Rigidity of Eigenvalues of Generalized Wigner Matrices (1007.4652v7)

Abstract: Consider $N\times N$ hermitian or symmetric random matrices $H$ with independent entries, where the distribution of the $(i,j)$ matrix element is given by the probability measure $\nu_{ij}$ with zero expectation and with variance $\sigma_{ij}^2$. We assume that the variances satisfy the normalization condition $\sum_{i} \sigma^2_{ij} = 1$ for all $j$ and that there is a positive constant $c$ such that $c\le N \sigma_{ij}^2 \le c^{-1}$. We further assume that the probability distributions $\nu_{ij}$ have a uniform subexponential decay. We prove that the Stieltjes transform of the empirical eigenvalue distribution of $H$ is given by the Wigner semicircle law uniformly up to the edges of the spectrum with an error of order $ (N \eta)^{-1}$ where $\eta$ is the imaginary part of the spectral parameter in the Stieltjes transform. There are three corollaries to this strong local semicircle law: (1) Rigidity of eigenvalues: If $\gamma_j =\gamma_{j,N}$ denotes the {\it classical location} of the $j$-th eigenvalue under the semicircle law ordered in increasing order, then the $j$-th eigenvalue $\lambda_j$ is close to $\gamma_j$ in the sense that for any $\xi>1$ there is a constant $L$ such that [\mathbb P \Big (\exists \, j : \; |\lambda_j-\gamma_j| \ge (\log N)^L \Big [ \min \big (\, j, N-j+1 \, \big) \Big ]^{-1/3} N^{-2/3} \Big) \le C\exp{\big[-c(\log N)^{\xi} \big]} ] for $N$ large enough. (2) The proof of the {\it Dyson's conjecture} \cite{Dy} which states that the time scale of the Dyson Brownian motion to reach local equilibrium is of order $N^{-1}$. (3) The edge universality holds in the sense that the probability distributions of the largest (and the smallest) eigenvalues of two generalized Wigner ensembles are the same in the large $N$ limit provided that the second moments of the two ensembles are identical.

• The paper shows that eigenvalues in generalized Wigner matrices are highly predictable, adhering closely to classical positions defined by the semicircle law.
• It establishes a precise local semicircle law with optimal error estimates and confirms edge universality through advanced stochastic analyses.
• The study employs rigorous probabilistic methods and Green function estimates to extend random matrix theory to ensembles with varied variance profiles.

Overview of "Rigidity of Eigenvalues of Generalized Wigner Matrices"

The paper "Rigidity of Eigenvalues of Generalized Wigner Matrices" by László Erdős, Horng-Tzer Yau, and Jun Yin offers a comprehensive mathematical analysis of the local behavior of eigenvalues for generalized Wigner matrices, building upon the foundations of random matrix theory and its applications.

Key Contributions

The paper extends the classical framework of Wigner matrices, which have fixed variance and standardized Gaussian entries, to a more generalized setting allowing for varied variance constraints on the matrix entries. The authors establish results concerning the local semicircle law, eigenvalue rigidity, and edge universality in this context, providing a substantial contribution to our understanding of eigenvalue behavior in more complex random matrix ensembles.

Local Semicircle Law: The paper proves a strong local semicircle law that provides optimal error estimates across the entire spectrum of generalized Wigner matrices, up to logarithmic factors. This result is significant as it contributes to the understanding of how the empirical spectral distribution adheres to the semicircle distribution in the proximity of the spectrum's edge as the matrix size increases.

Eigenvalue Rigidity: One of the primary results is the rigidity phenomenon, where eigenvalues show more predictability in their positions compared to classical random variables. The authors demonstrate that with overwhelming probability, eigenvalues are located very close to their classical predictions given by the semicircle law, with the deviation quantified precisely in terms of the matrix size $N$.

Edge Universality: The paper also tackles Dyson’s conjecture related to the universality of eigenvalue fluctuations at the spectral edges, commonly characterized by the Tracy-Widom distribution. In proving edge universality for generalized Wigner matrices, the paper shows that the distribution of the largest and smallest eigenvalues follows the same law in both generalized and standard Wigner ensembles.

Methodological Approach

To achieve these results, the paper employs a multi-step methodological framework:

• Local Semicircle Law: The authors initially derive a precise local semicircle law and Green function estimates for the empirical spectral distribution.
• Universality Proof: Through a series of stochastic differential equations, primarily the Dyson Brownian motion, they demonstrate how eigenvalues reach local equilibrium. This unification and estimation process removes dependency on specific matrix element distributions.
• Rigorous Probabilistic Bounds: The derivation heavily relies on proving high-probability bounds for the elements of the Green function related to these matrices, characterizing them using large deviation inequalities.

Implications and Future Directions

The implications of this paper are significant, broadening the scope of universality in random matrices beyond the traditional setup. It provides mathematical underpinnings for applications in physics, particularly quantum mechanics, where random matrices are used to describe energy levels of complex systems. Additionally, the methods introduced in this paper offer potential pathways for analyzing other forms of random matrix ensembles, such as those with dependent entries or non-Hermitian matrices.

Future research could focus on diminishing the reliance on variance constraints, thereby extending the universality principles further into other domains of random matrix theory. Additionally, this work provides a foundational basis for further explorations into the real-world applicability of spectral methods in complex systems, resonating across domains such as wireless communication and statistical physics.

Conclusion

The authors, by establishing eigenvalue rigidity and edge universality for a generalized setting of Wigner matrices, present advancements that address key conjectures in random matrix theory. Their approach, emphasizing stochastic analysis and precise eigenvalue behavior mapping, significantly contributes to the theoretical infrastructure necessary for expanding universality principles. Widening the application scope through this paper potentially marks future evolutions in both mathematical theory and its interdisciplinary applications.