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A phase transition for the limiting spectral density of random matrices. (English) Zbl 1287.60011

The authors study the spectral distribution of certain symmetric random matrices with correlated entries. The entries of the random matrices are assumed to be (in general, dependent) random variables with zero mean, unit variance and uniformly bounded moments of arbitrary order. The diagonals of these matrices are assumed to be stochastically independent random vectors. However, any two elements which are on the same diagonal are assumed to have correlation \(c_n\) depending only on \(n\). It is assumed that the limit \(c:=\lim_{n\to\infty} c_n\) exists. Under these assumptions it is shown that the empirical spectral distribution converges weakly to a non-random probability distribution \(\nu_c\) which does not depend on the distribution of the entries of the matrix. The proof relies on the moment method. The limiting distribution \(\nu_c\) is characterized as a certain free convolution of the Wigner semicircle law with the limiting spectral distribution of the Toeplitz random matrices derived by W. Bryc et al. [Ann. Probab. 34, No. 1, 1–38 (2006; Zbl 1094.15009)].

MSC:

60B20 Random matrices (probabilistic aspects)
60F15 Strong limit theorems
60K35 Interacting random processes; statistical mechanics type models; percolation theory
46L54 Free probability and free operator algebras

Citations:

Zbl 1094.15009
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