A phase transition for the limiting spectral density of random matrices

Abstract

We analyze the spectral distribution of symmetric random matrices with correlated entries. While we assume that the diagonals of these random matrices are stochastically independent, the elements of the diagonals are taken to be correlated. Depending on the strength of correlation the limiting spectral distribution is either the famous semicircle law or some other law, related to that derived for Toeplitz matrices by Bryc, Dembo and Jiang (2006).

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

“…We begin the proof with large deviations estimates that demonstrate the efficiency of the indicators X N,± defined in (8). They are immediate consequences of Hoeffding's inequality.…”

Sixty years of moments for random matrices

“…2. The proof in [19] uses the moment method. It allows the authors to give an expression for the moments of σ β in terms of m(β) (see (43)).…”

The semicircle law for matrices with ergodic entries

“…Similar attempts have been made in Schenker and Schulz-Baldes (2005), where a limited number of correlated entries are admitted, Götze and Tikhomirov (2006), where a martingale structure of the entries is imposed, Friesen and Löwe (2013b), where the diagonals of the (symmetric) matrices were filled with independent Markov chains or in Löwe and Schubert (2016), where the upper triangular part of the matrix is filled with a one-dimensional Markov chain. On the other hand, Friesen and Löwe (2013a) and Hochstättler et al (2016) study matrices where either the diagonals or the entire matrix is built of exchangeable random variables. 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).…”

“…On the other hand, Friesen and Löwe (2013a) and Hochstättler et al (2016) study matrices where either the diagonals or the entire matrix is built of exchangeable random variables. 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.…”

“…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.…”