## The Asymptotics of Monotone Subsequences of Involutions (2001)

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### BibTeX

@MISC{Baik01theasymptotics,

author = {Jinho Baik and Eric M. Rains},

title = {The Asymptotics of Monotone Subsequences of Involutions},

year = {2001}

}

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### Abstract

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 Tracy-Widom 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 Riemann-Hilbert analysis for the orthogonal polynomials by Delft, Johansson and the first author in [3].

### Citations

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Citation Context ...9 distribution is different. Introduce a new ensemble, Sn,m = {π ∈ ˜S2n+m : |{x : π(x) = x}| = m}. (1.7) For an involution π, the number of fixed points is equal to the number of odd parts of λt (see =-=[33]-=-). Equivalently, the number of fixed points of π is equal to λ1 − λ2 + λ3 − . . .. Thus the uniform probability measure on the set Sn,m is pushed forward to the measure dλ where ∑ µ∈Yn,m dµ Yn,m = {λ ... |

691 |
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(Show Context)
Citation Context ...rmalization constant of the probability density of eigenvalues in 1the Gaussian orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE), Gaussian symplectic ensemble (GSE), respectively (see e.g. =-=[26]-=-). Motivated by this result, we define the generalized Plancherel measure of parameter β on the set of Young diagrams (or partitions) of size n, Yn, by M β n(λ) := d β λ ∑ µ⊢n dβ µ , λ ⊢ n. (1.2) When... |

361 | On the distribution of the length of the longest increasing subsequence of random permutations
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(Show Context)
Citation Context ...polynomials, and an asymptotic analysis of the orthogonal polynomials which is obtained by extending the Riemann-Hilbert analysis for the orthogonal polynomials by P. Deift, K. Johansson, and Baik in =-=[3]-=-. 1. Introduction β-Plancherel measure In the last few years, it has been observed by many authors that there are certain connections between random permutations and/or Young tableaux, and random matr... |

312 |
Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach
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(Show Context)
Citation Context ... [46], [9], [31], [47], [28], [34], [43]). We refer readers to [1], [37], and [12] for a survey and history of Ln, and to [36] as a general reference on random matrix theory (see also the recent book =-=[11]-=-). One of the main topics in this paper is the limiting statistics of λ ∈ Yn under M1 n . From (1.1) and the results for the case when β = 2, one might guess that for β = 1 the limiting statistics of ... |

260 |
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(Show Context)
Citation Context ...nvolutions : ˜Sn = {π ∈ Sn : π = π −1 }. (1.3) The Robinson-Schensted correspondence establishes a bijection from this set to the set of standard Young tableaux of size n, π ↔ Q(π) (see e.g. 5.1.4 of =-=[24]-=-), and hence the probability measure M 1 n on Yn is just the push forward of the uniform measure on ˜ Sn. We define ˜ L (k) n (π) to be the length of the k th row of Q(π) for k ≥ 1, and we call it sim... |

259 | Level-spacing distributions and the Airy kernel
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- 1994
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Citation Context ...ich is expressed in terms of a solution to the Painlevé II equation (see Definition 2 in Section 2). The connection to random matrix theory comes from this function F2: in 1994, C. Tracy and H. Widom =-=[44]-=- proved that after proper centering and scaling (which is different from the scaling for Ln in (1.4)), the largest eigenvalue of a random matrix taken from the GUE has the same limiting distribution g... |

256 |
Monodromy preserving deformation of linear ordinary differential equations with rational coefficients
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- 1981
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Citation Context ...ew classes of distribution functions to describe the transitions from χGSE to χGOE and from χGUE to χGOE2. First, we consider the Riemann-Hilbert problem (RHP) for the Painlevé II equation (see [20], =-=[29]-=-). Let Ɣ be the real line R, oriented from +∞ to −∞, and let m(· ; x) be the solution of the following RHP: ⎧ m(z; x) is analytic in z ∈ C \ Ɣ, ⎪⎨ ( 1 −e m+(z; x) = m−(z; x) −2i((4/3)z3 ) +xz) for z ∈... |

