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Kenneth D.T-R McLaughlin

Bio: Kenneth D.T-R McLaughlin is an academic researcher from Colorado State University. The author has contributed to research in topics: Orthogonal polynomials & Discrete orthogonal polynomials. The author has an hindex of 24, co-authored 62 publications receiving 3864 citations. Previous affiliations of Kenneth D.T-R McLaughlin include Princeton University & Ohio State University.


Papers
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Journal ArticleDOI
TL;DR: In this article, asymptotics for orthogonal polynomials with respect to varying exponential weights wn(x)dx = e−nV(x)-dx on the line as n ∞ were considered.
Abstract: We consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn(x)dx = e−nV(x)dx on the line as n ∞. The potentials V are assumed to be real analytic, with sufficient growth at infinity. The principle results concern Plancherel-Rotach-type asymptotics for the orthogonal polynomials down to the axis. Using these asymptotics, we then prove universality for a variety of statistical quantities arising in the theory of random matrix models, some of which have been considered recently in [31] and also in [4]. An additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials (see also [4]). The orthogonal polynomial problem is formulated as a Riemann-Hilbert problem following [19, 20]. The Riemann-Hilbert problem is analyzed in turn using the steepest-descent method introduced in [12] and further developed in [11, 13]. A critical role in our method is played by the equilibrium measure dμV for V as analyzed in [8]. © 1999 John Wiley & Sons, Inc.

994 citations

Journal ArticleDOI
TL;DR: In this paper, asymptotics of orthogonal polynomials with respect to weights w(x)dx = e Q(x)-dx on the real line were considered.
Abstract: We consider asymptotics of orthogonal polynomials with respect to weights w(x)dx= e Q(x) dx on the real line, where Q(x)=∑ 2m k=0 qkx k , q2m> 0, denotes a polynomial of even order with positive leading coefficient. The orthogonal polynomial problem is formulated as a Riemann-Hilbert problem following [22, 23]. We employ the steepest-descent-type method introduced in [18] and further developed in [17, 19] in order to obtain uniform Plancherel-Rotach-type asymptotics in the entire complex plane, as well as asymptotic formulae for the zeros, the leading coefficients, and the recurrence coefficients of the orthogonal polynomials. c 1999 John Wiley & Sons, Inc.

724 citations

Journal ArticleDOI
TL;DR: In this paper, the authors used techniques from the theory of ODEs and also from inverse scattering theory to obtain a variety of results on the regularity and support properties of the equilibrium measure for logarithmic potentials on the finite interval?1, 1], in the presence of an external fieldV.

268 citations

Book
01 Jan 2003
TL;DR: In this paper, the authors presented the first asymptotic analysis via completely integrable techniques, of the focusing nonlinear Schrodinger equation in the semiclassical asyptotic regime.
Abstract: This book represents the first asymptotic analysis, via completely integrable techniques, of the initial value problem for the focusing nonlinear Schrodinger equation in the semiclassical asymptotic regime. This problem is a key model in nonlinear optical physics and has increasingly important applications in the telecommunications industry. The authors exploit complete integrability to establish pointwise asymptotics for this problem's solution in the semiclassical regime and explicit integration for the underlying nonlinear, elliptic, partial differential equations suspected of governing the semiclassical behavior. In doing so they also aim to explain the observed gradient catastrophe for the underlying nonlinear elliptic partial differential equations, and to set forth a detailed, pointwise asymptotic description of the violent oscillations that emerge following the gradient catastrophe. To achieve this, the authors have extended the reach of two powerful analytical techniques that have arisen through the asymptotic analysis of integrable systems: the Lax-Levermore-Venakides variational approach to singular limits in integrable systems, and Deift and Zhou's nonlinear Steepest-Descent/Stationary Phase method for the analysis of Riemann-Hilbert problems. In particular, they introduce a systematic procedure for handling certain Riemann-Hilbert problems with poles accumulating on curves in the plane. This book, which includes an appendix on the use of the Fredholm theory for Riemann-Hilbert problems in the Holder class, is intended for researchers and graduate students of applied mathematics and analysis, especially those with an interest in integrable systems, nonlinear waves, or complex analysis.

249 citations


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Posted Content
18 Dec 2005
TL;DR: In this paper, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed and orthogonality on the unit circle is not discussed.
Abstract: In this survey, different aspects of the theory of orthogonal polynomials of one (real or complex) variable are reviewed Orthogonal polynomials on the unit circle are not discussed

5,648 citations

Book
21 Dec 2009
TL;DR: The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial) as mentioned in this paper.
Abstract: The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader's understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence.

1,289 citations

Journal ArticleDOI
TL;DR: In this paper, the authors consider the problem of finding an increasing subsequence in a group of permutations of 1,2,..., N, and show that the longest increasing subsequences are 1 2 4 and 1 3 4, respectively.
Abstract: Let SN be the group of permutations of 1,2,..., N. If 7r E SN, we say that 7(i1),... , 7F(ik) is an increasing subsequence in 7r if il < i2 < ... < ik and 7r(ii) < 7r(i2) < ...< 7r(ik). Let 1N(r) be the length of the longest increasing subsequence. For example, if N = 5 and 7r is the permutation 5 1 3 2 4 (in one-line notation: thus 7r(1) = 5, 7r(2) = 1, ... ), then the longest increasing subsequences are 1 2 4 and 1 3 4, and N() = 3. Equip SN with uniform distribution,

1,265 citations

Journal ArticleDOI
TL;DR: In this article, the authors studied a random Groeth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process and showed that shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble.
Abstract: We study a certain random groeth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble.

1,031 citations