explicitly. We investigate various representations for this partition function: a statistical sum ..."
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Cited by 97 (6 self)
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We study N = 2 supersymmetric four dimensional gauge theories, in a certain N = 2 supergravity background, called Ωbackground. The partition function of the theory in the Ωbackground can be calculated explicitly. We investigate various representations for this partition function: a statistical sum over random partitions, a partition function of the ensemble of random curves, a free fermion correlator. These representations allow to derive rigorously the SeibergWitten geometry, the curves, the differentials, and the prepotential. We study pure N = 2 theory, as well as the theory with matter hypermultiplets in the fundamental or adjoint representations, and the five dimensional theory compactified
Random matrices and random permutations
 Internat. Math. Res. Notices
, 2000
"... We prove the conjecture of Baik, Deift, and Johansson which says that with respect to the Plancherel measure on the set of partitions λ of n, the rows λ1,λ2,λ3,... of λ behave, suitably scaled, like the 1st, 2nd, 3rd, and so on eigenvalues of a Gaussian random Hermitian matrix as n → ∞. Our proof is ..."
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Cited by 61 (7 self)
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We prove the conjecture of Baik, Deift, and Johansson which says that with respect to the Plancherel measure on the set of partitions λ of n, the rows λ1,λ2,λ3,... of λ behave, suitably scaled, like the 1st, 2nd, 3rd, and so on eigenvalues of a Gaussian random Hermitian matrix as n → ∞. Our proof is based on an interplay between maps on surfaces and ramified coverings of the sphere. We also establish a connection of this problem with intersection theory on the moduli spaces of curves. 1
Kerov’s central limit theorem for the Plancherel measure on Young diagrams
, 2003
"... Consider random Young diagrams with fixed number n of boxes, distributed according to the Plancherel measure Mn. That is, the weight Mn(λ) of a diagram λ equals dim 2 λ/n!, where dim λ denotes the dimension of the irreducible representation of the symmetric group Sn indexed by λ. As n → ∞, the boun ..."
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Cited by 40 (7 self)
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Consider random Young diagrams with fixed number n of boxes, distributed according to the Plancherel measure Mn. That is, the weight Mn(λ) of a diagram λ equals dim 2 λ/n!, where dim λ denotes the dimension of the irreducible representation of the symmetric group Sn indexed by λ. As n → ∞, the boundary of the (appropriately rescaled) random shape λ concentrates near a curve Ω (Logan– Shepp 1977, Vershik–Kerov 1977). In 1993, Kerov announced a remarkable theorem describing Gaussian fluctuations around the limit shape Ω. Here we propose a reconstruction of his proof. It is largely based on Kerov’s unpublished work notes, 1999.
Asymptotics of Numbers of Branched Coverings of a Torus and Volumes of Moduli Spaces of Holomorphic Differentials
, 2000
"... ..."
The Boundary of Young Lattice and Random Young Tableaux
, 1994
"... The description of characters of the infinite symmetric group S1 = lim \Gamma! Sn is considered in the paper as an asymptotical combinatorial problem. It is equivalent to the characterization problem of totally positive sequences. We derive the list of characters, first obtained by E. Thoma i ..."
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Cited by 22 (3 self)
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The description of characters of the infinite symmetric group S1 = lim \Gamma! Sn is considered in the paper as an asymptotical combinatorial problem. It is equivalent to the characterization problem of totally positive sequences. We derive the list of characters, first obtained by E. Thoma in 1964, using appropriate exact formulae for the number of skew Young tableaux. The character space is treated as a boundary of the Young lattice. The character problem for a locally finite group and the classical de Finetti theorem on exchangeable random sequences are two particular cases of a more general problem in potential theory on a graph, namely that of the explicite description of Martin boundary of this graph. We show that Selberg type integrals arise naturally in this general setup in connection with Pieri formula for Jack symmetric polynomials. Some remarkable connections between Plancherel measures of symmetric groups and topics in Analysis are indicated.
Extended SeibergWitten Theory and Integrable Hierarchy
"... The prepotential of the effective N = 2 superYangMills theory, perturbed in the ultraviolet by the descendents ∫ d4θ trΦk+1 of the singletrace chiral operators, is shown to be a particular taufunction of the quasiclassical Toda hierarchy. In the case of noncommutative U(1) theory (or U(N) theory ..."
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Cited by 16 (1 self)
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The prepotential of the effective N = 2 superYangMills theory, perturbed in the ultraviolet by the descendents ∫ d4θ trΦk+1 of the singletrace chiral operators, is shown to be a particular taufunction of the quasiclassical Toda hierarchy. In the case of noncommutative U(1) theory (or U(N) theory with 2N −2 fundamental hypermultiplets at the appropriate locus of the moduli space of vacua) or a theory on a single fractional D3 brane at the ADE singularity the hierarchy is the dispersionless Toda chain, and we present its explicit solution. Our results generalise the limit shape analysis of LoganSchepp and VershikKerov, support the prior work [1], which established the equivalence of these N = 2 theories with the topological A string on CP 1, and clarify the origin of the EguchiYang matrix integral. In the higher rank case we find an appropriate variant of the quasiclassical taufunction, show how the SeibergWitten curve is deformed by Toda flows, and fix the contact term ambiguity.
An explicit form for Kerov’s character polynomials
 Trans. Amer. Math. Soc
, 2005
"... Abstract. Kerov considered the normalized characters of irreducible representations of the symmetric group, evaluated on a cycle, as a polynomial in free cumulants. Biane has proved that this polynomial has integer coefficients, and made various conjectures. Recently, ´ Sniady has proved Biane’s co ..."
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Cited by 15 (4 self)
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Abstract. Kerov considered the normalized characters of irreducible representations of the symmetric group, evaluated on a cycle, as a polynomial in free cumulants. Biane has proved that this polynomial has integer coefficients, and made various conjectures. Recently, ´ Sniady has proved Biane’s conjectured explicit form for the first family of nontrivial terms in this polynomial. In this paper, we give an explicit expression for all terms in Kerov’s character polynomials. Our method is through Lagrange inversion. 1.
Increasing and decreasing subsequences and their variants
 Proceedings of International Congress of Mathematical Society
, 2006
"... Abstract.We survey the theory of increasing and decreasing subsequences of permutations. Enumeration problems in this area are closely related to the RSK algorithm. The asymptotic behavior of the expected value of the length is(w) of the longest increasing subsequence of a permutation w of 1, 2,..., ..."
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Cited by 13 (2 self)
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Abstract.We survey the theory of increasing and decreasing subsequences of permutations. Enumeration problems in this area are closely related to the RSK algorithm. The asymptotic behavior of the expected value of the length is(w) of the longest increasing subsequence of a permutation w of 1, 2,...,n was obtained by VershikKerov and (almost) by LoganShepp. The entire limiting distribution of is(w) was then determined by Baik, Deift, and Johansson. These techniques can be applied to other classes of permutations, such as involutions, and are related to the distribution of eigenvalues of elements of the classical groups. A number of generalizations and variations of increasing/decreasing subsequences are discussed, including the theory of pattern avoidance, unimodal and alternating subsequences, and crossings and nestings of matchings and set partitions.