The Semicircle Law for Matrices with Independent Diagonals

Friesen, Olga; Löwe, Matthias

doi:10.1007/s10959-011-0383-2

Cited by 14 publications

(42 citation statements)

References 6 publications

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“…In particular, in Friesen and Löwe (2013a) it was shown that there is a phase transition: If the correlation of the exchangeable random variables go to 0, the limit of the ESD is the semi-circle law, while otherwise it can be described in terms of a free convolution of the semi-circle law with a limiting law obtained in Bryc et al (2006). In this note we will answer a question by C. Deninger (private communication) and extend the results from Friesen and Löwe (2013a) and Friesen and Löwe (2013b) to ergodic sequences of random variables. We will see that the convergence of the ESD only depends on the correlations of the entries.…”

Section: Introductionmentioning

confidence: 54%

“…The ideas in the first section of the proof partially follow Friesen and Löwe (2013b) which in turn are based on considerations in Schenker and Schulz-Baldes (2005). For the reader's convenience and since we will need the arguments for the second part of the proof as well, we will illustrate the main steps.…”

Section: Proof Of Theorem 21mentioning

confidence: 99%

“…This is indeed well known. The proof can be found in Friesen and Löwe (2013b), Lemma 3.1 or Bryc et al (2006), Proposition 4.4. We hence obtain…”

Section: Proof Of Theorem 21mentioning

confidence: 99%

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The semicircle law for matrices with ergodic entries

Löwe

2018

Statistics & Probability Letters

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We study the empirical spectral distribution (ESD) of symmetric random matrices with ergodic entries on the diagonals. We observe that for entries with correlations that decay to 0, when the distance of the diagonal entries becomes large the limiting ESD is the well known semicircle law. If it does not decay to 0 (and have the same sign) the semicircle law cannot be the limit of the ESD. This is good agreement with results on exchangeable processes analysed in Friesen and Löwe (2013a) and Hochstättler et al. (2016).

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Section: Introductionmentioning

confidence: 54%

Section: Proof Of Theorem 21mentioning

confidence: 99%

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The semicircle law for matrices with ergodic entries

Löwe

2018

Statistics & Probability Letters

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“…Friesen and Löwe [18] consider matrix ensembles with independent diagonals generated by a sequence Y n of weakly correlated random variables. In their case the limit distribution is the semicircle law again.…”

Section: Decaying Correlationsmentioning

confidence: 99%

Sixty years of moments for random matrices

Kirsch

Kriecherbauer

2018

EMS Series of Congress Reports

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This is an elementary review, aimed at non-specialists, of results that have been obtained for the limiting distribution of eigenvalues and for the operator norms of real symmetric random matrices via the method of moments. This method goes back to a remarkable argument of Eugen Wigner some sixty years ago which works best for independent matrix entries, as far as symmetry permits, that are all centered and have the same variance. We then discuss variations of this classical result for ensembles for which the variance may depend on the distance of the matrix entry to the diagonal, including in particular the case of band random matrices, and/or for which the required independence of the matrix entries is replaced by some weaker condition. This includes results on ensembles with entries from Curie-Weiss random variables or from sequences of exchangeable random variables that have been obtained quite recently.

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“…A different approach to universality was taken in [16], [11] and [9]. In all these articles random matrices with correlated entries are studied.…”

Section: Introductionmentioning

confidence: 99%

On the limiting spectral density of random matrices filled with stochastic processes

Löwe

Schubert

2019

Random Operators and Stochastic Equations

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We discuss the limiting spectral density of real symmetric random matrices. Other than in standard random matrix theory the upper diagonal entries are not assumed to be independent, but we will fill them with the entries of a stochastic process. Under assumptions on this process, which are satisfied, e.g., by stationary Markov chains on finite sets, by stationary Gibbs measures on finite state spaces, or by Gaussian Markov processes, we show that the limiting spectral distribution depends on the way the matrix is filled with the stochastic process. If the filling is in a certain way compatible with the symmetry condition on the matrix, the limiting law of the empirical eigenvalue distribution is the well known semicircle law. For other fillings we show that the semi-circle law cannot be the limiting spectral density.2010 Mathematics Subject Classification. 60B20; 60F05.

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The Semicircle Law for Matrices with Independent Diagonals

Cited by 14 publications

References 6 publications

The semicircle law for matrices with ergodic entries

The semicircle law for matrices with ergodic entries

Sixty years of moments for random matrices

On the limiting spectral density of random matrices filled with stochastic processes

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