Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach

By Percy Deift. 2000

1,505 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Applied Mathematics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/d05f2f2b0e5d5fffb3f049650655cd9c734d1e79

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In Algebra and Number Theory

In Applied Mathematics

In Statistics and Probability

Citations in the corpus

Spectral statistics of Erdős–Rényi graphs I: Local semicircle law (2011, background): “ty in the interior of the spectrum and the edge universality near the spectral edge. The bulk and edge universalities for invariant ensembles have been extensively studied; see e.g. [2, 6, 7, 27] and [1, 4, 5] for a review. A key fact that attributes to the progress in the study of invariant ensembl”
Asymptotics of Orthogonal Polynomials for a Weight with a Jump on [−1,1] (2009, background): “From [10] (see also [8]) it follows that the coefficients can be found directly in terms of the matrix Y in (85): a(2)n = lim z→∞ z (2)Y12 (z, n) Y21 (z, n)”
Universality for mathematical and physical systems (2006, neutral): “There are many ensembles of random matrices that are of interest, and we refer the reader to the classic text of Mehta [Meh] for more information (see also [Dei1]).”
A Dynamical Approach to Random Matrix Theory (2017, methodology): “The detailed proof can be found in [8] or [37].”
Matrix models for beta ensembles (2002, methodology): “The Vandermonde determinant formula of Lemma 2.7 can also be obtained from the Heine formula, as presented in Deift [5], page 44.”
Continuum limits of random matrices and the Brownian carousel (2007, methodology): “Assuming certain growth conditions on V the corresponding problem in the β = 2 case can be treated using orthogonal polynomials and Riemann-Hilbert methods, see e.g. Deift (1999), Deift et al. (1999).”
Higher‐Dimensional Coulomb Gases and Renormalized Energy Functionals (2013, background): “For more details on these aspects, we refer the reader to [29], and for an introduction to the random matrix aspect to the texts [4, 21, 52].”
A New Approach to Universality Limits Involving Orthogonal Polynomials (2007, background): “Orthogonal polynomials play an important role in random matrix theory [3], [8].”
Tracy-Widom law for the extreme eigenvalues of sample correlation matrices (2011, methodology): “1 1 2 1 2 1 1 2 ... ... ... 1 2 1 2 1 2 1 , which is a Jacobi matrix with positive subdiagonal entries. Such a Jacobi matrix has simple eigenvalues, for example, see Proposition 2.40 of [5]. Next we turn to the proof of ii). We use X(p) to denote the submatrix of X with p-th row deleted, and ”
Asymptotic Zero Behavior of Laguerre Polynomials with Negative Parameter (2002, methodology): “ﬁes the same jump conditions on ΣT ∩ ∆ε(βj) as T does. (c) We have as n → ∞, Pj(z) = I +O 1 n N(z) uniformly for |z −βj| = ε. (3.40) The Riemann-Hilbert problems for P1 and P2 are solved as in [5, 7] with the use of Airy functions, see also [6, 21]. Note that P1 and P2, as well as N, depend on n.”
Asymptotics of Hermite–Padé Rational Approximants for Two Analytic Functions with Separated Pairs of Branch Points (Case of Genus 0) (2007, methodology): “Following [31, 26] we have Ub∗ = Eb∗Vb∗Ab∗ , (4.”