On the Limiting Spectral Density of Symmetric Random Matrices with Correlated Entries

Friesen, Olga; Löwe, Matthias

doi:10.1007/978-3-642-38806-4_1

Cited by 1 publication

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“…In [9] it is shown that, if the diagonals of X N are independent and the correlation between elements along a diagonal decays sufficiently quickly, again the limiting spectral distribution is the semi-circle law. This result has to be compared with the situation in [8] and [7], where the diagonals are still independent, but the random variables along a diagonal are exchangeable. Here again, for weak correlations one finds the semicircle as the limiting spectral density for the eigenvalues.…”

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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Section: Introductionmentioning

confidence: 99%