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THE KARDARPARISIZHANG EQUATION AND UNIVERSALITY CLASS
, 2011
"... Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or regularity) and expanding the breadth of its universality class. Over the past twenty five years a new univ ..."
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Cited by 8 (1 self)
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Brownian motion is a continuum scaling limit for a wide class of random processes, and there has been great success in developing a theory for its properties (such as distribution functions or regularity) and expanding the breadth of its universality class. Over the past twenty five years a new universality class has emerged to describe a host of important physical and probabilistic models (including one dimensional interface growth processes, interacting particle systems and polymers in random environments) which display characteristic, though unusual, scalings and new statistics. This class is called the KardarParisiZhang (KPZ) universality class and underlying it is, again, a continuum object – a nonlinear stochastic partial differential equation – known as the KPZ equation. The purpose of this survey is to explain the context for, as well as the content of a number of mathematical breakthroughs which have culminated in the derivation of the exact formula for the distribution function of the KPZ equation started with narrow wedge initial data. In particular we emphasize three topics: (1) The approximation of the KPZ equation through the weakly asymmetric simple exclusion process; (2) The derivation of the exact onepoint distribution of the solution to the KPZ equation with narrow wedge initial data; (3) Connections with directed polymers in random media. As the purpose of this article is to survey and review, we make precise statements but provide only heuristic arguments with indications of the technical complexities necessary to make such arguments mathematically rigorous.
THE CHEBOP SYSTEM FOR AUTOMATIC SOLUTION OF DIFFERENTIAL EQUATIONS
, 2008
"... In Matlab, it would be good to be able to solve a linear differential equation by typing u = L\f, wheref, u, andLare representations of the righthand side, the solution, and the differential operator with boundary conditions. Similarly it would be good to be able to exponentiate an operator with ex ..."
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Cited by 6 (5 self)
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In Matlab, it would be good to be able to solve a linear differential equation by typing u = L\f, wheref, u, andLare representations of the righthand side, the solution, and the differential operator with boundary conditions. Similarly it would be good to be able to exponentiate an operator with expm(L) or determine eigenvalues and eigenfunctions with eigs(L). A system is described in which such calculations are indeed possible, at least in one space dimension, based on the previously developed chebfun system in objectoriented Matlab. The algorithms involved amount to spectral collocation methods on Chebyshev grids of automatically determined resolution.
On the Numerical Evaluation of Distributions in Random Matrix Theory: A Review
, 2010
"... Abstract. In this paper we review and compare the numerical evaluation of those probability distributions in random matrix theory that are analytically represented in terms of Painlevé transcendents or Fredholm determinants. Concrete examples for the Gaussian and Laguerre (Wishart) βensembles and t ..."
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Cited by 6 (0 self)
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Abstract. In this paper we review and compare the numerical evaluation of those probability distributions in random matrix theory that are analytically represented in terms of Painlevé transcendents or Fredholm determinants. Concrete examples for the Gaussian and Laguerre (Wishart) βensembles and their various scaling limits are discussed. We argue that the numerical approximation of Fredholm determinants is the conceptually more simple and efficient of the two approaches, easily generalized to the computation of joint probabilities and correlations. Having the means for extensive numerical explorations at hand, we discovered new and surprising determinantal formulae for the kth largest (or smallest) level in the edge scaling limits of the Orthogonal and Symplectic Ensembles; formulae that in turn led to improved numerical evaluations. The paper comes with a toolbox of Matlab functions that facilitates further mathematical experiments by the reader.
2008, The Airy 1 process is not the limit of the largest eigenvalue in GOE matrix diffusion
"... Abstract Using a systematic approach to evaluate Fredholm determinants numerically, we provide convincing evidence that the Airy1process, arising as a limit law in stochastic surface growth, is not the limit law for the evolution of the largest eigenvalue in GOE matrix diffusion. ..."
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Cited by 5 (3 self)
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Abstract Using a systematic approach to evaluate Fredholm determinants numerically, we provide convincing evidence that the Airy1process, arising as a limit law in stochastic surface growth, is not the limit law for the evolution of the largest eigenvalue in GOE matrix diffusion.
ASYMPTOTIC INDEPENDENCE OF THE EXTREME EIGENVALUES OF GUE
, 902
"... Abstract. We give a short, operatortheoretic proof of the asymptotic independence (including a first correction term) of the minimal and maximal eigenvalue of the n × n Gaussian Unitary Ensemble in the large matrix limit n → ∞. This is done by representing the joint probability distribution of the ..."
