Gaussian fluctuations of eigenvalues in the GUE
By J. Gustavsson. 2004
187 citations recorded by the source corpus.
Abstract Under certain conditions on k we calculate the limit distribution of the kth largest eigenvalue, x k , of the Gaussian Unitary Ensemble (GUE). More specifically, if n is the dimension of a random matrix from the GUE and k is such that both n − k and k tends to infinity as n → ∞ then x k is normally distributed in the limit. We also consider the joint limit distribution of x k 1 ⋯ x k m where we require that n − k i and k i , 1 ⩽ i ⩽ m , tends to infinity with n. The result is an m-dimensional normal distribution.
Subfields: Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/4b91aec10868e1bc569499f20e8175595521f70a
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In Mathematical Physics
In Statistics and Probability
Citations in the corpus
- Random covariance matrices: Universality of local statistics of eigenvalues (2009, background): “As such, they provide an alternate approach to controlling correlation expressions such as (10), but we do not know how to convert such information back to control on individual eigenvalues or singular values in general, because the standard deviation of each eigenvalue can exceed (by a logarithmic ”
- Edge Universality of Beta Ensembles (2013, methodology): “The work of Gustavsson [34] for the special β = 2 and Gaussian case (i.e., the GUE case) indicates that in general the point-point correlation function decays only logarithmically in the bulk, i.e., 〈λi;λj〉µβ ∼ log |i − j|.”
- Universality of random matrices and local relaxation flow (2009, unknown): “In the bulk αm − γm is expected to be bounded by O(N−1+ε) (in the hermitian case it was proven in [14], see also [21]); near the edges one expects αm − γm ∼ O(N−2/3).”
- Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices (2012, background): “Gustafsson [17] has shown that for GUE and for y in the interior of the support of the limiting spectral density, and φy(x) = 1[y,∞)(x),”
- Random matrices: Universality of local eigenvalue statistics (2009, background): “, λik (Mn) in the bulk of the spectrum of a Wigner Hermitian matrix for any fixed k (the GUE case is treated in [26]).”
- Universality of Wigner random matrices: a survey of recent results (2010, background): “Prior to [96], apart from the GUE case [61, 74], for no other ensembles could the eigenvalues be located with a precision N−1+c0 for small c0, and such precision is needed to translate (1.”
- Random matrices: Sharp concentration of eigenvalues (2012, methodology): “By using a general result of Costin and Lebowitz [7], one can also obtain a central limit theorem for NI(Wn) as long as I is not too small; see [21].”
- The rate of convergence for spectra of GUE and LUE matrix ensembles (2005, methodology): “This approach has been successfully used by Gustavsson for GUE in [21] to estimate the expectation of the number of eigenvalues in an interval.”
- Rigidity of eigenvalues of generalized Wigner matrices (2010, background): “5) would be wrong since, at least for the classical GUE or GOE ensembles, the eigenvalues are known to fluctuate on a scale √ logN/N , see [25, 29].”
- Random matrix theory in statistics: A review (2014, background): “Tao and Vu (2010b, 2011) extended the result of Gustavsson (2005) for the GUE to a class of Hermitian Wigner matrices with nonGaussian entries whose first four moments match the Gaussian moments, which is an instance of the universality phenomena discussed in detail in Section 3.3.”