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

EMS Series of Congress Reports

2018

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.

“…The assumptions we made in Theorem 6.5 both on Z n and on the filling are but an example of the abstract assumptions given in [33]. These authors also show: Theorem 6.6.…”

Section: Decaying Correlationsmentioning

confidence: 73%

Section: Decaying Correlationsmentioning

confidence: 99%

See 1 more Smart Citation

Sixty years of moments for random matrices

EMS Series of Congress Reports

2018

“…For example, we could fill it row-wise or sub-diagonal by sub-diagonal. For other sequences of random variables the way of filling the matrix may be crucial (see [21]).…”

Section: Remark 22mentioning

confidence: 99%

Random matrices with exchangeable entries

2020

Rev. Math. Phys.

We consider ensembles of real symmetric band matrices with entries drawn from an infinite sequence of exchangeable random variables, as far as the symmetry of the matrices permits. In general the entries of the upper triangular parts of these matrices are correlated and no smallness or sparseness of these correlations is assumed. It is shown that the eigenvalue distribution measures still converge to a semicircle but with random scaling. We also investigate the asymptotic behavior of the corresponding ℓ 2 -operator norms.The key to our analysis is a generalisation of a classic result by de Finetti that allows to represent the underlying probability spaces as averages of Wigner band ensembles with entries that are not necessarily centred. Some of our results appear to be new even for such Wigner band matrices.

“…In [13,Theorem 5] it is shown that conditions (4) and (5) are well suited to apply the original ideas of Wigner [27,28] and Grenander [12] to prove the semicircle law via the method of moments (see also [2,3,18,21,22] and the monographs [1,4,23,25]). Matrix ensembles with correlated entries have already been considered in [5,7,8,10,11,14,19,24]. See [13] and the recent survey [16] for more information on these results.…”

Section: Observe That Pmentioning

confidence: 99%

Semicircle Law for Generalized Curie–Weiss Matrix Ensembles at Subcritical Temperature

2017

J Theor Probab

In [13] Hochstättler, Kirsch, and Warzel showed that the semicircle law holds for generalized Curie-Weiss matrix ensembles at or above the critical temperature. We extend their result to the case of subcritical temperatures for which the correlations between the matrix entries are stronger. Nevertheless, one may use the concept of approximately uncorrelated ensembles that was first introduced in [13]. In order to do so one needs to remove the average magnetization of the entries by an appropriate modification of the ensemble that turns out to be of rank 1 thus not changing the limiting spectral measure.