Bounds for the Stieltjes Transform and the Density of States of Wigner Matrices

Abstract

We consider ensembles of Wigner matrices, whose entries are (up to the symmetry constraints) independent and identically distributed random variables. We show the convergence of the Stieltjes transform towards the Stieltjes transform of the semicircle law on optimal scales and with the optimal rate. Our bounds improve previous results, in particular from [22,10], by removing the logarithmic corrections. As applications, we establish the convergence of the eigenvalue counting functions with the rate $(\log N)/N$ and the rigidity of the eigenvalues of Wigner matrices on the same scale. These bounds improve the results of [22,10,23].

New title, former title "Optimal Bounds on the Stieltjes Transform of Wigner Matrices". Updated references

References in corpus (6)

• Local semicircle law and complete delocalization for Wigner random matrices
• Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices
• Averaging Fluctuations in Resolvents of Random Band Matrices
• A Necessary and Sufficient Condition for Edge Universality of Wigner matrices
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Cited by in corpus (2)

• Maximum of the characteristic polynomial of random unitary matrices
• Local semicircle law for random regular graphs