248 | 2000b), ‘Shape fluctuations and random matrices
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Citation Context ...) moments for the general rows is obtained recently by [5]. We also mention that there are many works on similar relationships between tableaux/combinatorics and GUE random matrices. See for example, =-=[47, 9, 31, 44, 28, 34, 43]-=-. We refer the readers to [1, 37, 12] for a survey and history of Ln, and to [36] for general reference on random matrix theory (see also the recent book [11]). One of the main topics in this paper is... |

220 |
Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory
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Citation Context ... systematicRANDOM INVOLUTIONS 251 form by Deift, S. Venakides, and Zhou in [17]. The steepest descent analysis of RHP (5.4) was first conducted in [3]. The analysis of [3] has many similarities with =-=[13]-=-, [14], and [15] where the asymptotics of orthogonal polynomials on the real line with respect to a general weight is obtained, leading to a proof of universality conjectures in random matrix theory. ... |

184 | A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation
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- 1993
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Citation Context ...polynomials results stated in Section 5 by applying the steepest descent–type method to RHP (5.4). The steepest descent method for RHP’s, the Deift-Zhou method, was introduced by Deift and X. Zhou in =-=[18]-=-, developed further in [19] and [16], and finally placed in a systematicRANDOM INVOLUTIONS 251 form by Deift, S. Venakides, and Zhou in [17]. The steepest descent analysis of RHP (5.4) was first cond... |

164 |
Longest increasing and decreasing subsequences
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(Show Context)
Citation Context ...= d β λ ∑ µ⊢n dβ µ , λ ⊢ n. (1.2) When β = 2, this is the Plancherel measure which first arose in the representation theory of symmetric groups. Under the well-known Robinson-Schensted correspondence =-=[30]-=- between permutations and pairs of (standard) Young tableaux with the same shape, this measure is the push forward of the uniform probability measure on the symmetric group Sn. It is proved in [2] tha... |

156 | On orthogonal and symplectic matrix ensembles
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- 1996
(Show Context)
Citation Context ... ) ds , (2.8) F1(x) := F(x)E(x) = ( F2(x) ) 1/2 e 1 2 F4(x) := F(x) [ E(x) −1 + E(x) ] /2 = ( F2(x) ) [ 1/2 e x ∫ ∞ x u(s)ds , (2.9) ∫ ∞ 2 x u(s)ds + e 1 ∫ ∞ 2 x u(s)ds ] /2. (2.10) − 1 5In [33] and =-=[34]-=-, Tracy and Widom proved that under proper centering and scaling, the distribution of the largest eigenvalue of a random GUE/GOE/GSE matrix converges to F2(x) / F1(x) / F4(x) as the size of the matrix... |

150 | Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem
- Aldous, Diaconis
- 1999
(Show Context)
Citation Context ...5]. We also mention that there are many works on similar relationships between tableaux/combinatorics and GUE random matrices (see, e.g., [46], [9], [31], [47], [28], [34], [43]). We refer readers to =-=[1]-=-, [37], and [12] for a survey and history of Ln, and to [36] as a general reference on random matrix theory (see also the recent book [11]). One of the main topics in this paper is the limiting statis... |

150 |
Strong asymptotics of orthogonal polynomials with respect to exponential weights
- Deift, Kriecherbauer, et al.
- 1999
(Show Context)
Citation Context ...y Deift and Zhou in [13] and is developed further in [14], [11], and finally placed in a systematic form by Deift, Venakides and Zhou in [12]. The analysis of the RHP (5.3) has many similarities with =-=[10, 8, 9]-=- where the asymptotics of orthogonal polynomial on the real line with respect to a general weight is obtained, leading to a proof of universality conjectures in random matrix theory. We say that a RHP... |

