[Paper Review] Optimal Bounds on the Stieltjes Transform of Wigner Matrices

Claudio Cacciapuoti, Anna Maltsev+1 more|arXiv (Cornell University)|Nov 2, 2013

Random Matrices and Applications24 references3 citations

TL;DR

This paper establishes optimal convergence bounds for the Stieltjes transform of Wigner matrices, removing logarithmic corrections present in prior work. It proves convergence at the optimal scale with rate $(\log N)/N$, enabling improved eigenvalue rigidity and counting function estimates.

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].

Motivation & Objective

• To establish optimal convergence rates for the Stieltjes transform of Wigner matrices on the finest possible scale.
• To remove logarithmic corrections that previously limited the sharpness of convergence bounds in random matrix theory.
• To derive improved bounds for the eigenvalue counting function and eigenvalue rigidity using the refined Stieltjes transform analysis.
• To extend the precision of spectral statistics results beyond prior works such as [22, 10, 23].

Proposed method

• Analyzes the Stieltjes transform of Wigner matrices using advanced probabilistic and spectral techniques.
• Applies moment methods and isotropic local laws to control the resolvent and its trace.
• Employs a self-averaging argument to eliminate logarithmic corrections in convergence rates.
• Establishes uniform bounds on the Stieltjes transform over the entire spectral domain.
• Uses the convergence of the Stieltjes transform to derive quantitative estimates for eigenvalue statistics.
• Applies the optimal bounds to derive the $(\log N)/N$ rate for eigenvalue counting functions and eigenvalue rigidity.

Experimental results

Research questions

• RQ1 Can the convergence rate of the Stieltjes transform of Wigner matrices be improved beyond logarithmic corrections?
• RQ2 What is the optimal scale for the convergence of the Stieltjes transform in the context of Wigner matrices?
• RQ3 How do improved Stieltjes transform bounds affect the precision of eigenvalue counting function estimates?
• RQ4 To what extent can eigenvalue rigidity be strengthened using sharper Stieltjes transform convergence?
• RQ5 Can the results from [22, 10, 23] be improved by refining the analysis of the Stieltjes transform?

Key findings

• The Stieltjes transform of Wigner matrices converges to the Stieltjes transform of the semicircle law at the optimal scale without logarithmic corrections.
• The convergence rate of the eigenvalue counting function is improved to $(\log N)/N$, matching the optimal possible rate.
• Eigenvalue rigidity is established on the same optimal scale, improving prior results from [22, 10, 23].
• The removal of logarithmic corrections leads to sharper bounds in the local law and isotropic estimates.
• The refined Stieltjes transform bounds enable tighter control over the spectral distribution at mesoscopic scales.
• The results provide a foundation for improved analysis of eigenvalue fluctuations and extremal eigenvalues in Wigner ensembles.