Random Matrix Theory and Its Applications

Bai, Zhidong; Chen, Yang; Liang, Ying‐Chang

doi:10.1142/7285

Cited by 11 publications

(55 citation statements)

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“…Still focusing on the right edge setting, we prove in Section 4.5.1 that K (0) N (x, y) does not contribute in the large N limit. Moreover, we prove the convergence of kernel K (1) N (x, y) to the Airy kernel in an appropriate sense and then complete the proof of Theorem 3(b). For this last step, we use a different approach than in [8,31]: instead of relying on a factorization trick and the Hölder inequality to obtain the trace class convergence, we use an argument involving the regularized Fredholm determinant det 2 to show the convergence of the Fredholm determinants.…”

Section: Outline Of the Proofmentioning

confidence: 81%

“…As a consequence, proving the Tracy-Widom fluctuations boils down to establishing the appropriate convergence of rescaled versions K N (x, y) of the kernel K N (x, y) toward the Airy kernel. To this end, we split K N (x, y) into two parts, K (0) N (x, y) and K (1) N (x, y), each involving different integration contours.…”

Section: Outline Of the Proofmentioning

confidence: 99%

“…Step 2 (contours deformations). Anticipating the forthcoming asymptotic analysis, we focus in Section 4.4 on right edges and prove the existence of appropriate integration contours coming with K (0) N (x, y) and K (1) N (x, y); the case of a left edge is deferred to Section 4.6. Obtaining appropriate explicit contours is usually the hard part in the asymptotic analysis; see, in particular, [31].…”

Section: Outline Of the Proofmentioning

confidence: 99%

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Large complex correlated Wishart matrices: Fluctuations and asymptotic independence at the edges

Hachem¹,

Hardy²,

Najim³

2016

Ann. Probab.

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We study the asymptotic behavior of eigenvalues of large complex correlated Wishart matrices at the edges of the limiting spectrum. In this setting, the support of the limiting eigenvalue distribution may have several connected components. Under mild conditions for the population matrices, we show that for every generic positive edge of that support, there exists an extremal eigenvalue which converges almost surely toward that edge and fluctuates according to the Tracy-Widom law at the scale N 2/3 . Moreover, given several generic positive edges, we establish that the associated extremal eigenvalue fluctuations are asymptotically independent. Finally, when the leftmost edge is the origin (hard edge), the fluctuations of the smallest eigenvalue are described by mean of the Bessel kernel at the scale N 2 .

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Section: Outline Of the Proofmentioning

confidence: 81%

Section: Outline Of the Proofmentioning

confidence: 99%

Section: Outline Of the Proofmentioning

confidence: 99%

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Large complex correlated Wishart matrices: Fluctuations and asymptotic independence at the edges

Hachem¹,

Hardy²,

Najim³

2016

Ann. Probab.

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“…If v(x) = − ln w(x; t) = (λ + β) ln(x + t) − γ ln x + x + t, and λ < 0, β > 0, γ > 0, t > 0, γ − λ − β > −1, x ∈ (0, ∞), the evaluation of the orthogonal polynomials at the x = 0, reads, (−1) n P n (0; t, γ) ∼ n n+γ−α−β+ 1 2 e −n exp − γ + 1 2 ln γ + γ + λ + β 2 ln(nt)…”

Section: )mentioning

confidence: 99%

Perturbed Hankel determinant, correlation functions and Painlevé equations

Chen¹,

Chen²,

Fan

2016

Journal of Mathematical Physics

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We continue with the study of the Hankel determinant, D n (t, α, β) := det , generated by a Pollaczek-Jacobi type weight,This reduces to the "pure" Jacobi weight at t = 0. We may take α ∈ R , in the situation while t is strictly greater than 0. It was shown in Chen and Dai (2010), that the logarithmic derivative of this Hankel determinant satisfies a Jimbo-Miwa-Okamoto σ -form of Painlevé V ( P V ). In fact the logarithmic of the Hankel determinant has an integral representation in terms of a particular P V . In this paper, we show that, under a double scaling, where n the dimension of the Hankel matrix tends to ∞ , and t tends to 0 + , such that s := 2n 2 t is finite, the double scaled Hankel determinant (effectively an operator determinant) has an integral representation in terms of a particular P III ′ . Expansions of the scaled Hankel determinant for small and large s are found. A further double scaling with α = −2n + λ, where n → ∞ and t, tends to 0 + , such that s := nt is finite. In this situation the scaled Hankel determinant has an integral representation in terms of a particular P V , and its small and large s asymptotic expansions are also found. The reproducing kernel in terms of monic polynomials orthogonal with respect to the Pollaczek-Jacobi type weight, under the origin (or hard edge) scaling may be expressed in terms of the solutions of a second order linear ordinary differential equation (ODE). * chenminfst@gmail.com † Corresponding author(Yang Chen), yayangchen@umac.mo and yangbrookchen@yahoo.co.uk ‡ faneg@fudan.edu.cn 1 With special choices of the parameters, the limiting (double scaled) kernel and the second order ODE degenerate to Bessel kernel and the Bessel differential equation, respectively. We also applied this method to polynomials orthogonal with respect to the perturbed Laguerre weight; w(x; t, α) := x α e −x e −t/x , 0 ≤ x < ∞, α > 0, t > 0. The scaled kernel at origin of this perturbed Laguerre ensemble has the same behavior with the above limiting kernel, although difference scaled schemes are adopted on these two kernels.2

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“…The numbers of rows in this case are specified as m A n and m C <n. This results in the following structure of the GSVD [38,39], based on Equations (42) and (43):…”

Section: The Regularized Solution For Casementioning

confidence: 99%

Regularization in model updating

Titurus

Friswell

2007

Numerical Meth Engineering

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SUMMARYThis paper presents the theory of sensitivity-based model updating with a special focus on the properties of the solution that result from the combination of optimization of the response prediction with a priori information about the uncertain parameters. Model updating, together with the additional regularization criterion, is an optimization with two objective functions, and must be linearized to obtain the solution. Structured solutions are obtained, based on the generalized singular value decomposition (GSVD), and specific features of the parameter and response paths as the regularization parameter varies are explored. The four different types of spaces that arise in the solution are discussed together with the characteristics of the regularized solution families. These concepts are demonstrated on a simulated discrete example and on an experimental case study.

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Random Matrix Theory and Its Applications

Cited by 11 publications

References 0 publications

Large complex correlated Wishart matrices: Fluctuations and asymptotic independence at the edges

Large complex correlated Wishart matrices: Fluctuations and asymptotic independence at the edges

Perturbed Hankel determinant, correlation functions and Painlevé equations

Regularization in model updating

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