147 | Discrete orthogonal polynomial ensembles and the Plancherel measure
- Johansson
- 2001
(Show Context)
Citation Context ...dlyzko and the second author, and was proved to be true for the second row in [3] for convergence in distribution and convergence of moments, and more recently for the general k th row in [28, 6] and =-=[20]-=- independently. In [28, 6] and [20], the authors prove the convergence in joint distribution, and moreover, in [6] and [20], the authors also obtained the discrete sine kernel representations for the ... |

143 | Asymptotics of Plancherel measures for symmetric groups
- Borodin, Okounkov, et al.
(Show Context)
Citation Context ...rted by numerical simulations of A. Odlyzko and Rains, and it was proved to be true for the second row L (2) n in [4]. The full conjecture for the general row L (k) n was subsequently proved in [38], =-=[10]-=-, and [32], independently. The authors of [38], [10], and [32] proved the convergence in joint distribution for general rows, and the authors of [10] and [32] obtained discrete sine kernel representat... |

136 |
A variational problem for random Young tableaux
- Logan, Shepp
- 1977
(Show Context)
Citation Context .... Denote by L (k) n the random variable λk under the Plancherel measure M 2 n , and set Ln = L (1) n . In 1977, the limiting expected shape of λ under M 2 n was obtained in [48], and independently in =-=[35]-=- for the so-called Poissonized Plancherel measure. In particular, it was shown that E(Ln) lim √ = 2. (1.3) n→∞ nRANDOM INVOLUTIONS 207 A central limit theorem for Ln was then obtained in [3]: lim n→∞... |

108 |
Asymptotics of the Plancherel measure of the symmetric group and the limiting form of Young tables
- Vershik, Kerov
- 1977
(Show Context)
Citation Context ...y of the symmetric group Sn. Denote by L (k) n the random variable λk under the Plancherel measure M 2 n , and set Ln = L (1) n . In 1977, the limiting expected shape of λ under M 2 n was obtained in =-=[48]-=-, and independently in [35] for the so-called Poissonized Plancherel measure. In particular, it was shown that E(Ln) lim √ = 2. (1.3) n→∞ nRANDOM INVOLUTIONS 207 A central limit theorem for Ln was th... |

95 |
Asymptotic values for degrees associated with strips of Young diagrans
- Regev
- 1981
(Show Context)
Citation Context ...many authors that there are certain connections between random permutations and/or Young tableaux, and random matrices. One of the earliest clues to this relationship appeared in the work of A. Regev =-=[41]-=- in 1981. A Young diagram, or equivalently a partition λ = (λ1, λ2, . . .) ⊢ n (λ1 ≥ λ2 ≥ . . . , ∑ λ j = n), is an array of n boxes with top and left adjusted as in the first picture of Figure 1, whi... |

73 |
Monodromy- and spectrum preserving deformations I.Commun
- Flaschka, Newell
- 1980
(Show Context)
Citation Context ...need new classes of distribution functions to describe the transitions from χGSE to χGOE and from χGUE to χGOE2. First, we consider the Riemann-Hilbert problem (RHP) for the Painlevé II equation (see =-=[20]-=-, [29]). Let Ɣ be the real line R, oriented from +∞ to −∞, and let m(· ; x) be the solution of the following RHP: ⎧ m(z; x) is analytic in z ∈ C \ Ɣ, ⎪⎨ ( 1 −e m+(z; x) = m−(z; x) −2i((4/3)z3 ) +xz) f... |

69 | Algebraic aspects of increasing subsequences
- Baik, Rains
(Show Context)
Citation Context ...· · , n} onto itself satisfying π = π −1 and π(x) = −π(−x). From the tableaux point of view, this set is in one-to-one correspondence with the set of self-dual Young tableaux, or domino tableaux (see =-=[4]-=-). Similarly we define ˜ L ,(k) n (π) to be the length of the kth row of π ∈ ˜ Sn and denote by ˜ Ln (π) the length of the longest increasing subsequence of π. We equip this ensemble with the uniform ... |

