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57
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
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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:
On the distribution of the largest eigenvalue in principal components analysis
 Ann. Statist
, 2001
"... Let x �1 � denote the square of the largest singular value of an n × p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x �1 � is the largest principal component variance of the covariance matrix X ′ X, or the largest eigenvalue of a pvariate Wishart distribu ..."
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Cited by 197 (2 self)
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Let x �1 � denote the square of the largest singular value of an n × p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x �1 � is the largest principal component variance of the covariance matrix X ′ X, or the largest eigenvalue of a pvariate Wishart distribution on n degrees of freedom with identity covariance. Consider the limit of large p and n with n/p = γ ≥ 1. When centered by µ p = � √ n − 1 + √ p � 2 and scaled by σ p = � √ n − 1 + √ p��1 / √ n − 1 + 1 / √ p � 1/3 � the distribution of x �1 � approaches the Tracy–Widom lawof order 1, which is defined in terms of the Painlevé II differential equation and can be numerically evaluated and tabulated in software. Simulations showthe approximation to be informative for n and p as small as 5. The limit is derived via a corresponding result for complex Wishart matrices using methods from random matrix theory. The result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts. 1. Introduction. The
Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices
, 2008
"... ..."
Limiting Distributions for a Polynuclear Growth Model With External Sources
, 2000
"... The purpose of this paper is to investigate the limiting distribution functions for a polynuclear growth model with two external sources, which was considered by Pr ahofer and Spohn in [13]. Depending on the strength of the sources, the limiting distribution functions are either the TracyWidom func ..."
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Cited by 59 (9 self)
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The purpose of this paper is to investigate the limiting distribution functions for a polynuclear growth model with two external sources, which was considered by Pr ahofer and Spohn in [13]. Depending on the strength of the sources, the limiting distribution functions are either the TracyWidom functions of random matrix theory, or a new explicit function which has the special property that its mean is zero. Moreover, we obtain transition functions between pairs of the above distribution functions in suitably scaled limits. There are also similar results for a discrete totally asymmetric exclusion process.
On the distributions of the lengths of the longest monotone subsequences in random words
"... We consider the distributions of the lengths of the longest weakly increasing and strongly decreasing subsequences in words of length N from an alphabet of k letters. (In the limit as k → ∞ these become the corresponding distributions for permutations on N letters.) We find Toeplitz determinant rep ..."
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Cited by 50 (9 self)
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We consider the distributions of the lengths of the longest weakly increasing and strongly decreasing subsequences in words of length N from an alphabet of k letters. (In the limit as k → ∞ these become the corresponding distributions for permutations on N letters.) We find Toeplitz determinant representations for the exponential generating functions (on N) of these distribution functions and show that they are expressible in terms of solutions of Painlevé V equations. We show further that in the weakly increasing case the generating function gives the distribution of the smallest eigenvalue in the k×k Laguerre random matrix ensemble and that the distribution itself has, after centering and normalizing, an N → ∞ limit which is equal to the distribution function for the largest eigenvalue in the Gaussian Unitary Ensemble of k × k hermitian matrices of trace zero. I.
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].
On the Distribution of the Largest Principal Component
 ANN. STATIST
, 2000
"... Let x (1) denote square of the largest singular value of an n p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x (1) is the largest principal component of the covariance matrix X 0 X, or the largest eigenvalue of a p variate Wishart distribution on n degr ..."
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Cited by 48 (0 self)
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Let x (1) denote square of the largest singular value of an n p matrix X, all of whose entries are independent standard Gaussian variates. Equivalently, x (1) is the largest principal component of the covariance matrix X 0 X, or the largest eigenvalue of a p variate Wishart distribution on n degrees of freedom with identity covariance. Consider the limit of large p and n with n=p = 1: When centered by p = ( p n 1+ p p) 2 and scaled by p = ( p n 1+ p p)(1= p n 1+1= p p) 1=3 the distribution of x (1) approaches the TracyWidom law of order 1, which is dened in terms of the Painleve II dierential equation, and can be numerically evaluated and tabulated in software. Simulations show the approximation to be informative for n and p as small as 5. The limit is derived via a corresponding result for complex Wishart matrices using methods from random matrix theory. The result suggests that some aspects of large p multivariate distribution theory may be easier to ...
Double scaling limit in the random matrix model: the RiemannHilbert approach
"... Abstract. We derive the double scaling limit of eigenvalue correlations in the random matrix model at critical points and we relate it to a nonlinear hierarchy of ordinary differential equations. 1. ..."
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Cited by 40 (7 self)
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Abstract. We derive the double scaling limit of eigenvalue correlations in the random matrix model at critical points and we relate it to a nonlinear hierarchy of ordinary differential equations. 1.
Exact scaling functions for onedimensional stationary KPZ growth
, 2004
"... With deep appreciation dedicated to Giovanni JonaLasinio at the occasion of his 70th birthday. We determine the stationary twopoint correlation function of the onedimensional KPZ equation through the scaling limit of a solvable microscopic model, the polynuclear growth model. The equivalence to a ..."
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Cited by 27 (5 self)
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With deep appreciation dedicated to Giovanni JonaLasinio at the occasion of his 70th birthday. We determine the stationary twopoint correlation function of the onedimensional KPZ equation through the scaling limit of a solvable microscopic model, the polynuclear growth model. The equivalence to a directed polymer problem with specific boundary conditions allows one to express the corresponding scaling function in terms of the solution to a RiemannHilbert problem related to the Painlevé II equation. We solve these equations numerically with very high precision and compare our, up to numerical rounding exact, result with the prediction of Colaiori and Moore [1] obtained from the mode coupling approximation. 1
Symmetrized random permutations
 Random Matrix Models and Their Applications, volume 40 of Mathematical Sciences Research Institute Publications
, 2001
"... Suppose that we are selecting n points, p1, p2, · · · , pn, at random in a rectangle, say R = [0, 1] × [0, 1] (see ..."
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Cited by 27 (5 self)
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Suppose that we are selecting n points, p1, p2, · · · , pn, at random in a rectangle, say R = [0, 1] × [0, 1] (see