Bounds on the norm of Wigner-type random matrices

2018/02/14 by Erdős, László, Mühlbacher, Peter
#60B20 (Primary) 60J10 (Secondary) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1802.05175

Abstract

We consider a Wigner-type ensemble, i.e. large hermitian N× N random matrices H=H^* with centered independent entries and with a general matrix of variances S_xy=\mathbb E|H_xy|². The norm of H 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 S that substantially improves the earlier bound 2‖ S‖^1/2_∞ given in [arXiv:1506.05098]. 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.

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