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113
On the distribution of the length of the longest increasing subsequence of random permutations
 J. Amer. Math. Soc
, 1999
"... Let SN be the group of permutations of 1, 2,...,N. If π ∈ SN,wesaythat π(i1),...,π(ik) is an increasing subsequence in π if i1
Abstract

Cited by 347 (28 self)
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Let SN be the group of permutations of 1, 2,...,N. If π ∈ SN,wesaythat π(i1),...,π(ik) is an increasing subsequence in π if i1 <i2 <·· · <ikand π(i1) < π(i2) < ···<π(ik). Let lN (π) be the length of the longest increasing subsequence. For example, if N =5andπis the permutation 5 1 3 2 4 (in oneline notation:
Longest increasing subsequences: from patience sorting to the BaikDeiftJohansson theorem
 Bull. Amer. Math. Soc. (N.S
, 1999
"... Abstract. We describe a simple oneperson card game, patience sorting. Its analysis leads to a broad circle of ideas linking Young tableaux with the longest increasing subsequence of a random permutation via the Schensted correspondence. A recent highlight of this area is the work of BaikDeiftJoha ..."
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Cited by 137 (2 self)
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Abstract. We describe a simple oneperson card game, patience sorting. Its analysis leads to a broad circle of ideas linking Young tableaux with the longest increasing subsequence of a random permutation via the Schensted correspondence. A recent highlight of this area is the work of BaikDeiftJohansson which yields limiting probability laws via hard analysis of Toeplitz determinants. 1.
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 137 (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
Hammersley's Interacting Particle Process and Longest Increasing Subsequences
 Probab. Th. Rel. Fields
, 1995
"... In a famous paper [8] Hammersley investigated the length Ln of the longest increasing subsequence of a random npermutation. Implicit in that paper is a certain onedimensional continuousspace interacting particle process. By studying a hydrodynamical limit for Hammersley's process we show by fairl ..."
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Cited by 68 (3 self)
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In a famous paper [8] Hammersley investigated the length Ln of the longest increasing subsequence of a random npermutation. Implicit in that paper is a certain onedimensional continuousspace interacting particle process. By studying a hydrodynamical limit for Hammersley's process we show by fairly "soft" arguments that limn \Gamma1=2 ELn = 2. This is a known result, but previous proofs (Vershik  Kerov [14]; Logan  Shepp [11]) relied on hard analysis of combinatorial asymptotics. Mathematics subject classification. 60C05, 60K35. Running title. Hammersley's process. Research supported by N.S.F. Grant MCS 9224857 and the Miller Institute for Basic Research in Science y Research supported by N.S.F. Grant DMS9204864 1 Introduction An increasing subsequence i 1 ; i 2 ; : : : ; i k of a permutation i ! ß(i) is a subsequence such that i 1 ! i 2 ! : : : ! i k ; ß(i 1 ) ! ß(i 2 ) ! : : : ß(i k ): For instance, the permutation 7 2 8 1 3 4 10 6 9 5 (1) (for which ß(1) = 4; ß(2) =...
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 62 (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
J.Propp, The shape of a typical boxed plane partition
 J. of Math
, 1998
"... Abstract. Using a calculus of variations approach, we determine the shape of a typical plane partition in a large box (i.e., a plane partition chosen at random according to the uniform distribution on all plane partitions whose solid Young diagrams fit inside the box). Equivalently, we describe the ..."
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Cited by 51 (5 self)
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Abstract. Using a calculus of variations approach, we determine the shape of a typical plane partition in a large box (i.e., a plane partition chosen at random according to the uniform distribution on all plane partitions whose solid Young diagrams fit inside the box). Equivalently, we describe the distribution of the three different orientations of lozenges in a random lozenge tiling of a large hexagon. We prove a generalization of the classical formula of MacMahon for the number of plane partitions in a box; for each of the possible ways in which the tilings of a region can behave when restricted to certain lines, our formula tells the number of tilings that behave in that way. When we take a suitable limit, this formula gives us a functional which we must maximize to determine the asymptotic behavior of a plane partition in a box. Once the variational problem has been set up, we analyze it using a modification of the methods employed by Logan and Shepp and by Vershik and Kerov in their studies of random Young tableaux. 1.
The Asymptotics of Monotone Subsequences of Involutions
, 2001
"... We compute the limiting distributions of the lengths of the longest monotone subsequences of random (signed) involutions with or without conditions on the number of fixed points (and negated points) as the sizes of the involutions tend to infinity. The resulting distributions axe, depending on the ..."
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Cited by 50 (5 self)
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We compute the limiting distributions of the lengths of the longest monotone subsequences of random (signed) involutions with or without conditions on the number of fixed points (and negated points) as the sizes of the involutions tend to infinity. The resulting distributions axe, depending on the number of fixed points, (1) the TracyWidom distributions for the laxgest eigenvalues of random GOE, GUE, GSE matrices, (2) the normal distribution, or (3) new classes of distributions which interpolate between pairs of the Tracy Widom distributions. We also consider the second rows of the corresponding Young diagrams. In each case the convergence of moments is also shown. The proof is based on the algebraic work of the authors in [7] which establishes a connection between the statistics of random involutions and a family of orthogonal polynomials, and an asymptotic analysis of the orthogonal polynomials which is obtained by extending the RiemannHilbert analysis for the orthogonal polynomials by Delft, Johansson and the first author in [3].
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 41 (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.