From Gumbel to Tracy-Widom

Johansson, Kurt

doi:10.1007/s00440-006-0012-7

Cited by 100 publications

(126 citation statements)

References 31 publications

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“…Hence the family U β (ξ) interpolates between the Gumbel and Tracy-Widom distributions, compare [11]. We can also consider the process k → G(N +k, N −k), |k| < N .…”

Section: Introduction and Resultsmentioning

confidence: 99%

On some special directed last-passage percolation models

Johansson¹

2008

Integrable Systems and Random Matrices

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“…Hence the family U β (ξ) interpolates between the Gumbel and Tracy-Widom distributions, compare [11]. We can also consider the process k → G(N +k, N −k), |k| < N .…”

Section: Introduction and Resultsmentioning

confidence: 99%

On some special directed last-passage percolation models

Johansson¹

2008

Integrable Systems and Random Matrices

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“…There is a constant C, such that for k ∈ 1, 28) with (ξ, ν)-high probability, on Ω V . 29) with high probability on Ω V ; see Lemma 5.1. Next, recall from Proposition 4.6, we have with high probability on…”

Section: Preliminariesmentioning

confidence: 98%

“…If W belongs to the Gaussian Unitary ensemble (GUE), the model (1.1) is called the deformed GUE. It was shown in [29,43] that the edge eigenvalues of the deformed GUE are governed by the Tracy-Widom distribution for λ ≪ N −1/6 . At λ ∼ N −1/6 the fluctuations of the edge eigenvalues change from the Tracy-Widom to a Gaussian distribution.…”

Section: Introductionmentioning

confidence: 99%

“…At λ ∼ N −1/6 the fluctuations of the edge eigenvalues change from the Tracy-Widom to a Gaussian distribution. More precisely, Johansson showed in [29] that the limiting distribution of the edge eigenvalues for λ = αN −1/6 is given by the convolution of the Tracy-Widom and the centered Gaussian distribution, with variance depending on α. These results have not been established for the Gaussian orthogonal ensemble (GOE) or for general Wigner matrices.…”

Section: Introductionmentioning

confidence: 99%

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Extremal eigenvalues and eigenvectors of deformed Wigner matrices

Lee

Schnelli

2015

Probab. Theory Relat. Fields

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We consider random matrices of the form H = W + λV , λ ∈ R + , where W is a real symmetric or complex Hermitian Wigner matrix of size N and V is a real bounded diagonal random matrix of size N with i.i.d. entries that are independent of W . We assume subexponential decay of the distribution of the matrix entries of W and we choose λ ∼ 1, so that the eigenvalues of W and λV are typically of the same order. Further, we assume that the density of the entries of V is supported on a single interval and is convex near the edges of its support. In this paper we prove that there is λ+ ∈ R + such that the largest eigenvalues of H are in the limit of large N determined by the order statistics of V for λ > λ+. In particular, the largest eigenvalue of H has a Weibull distribution in the limit N → ∞ if λ > λ+. Moreover, for N sufficiently large, we show that the eigenvectors associated to the largest eigenvalues are partially localized for λ > λ+, while they are completely delocalized for λ < λ+. Similar results hold for the lowest eigenvalues.

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“…(5.1) are skew-orthogonal in the complex plane with respect to the weight function F (1) (z 1 , z 2 ) in eq. (2.7) 11) and are given by [38] …”

Section: Hermitian and Strongly Non-hermitian Limitsmentioning

confidence: 99%

The Interpolating Airy Kernels for the $$\beta =1$$ β = 1 and $$\beta =4$$ β = 4 Elliptic Ginibre Ensembles

Akemann¹,

Phillips²

2014

J Stat Phys

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We consider two families of non-Hermitian Gaussian random matrices, namely the elliptical Ginibre ensembles of asymmetric N -by-N matrices with Dyson index β = 1 (real elements) and with β = 4 (quaternion-real elements). Both ensembles have already been solved for finite N using the method of skew-orthogonal polynomials, given for these particular ensembles in terms of Hermite polynomials in the complex plane. In this paper we investigate the microscopic weakly non-Hermitian large-N limit of each ensemble in the vicinity of the largest or smallest real eigenvalue. Specifically, we derive the limiting matrix-kernels for each case, from which all the eigenvalue correlation functions can be determined. We call these new kernels the "interpolating" Airy kernels, since we can recover -as opposing limiting cases -not only the well-known Airy kernels for the Hermitian ensembles, but also the complementary error function and Poisson kernels for the maximally non-Hermitian ensembles at the edge of the spectrum. Together with the known interpolating Airy kernel for β = 2, which we rederive here as well, this completes the analysis of all three elliptical Ginibre ensembles in the microscopic scaling limit at the spectral edge.

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From Gumbel to Tracy-Widom

Cited by 100 publications

References 31 publications

On some special directed last-passage percolation models

On some special directed last-passage percolation models

Extremal eigenvalues and eigenvectors of deformed Wigner matrices

The Interpolating Airy Kernels for the $$\beta =1$$ β = 1 and $$\beta =4$$ β = 4 Elliptic Ginibre Ensembles

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