[Paper Review] Fluctuations Of Linear Spectral Statistics Of Deformed Wigner Matrices

Sandrine Dallaporta, Maxime Février|arXiv (Cornell University)|Mar 27, 2019

Random Matrices and Applications52 references8 citations

TL;DR

This paper establishes Gaussian fluctuations for linear spectral statistics of deformed Wigner matrices—Wigner matrices perturbed by deterministic diagonal matrices—showing convergence to a Gaussian distribution with explicit bias and variance terms derived from free probability and resolvent trace analysis. It extends fluctuation results to non-Gaussian entries and proves asymptotic infinitesimal freeness between independent GUE matrices and deterministic matrices.

ABSTRACT

We investigate the fluctuations of linear spectral statistics of a Wigner matrix $W\_N$ deformed by a deterministic diagonal perturbation $D\_N$, around a deterministic equivalent which can be expressed in terms of the free convolution between a semicircular distribution and the empirical spectral measure of $D\_N$. We obtain Gaussian fluctuations for test functions in $\mathcal{C}\_c^7(\mathbb{R})$ ($\mathcal{C}\_c^2(\mathbb{R})$ for fluctuations around the mean). Furthermore, we provide as a tool a general method inspired from Shcherbina and Johansson to extend the convergence of the bias if there is a bound on the bias of the trace of the resolvent of a random matrix. Finally, we state and prove an asymptotic infinitesimal freeness result for independent GUE matrices together with a family of deterministic matrices, generalizing the main result from [Shl18].

Motivation & Objective

• To analyze the fluctuations of linear spectral statistics of Wigner matrices deformed by deterministic diagonal matrices.
• To derive explicit expressions for the bias and variance of these fluctuations using free convolution and resolvent trace techniques.
• To extend fluctuation results beyond Gaussian entries to general i.i.d. entries with finite fourth moments.
• To establish an asymptotic infinitesimal freeness result between independent GUE matrices and deterministic matrices.
• To provide a general method for extending convergence of the resolvent trace bias using bounds on trace fluctuations.

Proposed method

• Uses the Stieltjes transform and resolvent trace to analyze linear spectral statistics of deformed Wigner matrices.
• Applies free probability tools, particularly free convolution, to express the deterministic equivalent of the spectral measure.
• Employs the Wick formula and combinatorial analysis of non-crossing partitions to compute moments of GUE matrices.
• Derives bias and variance terms via asymptotic expansion of the trace of the resolvent, leveraging bounds on its fluctuations.
• Uses the Kreweras complementation map and cyclic permutations to relate combinatorial structures in free probability to matrix traces.
• Establishes asymptotic infinitesimal freeness by comparing the $*$-distribution of the matrix tuple to the free convolution of semicircular and deterministic spectral measures.

Experimental results

Research questions

• RQ1 What is the limiting distribution of linear spectral statistics of deformed Wigner matrices with general i.i.d. entries?
• RQ2 How do the bias and variance of these statistics depend on the fourth moments of the matrix entries and the regularity of the test function?
• RQ3 Can the convergence of the resolvent trace bias be extended under boundedness assumptions on the trace fluctuations?
• RQ4 Under what conditions do independent GUE matrices and deterministic matrices become asymptotically infinitesimally free?
• RQ5 What is the precise form of the deterministic equivalent for the spectral measure of a deformed Wigner matrix?

Key findings

• For test functions in $\mathcal{C}_{c}^{7}(\mathbb{R})$, linear spectral statistics of deformed Wigner matrices exhibit $N^{-1}$-scale Gaussian fluctuations with explicit bias and variance.
• The limiting variance $V_0(\varphi)$ is expressed via the Stieltjes transform of the semicircular law and involves terms dependent on the fourth moments of the matrix entries.
• The bias term $b_0(\varphi)$ depends on the derivative of the Stieltjes transform and incorporates the variance profile and excess kurtosis of the entries.
• The paper provides a general method to extend convergence of the resolvent trace bias under boundedness assumptions on the trace fluctuations.
• Asymptotic infinitesimal freeness holds between independent GUE matrices and a family of deterministic matrices, with the joint $*$-distribution converging to the free convolution of the semicircular and deterministic spectral measures at $O(N^{-2})$ error.
• The result generalizes Shcherbina's work [Shl18] by extending infinitesimal freeness to non-Gaussian entries and more general deterministic matrix families.