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Cited by 3 (2 self)
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Abstract. We give a short, operatortheoretic proof of the asymptotic independence (including a first correction term) of the minimal and maximal eigenvalue of the n × n Gaussian Unitary Ensemble in the large matrix limit n → ∞. This is done by representing the joint probability distribution of the extreme eigenvalues as the Fredholm determinant of an operator matrix that asymptotically becomes diagonal. As a corollary we obtain that the correlation of the extreme eigenvalues asymptotically behaves like n −2/3 /4σ 2, where σ 2 denotes the variance of the Tracy–Widom distribution. While we conjecture that the extreme eigenvalues are asymptotically independent for Wigner random hermitian matrix ensembles in general, the actual constant in the asymptotic behavior of the correlation turns out to be specific and can thus be used to distinguish the Gaussian Unitary Ensemble statistically from other Wigner ensembles.
Review notes Introductory Fredholm theory and computation
, 2010
"... Abstract We provide an introduction to Fredholm theory and discuss using the Fredholm determinant to compute purepoint spectra. ..."
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Abstract We provide an introduction to Fredholm theory and discuss using the Fredholm determinant to compute purepoint spectra.
ASYMPTOTIC INDEPENDENCE OF THE EXTREME EIGENVALUES OF GUE FOLKMAR BORNEMANN ∗
, 902
"... Abstract. We give a short, operatortheoretic proof of the asymptotic independence of the minimal and maximal eigenvalue of the n × n Gaussian Unitary Ensemble in the large matrix limit n → ∞. This is done by representing the joint probability distribution of those extreme eigenvalues as the Fredhol ..."
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Abstract. We give a short, operatortheoretic proof of the asymptotic independence of the minimal and maximal eigenvalue of the n × n Gaussian Unitary Ensemble in the large matrix limit n → ∞. This is done by representing the joint probability distribution of those extreme eigenvalues as the Fredholm determinant of an operator matrix that asymptotically becomes diagonal. The method is amenable to explicitly establish the leading order term of an asymptotic expansion. As a corollary we obtain that the correlation of the extreme eigenvalues asymptotically behaves like n −2/3 /4σ 2, where σ 2 denotes the variance of the Tracy–Widom distribution. We consider the n × n Gaussian Unitary Ensemble (GUE) with the joint probability distribution of its (unordered) eigenvalues given by pn(λ1,..., λn) = cne −λ2 1 −···−λ2 n ∏i<j λ i − λ j  2 and denote the induced random variables of the minimal and maximal eigenvalue by λ (n) min and λ(n) max. Bianchi, Debbah and Najim (2008) have recently shown the asymptotic independence of the edgescaled extreme eigenvalues, that is, P ˜λ (n) min � x, ˜λ (n) max � y = P ˜λ (n) min
ON THE CONVERGENCE RATES OF GAUSS AND CLENSHAW–CURTIS QUADRATURE FOR FUNCTIONS OF LIMITED REGULARITY ∗
"... Abstract. We study the optimal general rate of convergence of the npoint quadrature rules of Gauss and Clenshaw–Curtis when applied to functions of limited regularity: if the Chebyshev coefficients decay at a rate O(n−s−1)forsomes>0, Clenshaw–Curtis and Gauss quadrature inherit exactly this rate. T ..."
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Abstract. We study the optimal general rate of convergence of the npoint quadrature rules of Gauss and Clenshaw–Curtis when applied to functions of limited regularity: if the Chebyshev coefficients decay at a rate O(n−s−1)forsomes>0, Clenshaw–Curtis and Gauss quadrature inherit exactly this rate. The proof (for Gauss, if 0 <s<2, there is numerical evidence only) is based on work of Curtis, Johnson, Riess, and Rabinowitz from the early 1970s and on a refined estimate for Gauss quadrature applied to Chebyshev polynomials due to Petras (1995). The convergence rate of both quadrature rules is up to one power of n better than polynomial best approximation; hence, the classical proof strategy that bounds the error of a quadrature rule with positive weights by polynomial best approximation is doomed to fail in establishing the optimal rate. Key words. Gauss and Clenshaw–Curtis quadrature, Chebyshev expansion, convergence rate
DOI 10.1007/s1095501101555 Asymptotics for the Covariance of the Airy2 Process
"... this paper we compute some of the higher order terms in the asymptotic behavior of the two point function P(A2(0) ≤ s1, A2(t) ≤ s2), extending the previous work of Adler and van Moerbeke (arXiv:math.PR/0302329; Ann. Probab. 33, 1326–1361, 2005)and Widom (J. Stat. Phys. 115, 1129–1134, 2004). We pr ..."
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this paper we compute some of the higher order terms in the asymptotic behavior of the two point function P(A2(0) ≤ s1, A2(t) ≤ s2), extending the previous work of Adler and van Moerbeke (arXiv:math.PR/0302329; Ann. Probab. 33, 1326–1361, 2005)and Widom (J. Stat. Phys. 115, 1129–1134, 2004). We prove that it is possible to represent any order asymptotic approximation as a polynomial and integrals of the Painlevé II function q and its derivative q ′. Further, for up to tenth order we give this asymptotic approximation as a linear combination of the TracyWidom GUE density function f2 and its derivatives. As a corollary to this, the asymptotic covariance is expressed up to tenth order in terms of the moments of the TracyWidom GUE distribution.