Large deviations for the largest eigenvalue of rank one deformations of Gaussian ensembles

Maïda, Mylène

doi:10.1214/ejp.v12-438

Cited by 49 publications

(75 citation statements)

References 16 publications

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“…In this regime the typical value µ typ M of the smallest eigenvalue is out of bulk and equals to µ 0 (θ, β), see (52) and (74). Given the functions:…”

Section: Large Deviation Function Optimized Over U At Fixed θmentioning

confidence: 95%

Distribution of rare saddles in the p -spin energy landscape

Ros

2020

J. Phys. A: Math. Theor.

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We compute the statistical distribution of index-1 saddles surrounding a given local minimum of the p-spin energy landscape, as a function of their distance to the minimum in configuration space and of the energy of the latter. We identify the saddles also in the region of configuration space in which they are subdominant in number (i.e., rare) with respect to local minima, by computing large deviation probabilities of the extremal eigenvalues of their Hessian. As an independent result, we determine the joint large deviation probability of the smallest eigenvalue and eigenvector of a GOE matrix perturbed with both an additive and multiplicative finite-rank perturbation.

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“…In this regime the typical value µ typ M of the smallest eigenvalue is out of bulk and equals to µ 0 (θ, β), see (52) and (74). Given the functions:…”

Section: Large Deviation Function Optimized Over U At Fixed θmentioning

confidence: 95%

Distribution of rare saddles in the p -spin energy landscape

Ros

2020

J. Phys. A: Math. Theor.

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“…In the case of the GUE, this was due to Peché, 37 while a rigorous study of the GOE case can be found in the work of Maida. 30 The paper by Baik et al 5 proved the phase transition property relating to ͑1.2͒ in the complex case with ⌺ given by ⌺ = diag͑͑b͒ r , ͑1͒ m−r ͒ ͓͑1.4͒ corresponds to r =1͔. Subsequent studies by Baik and Silverstein, 6 Paul, 36 and Bai and Yao 4 considered the real case.…”

Section: B Related Workmentioning

confidence: 99%

Eigenvalue separation in some random matrix models

Bassler

Forrester

Frankel

2009

Journal of Mathematical Physics

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The eigenvalue density for members of the Gaussian orthogonal and unitary ensembles follows the Wigner semicircle law. If the Gaussian entries are all shifted by a constant amount s / ͑2N͒ 1/2 , where N is the size of the matrix, in the large N limit a single eigenvalue will separate from the support of the Wigner semicircle provided s Ͼ 1. In this study, using an asymptotic analysis of the secular equation for the eigenvalue condition, we compare this effect to analogous effects occurring in general variance Wishart matrices and matrices from the shifted mean chiral ensemble. We undertake an analogous comparative study of eigenvalue separation properties when the sizes of the matrices are fixed and s → ϱ, and higher rank analogs of this setting. This is done using exact expressions for eigenvalue probability densities in terms of generalized hypergeometric functions and using the interpretation of the latter as a Green function in the Dyson Brownian motion model. For the shifted mean Gaussian unitary ensemble and its analogs, an alternative approach is to use exact expressions for the correlation functions in terms of classical orthogonal polynomials and associated multiple generalizations. By using these exact expressions to compute and plot the eigenvalue density, illustrations of the various eigenvalue separation effects are obtained.

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“…Therefore we only need to estimate small ball probabilities around x = 2. Asμ X N concentrates at the scale N, and ||X N || is exponentially tight at the scale N by Assumption 1.2 it is enough to show that for any K > 0, [16,Proposition 2.1], we know that the spherical integral is continuous, more precisely, for N large enough and any X N ∈ V δ,x ,…”

Section: Remark 13mentioning

confidence: 99%

Large deviations principle for the largest eigenvalue of Wigner matrices without Gaussian tails

Augeri¹

2016

Electron. J. Probab.

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We establish large deviations estimates for the largest eigenvalue of Wigner matrices with sub-Gaussian entries. We estimate the probability that the largest eigenvalue is close to some value large enough and show that if the entries do not have sharp sub-Gaussian tails, the rate function is strictly smaller than the rate function for Gaussian entries. This contrasts with [12] where it was shown that the law of the largest eigenvalue of Wigner matrices with entries with sharp sub-Gaussian tails obeys a large deviation principle with the same rate function than in the Gaussian case.

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Large deviations for the largest eigenvalue of rank one deformations of Gaussian ensembles

Cited by 49 publications

References 16 publications

Distribution of rare saddles in the p -spin energy landscape

Distribution of rare saddles in the p -spin energy landscape

Eigenvalue separation in some random matrix models

Large deviations principle for the largest eigenvalue of Wigner matrices without Gaussian tails

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