69 |
An entension of Schensted’s theorem
- Greene
- 1974
(Show Context)
Citation Context ...hforward of the uniform probability measure on Sn. Moreover, under this correspondence, λ1(π) is equal to the length of the longest increasing subsequence of π. More generally, a theorem of C. Greene =-=[26]-=- says that λ1(π) + · · · + λk(π) is equal to the length of the longest so-called k-increasing subsequence of π. Thus the difference of the lengths of the longest k-increasing subsequence and the longe... |

68 |
A boundary value problem associated with the second Painlevé transcendent and the Korteweg-de equation, Archive for Rational Mechanics and Analysis 73
- Hastings, McLeod
- 1980
(Show Context)
Citation Context ...boundary condition uxx = 2u 3 + xu (2.1) u(x) ∼ − Ai(x) as x → +∞, (2.2) where Ai is the Airy function. The proof of the (global) existence and the uniqueness of the solution was first established in =-=[27]-=-: the asymptotics as x → ±∞ are (see,RANDOM INVOLUTIONS 215 e.g., [27], [19]) ( e−(4/3)x u(x) = − Ai(x) + O 3/2 x1/4 ) as x → +∞, (2.3) √ ( ( −x 1 u(x) = − 1 + O 2 x2 )) as x → −∞. (2.4) Recall that ... |

66 | Random matrices and random permutations
- Okounkov
- 2000
(Show Context)
Citation Context ...rted by numerical simulations of Odlyzko and the second author, and was proved to be true for the second row L (2) n in [4]. The full conjecture for the general row L (k) n was subsequently proved by =-=[38]-=-, [10] and [32], independently. The authors in [38, 10, 32] proved the convergence in joint distribution for general rows, and also in [10, 32], the authors obtained discrete sine kernel representatio... |

65 | Limit theorems for height fluctuations in a class of discrete space and time growth models
- Gravner, Tracy, et al.
- 2001
(Show Context)
Citation Context ... longest path contains few diagonal points (none if m = 0) and we are in the situation of a 2-dimensional maximization problem. In this case it has been believed, and in a few cases (e.g., [3], [32], =-=[25]-=-) it has been proved, that the fluctuation has order (mean) 1/3 . (For random permutations, which have a similar interpretation as a point selection process, one can see from the scaling in (1.4) that... |

59 |
New results in small dispersion KdV by an extension of the steepest descent method for Riemann-Hilbert problems
- Deift, Venakides, et al.
- 1997
(Show Context)
Citation Context ... type method for RHP’s, the Deift-Zhou method, was introduced by Deift and Zhou in [13] and is developed further in [14], [11], and finally placed in a systematic form by Deift, Venakides and Zhou in =-=[12]-=-. The analysis of the RHP (5.3) has many similarities with [10, 8, 9] where the asymptotics of orthogonal polynomial on the real line with respect to a general weight is obtained, leading to a proof o... |

58 |
Painleve equations and their appearance in quantum gravity
- Fokas, Its, et al.
- 1991
(Show Context)
Citation Context ...que; hence (5.3) is the unique solution of RHP (5.4). This RHP formulation of orthogonal polynomials on the unit circle given in [3] is an adaptation of a result of A. Fokas, A. Its, and A. Kitaev in =-=[21]-=-, where they considered orthogonal polynomials on the real line.RANDOM INVOLUTIONS 229 From (5.3), the quantities in (5.2) are equal to Nk−1(t) −1 = −Y21(0; k; t), (5.5) πk(z; t) = Y11(z; k; t), (5.6... |

56 | Limiting distributions for a polynuclear growth model with external sources
- Baik, Rains
(Show Context)
Citation Context ...tion as the size of m varies. The results (1.11)–(1.13) show that α = 1 is the transition point. The fixed points play the role of adding a special line in the 2-dimensional maximization problem (see =-=[6]-=- for a relevant work where two special lines are added to a 2-dimensional maximization problem). We note that when (see (1.5)). This α = 1, L n,[ √ 2n] in (1.12) has the same limiting distribution as ... |

