Singularities of the density of states of random Gram matrices. (English) Zbl 1378.60020

Alt, Johannes

Electron. Commun. Probab. 22, Paper No. 63, 13 p. (2017).

Summary: For large random matrices \(X\) with independent, centered entries but not necessarily identical variances, the eigenvalue density of \(XX^*\) is well-approximated by a deterministic measure on \(\mathbb{R}\). We show that the density of this measure has only square and cubic-root singularities away from zero. We also extend the bulk local law in [the author et al., Electron. J. Probab. 22, Paper No. 25, 41 p. (2017; Zbl 1376.60014)] to the vicinity of these singularities.

MSC:

60B20	Random matrices (probabilistic aspects)
15B52	Random matrices (algebraic aspects)

Keywords:

local law; Dyson equation; square-root edge; cubic cusp; general variance profile

Citations:

Zbl 1376.60014