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Proof methods in random matrix theory
Probability Surveys, 2023 In this survey article, we give an introduction to two methods of proof in random matrix theory: The method of moments and the Stieltjes transform method. We thoroughly develop these methods and apply them to show both the semicircle law and the Marchenko-Pastur law for random matrices with independent entries.
Werner Kirsch
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2017
Random matrix theory deals with the study of matrix-valued random variables. It is conventionally considered that random matrix theory dates back to the work of Wishart in 1928 [1] on the properties of matrices of the type XX † with X ε ℂ N×n a random matrix with independent Gaussian entries with zero mean and equal variance.
Couillet, Romain, Debbah, Merouane
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Random matrix theory deals with the study of matrix-valued random variables. It is conventionally considered that random matrix theory dates back to the work of Wishart in 1928 [1] on the properties of matrices of the type XX † with X ε ℂ N×n a random matrix with independent Gaussian entries with zero mean and equal variance.
Couillet, Romain, Debbah, Merouane
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2016
In this chapter the Gaussian random matrix ensembles are investigated. We determine their Green’s functions and show that for small energy differences a soft mode appears. As a consequence, the non-linear sigma-model is introduced and the level correlations are determined.
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In this chapter the Gaussian random matrix ensembles are investigated. We determine their Green’s functions and show that for small energy differences a soft mode appears. As a consequence, the non-linear sigma-model is introduced and the level correlations are determined.
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Acta Numerica, 2005
Random matrix theory is now a big subject with applications in many disciplines of science, engineering and finance. This article is a survey specifically oriented towards the needs and interests of a numerical analyst. This survey includes some original material not found anywhere else.
Alan Edelman, N. Raj Rao
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Random matrix theory is now a big subject with applications in many disciplines of science, engineering and finance. This article is a survey specifically oriented towards the needs and interests of a numerical analyst. This survey includes some original material not found anywhere else.
Alan Edelman, N. Raj Rao
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Random Matrix Theory and Its Applications
Statistical Science, 2021zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Random Matrix Theory and Wireless Communications
Foundations and Trends® in Communications and Information Theory, 2004Random matrix theory has found many applications in physics, statistics and engineering since its inception. Although early developments were motivated by practical experimental problems, random matrices are now used in fields as diverse as Riemann hypothesis, stochastic differential equations, condensed matter physics, statistical physics, chaotic ...
Antonia M. Tulino, Sergio Verdú
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2004
In this chapter, we will work not with \(\mathrm{GL}(n, \mathbb{C})\) but with its compact subgroup U(n). As in the previous chapters, we will consider elements of \(\mathcal{R}_{k}\) as generalized characters on S k . If \(\mathbf{f} \in \mathcal{R}_{k}\), then \(f ={ \mathrm{ch}}^{(n)}(\mathbf{f}) \in \varLambda _{k}^{(n)}\) is a symmetric polynomial
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In this chapter, we will work not with \(\mathrm{GL}(n, \mathbb{C})\) but with its compact subgroup U(n). As in the previous chapters, we will consider elements of \(\mathcal{R}_{k}\) as generalized characters on S k . If \(\mathbf{f} \in \mathcal{R}_{k}\), then \(f ={ \mathrm{ch}}^{(n)}(\mathbf{f}) \in \varLambda _{k}^{(n)}\) is a symmetric polynomial
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2001
A wealth of empirical and numerical evidence suggests universality for local fluctuations in quantum energy or quasi-energy spectra of systems that display global chaos in their classical phase spaces. Exceptions apart, all such Hamiltonian matrices of sufficiently large dimension yield the same spectral fluctuations provided they have the same group ...
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A wealth of empirical and numerical evidence suggests universality for local fluctuations in quantum energy or quasi-energy spectra of systems that display global chaos in their classical phase spaces. Exceptions apart, all such Hamiltonian matrices of sufficiently large dimension yield the same spectral fluctuations provided they have the same group ...
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1992
Before about 1956, there was no systematic statistical theory of nuclear energy level structure. There was a shortage of close spacings in experimentally obtained energy levels which was generally dismissed as being due to instrumental resolution failings.
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Before about 1956, there was no systematic statistical theory of nuclear energy level structure. There was a shortage of close spacings in experimentally obtained energy levels which was generally dismissed as being due to instrumental resolution failings.
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