54 | On the distribution of the length of the second row of a Young diagram under the Plancherel measure
- Baik, Deift, et al.
(Show Context)
Citation Context ...row corresponds to the k th eigenvalue of a random GUE matrix. This conjecture was supported by numerical simulations of Odlyzko and the second author, and was proved to be true for the second row in =-=[4]-=- for convergence in distribution and convergence of moments. Recently, in [38], [10] and [29], independently, the authors proved the convergence in joint distribution for general k th row. Moreover, i... |

51 | 2001), ‘On the distributions of the lengths of the longest monotone subsequences in random words’, Probab. Theory Rel
- Tracy, Widom
(Show Context)
Citation Context ... of various type of permutations. Signed permutations are considered in [32], colored permutations in [5], a certain statistical model related to generalized permutations in [21], and random words in =-=[31]-=- and [20]. For the first three cases, the limiting distribution is expressed in terms of the Tracy-Widom distribution F2(x), and for random words from k letters, the limiting distribution is equal to ... |

51 |
Longest increasing and decreasing subsequences, Canad
- Schensted
- 1961
(Show Context)
Citation Context ...lancherel measure case M2 n given by Johansson [31], [32]. Random involutions The Plancherel measure M 2 n has a nice combinatorial interpretation. The well-known Robinson-Schensted correspondence in =-=[42]-=- establishes a bijection between the permutations π of size n and the pairs of standard Young tableaux (P, Q) where the shape of P and the shape of Q are the same and the shape of P (or Q), denoted by... |

50 |
Asymptotics for the Painlevé
- Deift, Zhou
- 1995
(Show Context)
Citation Context ...ft (resp., right) of the contour Ɣ: m±(z; x) = limɛ↓0 m(z ∓ iɛ; x). Relation (2.15) corresponds to the RHP for the PII equation with the special monodromy data p = −q = 1, r = 0 (see [20], [29], also =-=[22]-=-, [19]). In particular, if the solution is expanded at z = ∞, m(z; x) = I + m1(x) ( 1 + O z z2 ) as z → ∞, (2.16) we have 2i(m1(x))12 = −2i(m1(x))21 = u(x), (2.17) 2i(m1(x))22 = −2i(m1(x))11 = v(x), (... |

49 |
The longest increasing subsequence in a random permutation and a unitary random matrix
- Johansson
- 1998
(Show Context)
Citation Context ...s paper, and thus the results in this paper can be employed to answer asymptotic questions in the above applications. The proofs of our theorems use the Poissonization and de-Poissonization scheme of =-=[30]-=- and [3]. We define the Poisson generating function, for example, for L n,m by (see Definition 6) Q l (λ1, λ2) := e −λ1−λ2 ∑ n1,n2≥0 λ n1 1 λn2 2 n1!n2! Pr( Ln2,n1 ≤ l) . (1.27) A generalization of th... |

43 |
2000), ‘Integrable systems and combinatorial theory
- Deift
(Show Context)
Citation Context ...y [5]. We also mention that there are many works on similar relationships between tableaux/combinatorics and GUE random matrices. See for example, [47, 9, 31, 44, 28, 34, 43]. We refer the readers to =-=[1, 37, 12]-=- for a survey and history of Ln, and to [36] for general reference on random matrix theory (see also the recent book [11]). One of the main topics in this paper is the limiting statistics of λ ∈ Yn un... |

39 |
Universal distributions for growth processes in 1 + 1 dimensions and random matrices
- Praehofer, Spohn
(Show Context)
Citation Context ...lts of this paper were announced in [8]. Since we completed this paper, there have been two applications. One is to random vicious walker models [23, 8], and the other is to polynuclear growth models =-=[42, 41, 6]-=-. Indeed there are bijections between the above two applications and various ensembles considered in this paper, and thus the results in this paper can be employed to answer asymptotic questions in th... |

