Bounds on the norm of Wigner-type random matrices

Abstract

We consider a Wigner-type ensemble, i.e. large hermitian [Formula: see text] random matrices [Formula: see text] with centered independent entries and with a general matrix of variances [Formula: see text]. The norm of [Formula: see text] is asymptotically given by the maximum of the support of the self-consistent density of states. We establish a bound on this maximum in terms of norms of powers of [Formula: see text] that substantially improves the earlier bound [Formula: see text] given in [O. Ajanki, L. Erdős and T. Krüger, Universality for general Wigner-type matrices, Prob. Theor. Rel. Fields 169 (2017) 667–727]. The key element of the proof is an effective Markov chain approximation for the contributions of the weighted Dyck paths appearing in the iterative solution of the corresponding Dyson equation.

Locations

• Random Matrices Theory and Application - View
• arXiv (Cornell University) - View - PDF