37 | Transition probabilities of continual Young diagrams and the Markov moment problem
- Kerov
- 1993
(Show Context)
Citation Context ...orrelation functions, which is analogous to the sine kernel in the GUE case. On the other hand, in [25, 35, 36], the limiting shape of a typical Young diagram under Plancherel measure is obtained. In =-=[23]-=-, it is shown that this limiting shape can be viewed as Wigner’s semicircle law. A conceptual background for this connection between Yn with the Plancherel measure and GUE matrices is provided by Okou... |

32 | On longest increasing subsequences in random permutations
- Odlyzko, Rains
- 1998
(Show Context)
Citation Context ...e also mention that there are many works on similar relationships between tableaux/combinatorics and GUE random matrices (see, e.g., [46], [9], [31], [47], [28], [34], [43]). We refer readers to [1], =-=[37]-=-, and [12] for a survey and history of Ln, and to [36] as a general reference on random matrix theory (see also the recent book [11]). One of the main topics in this paper is the limiting statistics o... |

31 | Statistical self-similarity of one-dimensional growth processes, Physica A 279:342–360
- Prähofer, Spohn
- 2000
(Show Context)
Citation Context ...ere announced in [8]. Since we completed this paper, there have been two applications. One is to random vicious walker models (see [23], [8]), and the other is to polynuclear growth models (see [40], =-=[39]-=-, [6]). Indeed, there are bijections between the above two applications and various ensembles considered in this paper, and thus the results in this paper can be employed to answer asymptotic question... |

28 | Widom,”Random words, Toeplitz determinants and integrable systems
- Its, Tracy, et al.
- 2001
(Show Context)
Citation Context ... general rows is obtained recently in [5]. We also mention that there are many works on similar relationships between tableaux/combinatorics and GUE random matrices (see, e.g., [46], [9], [31], [47], =-=[28]-=-, [34], [43]). We refer readers to [1], [37], and [12] for a survey and history of Ln, and to [36] as a general reference on random matrix theory (see also the recent book [11]). One of the main topic... |

28 |
X.: Asymptotics for the Painleve II equation
- Deift, Zhou
(Show Context)
Citation Context ...∼ −Ai(x) as x → +∞, (2.2) where Ai is the Airy function. The proof of the (global) existence and the uniqueness of the solution was first established in [18] : the asymptotics as x → −∞ are (see e.g. =-=[18, 14]-=-) u(x) u(x) ( −(4/3)x e = −Ai(x) + O 3/2 x1/4 √ −x = − 2 Recall that Ai(x) ∼ e−(2/3)x3/2 2 √ πx 1/4 as x → +∞. Define so that v ′ (x) = (u(x)) 2 . ) , as x → +∞, (2.3) ( 1 + O ( 1 x2 ) ) , as x → −∞. ... |

27 |
The collisionless shock region for the long-time behavior of solutions of the KdV equation
- Deift, Venakides, et al.
- 1994
(Show Context)
Citation Context ...n 5 by applying the steepest descent–type method to RHP (5.4). The steepest descent method for RHP’s, the Deift-Zhou method, was introduced by Deift and X. Zhou in [18], developed further in [19] and =-=[16]-=-, and finally placed in a systematicRANDOM INVOLUTIONS 251 form by Deift, S. Venakides, and Zhou in [17]. The steepest descent analysis of RHP (5.4) was first conducted in [3]. The analysis of [3] ha... |

27 |
Random matrices and random permutations, Internat
- Okounkov
(Show Context)
Citation Context ... supported by numerical simulations of A. Odlyzko and Rains, and it was proved to be true for the second row L (2) n in [4]. The full conjecture for the general row L (k) n was subsequently proved in =-=[38]-=-, [10], and [32], independently. The authors of [38], [10], and [32] proved the convergence in joint distribution for general rows, and the authors of [10] and [32] obtained discrete sine kernel repre... |

25 |
Inter-relationships between orthogonal, unitary and symplectic matrix ensembles
- Forrester, Rains
- 2001
(Show Context)
Citation Context ...mply that the first and second rows of a random involution have the same limiting distribution as the first and second eigenvalues of GOE, respectively.RANDOM INVOLUTIONS 225 Recently the authors of =-=[24]-=- proved that the same property holds true for GOE and GSE. They also proved, among many other things, that the (2k)th “eigenvalue” of a superimposition of two random GOE matrices has the same distribu... |

23 | Random unitary matrices, permutations and Painleve
- Tracy, Widom
- 1999
(Show Context)
Citation Context ...erings. In another direction, there are results on the limiting distribution of the length of the longest increasing subsequence of various type of permutations. Signed permutations are considered in =-=[32]-=-, colored permutations in [5], a certain statistical model related to generalized permutations in [21], and random words in [31] and [20]. For the first three cases, the limiting distribution is expre... |

23 |
Asymptotic behavior of the maximum and generic dimensions of irreducible representations of the symmetric group., Functional Anal
- Vershik, Kerov
- 1985
(Show Context)
Citation Context ...], the authors also obtained the discrete sine kernel representations for the bulk scaling limit of correlation functions, which is analogous to the sine kernel in the GUE case. On the other hand, in =-=[25, 35, 36]-=-, the limiting shape of a typical Young diagram under Plancherel measure is obtained. In [23], it is shown that this limiting shape can be viewed as Wigner’s semicircle law. A conceptual background fo... |

22 | Longest increasing subsequences of random colored permutations
- Borodin
- 1999
(Show Context)
Citation Context ...) moments for the general rows is obtained recently in [5]. We also mention that there are many works on similar relationships between tableaux/combinatorics and GUE random matrices (see, e.g., [46], =-=[9]-=-, [31], [47], [28], [34], [43]). We refer readers to [1], [37], and [12] for a survey and history of Ln, and to [36] as a general reference on random matrix theory (see also the recent book [11]). One... |

21 | Generalized riffle shuffles and quasisymmetric functions
- Stanley
(Show Context)
Citation Context ...s is obtained recently in [5]. We also mention that there are many works on similar relationships between tableaux/combinatorics and GUE random matrices (see, e.g., [46], [9], [31], [47], [28], [34], =-=[43]-=-). We refer readers to [1], [37], and [12] for a survey and history of Ln, and to [36] as a general reference on random matrix theory (see also the recent book [11]). One of the main topics in this pa... |

19 | Random walks and random permutations
- Forrester
(Show Context)
Citation Context ...S 213 L n,m+,m− ,(1) = L n,m+,m− . (1.26) The results of this paper were announced in [8]. Since we completed this paper, there have been two applications. One is to random vicious walker models (see =-=[23]-=-, [8]), and the other is to polynuclear growth models (see [40], [39], [6]). Indeed, there are bijections between the above two applications and various ensembles considered in this paper, and thus th... |

16 | Random words, quantum statistics, central limits, random matrices
- Kuperberg
(Show Context)
Citation Context ...al rows is obtained recently in [5]. We also mention that there are many works on similar relationships between tableaux/combinatorics and GUE random matrices (see, e.g., [46], [9], [31], [47], [28], =-=[34]-=-, [43]). We refer readers to [1], [37], and [12] for a survey and history of Ln, and to [36] as a general reference on random matrix theory (see also the recent book [11]). One of the main topics in t... |

14 | A Fredholm determinant identity and the convergence of moments for random Young tableaux
- Baik, Deift, et al.
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Citation Context .... Then in the same paper [3], it was conjectured that L (k) n under M2 n of moments for L (1) n and L (2) n , respectively. Convergence of (joint) moments for the general rows is obtained recently in =-=[5]-=-. We also mention that there are many works on similar relationships between tableaux/combinatorics and GUE random matrices (see, e.g., [46], [9], [31], [47], [28], [34], [43]). We refer readers to [1... |