Bulk universality for generalized Wigner matrices (1001.3453v8)

Abstract: Consider $N\times N$ Hermitian or symmetric random matrices $H$ where the distribution of the $(i,j)$ matrix element is given by a probability measure $\nu_{ij}$ with a subexponential decay. Let $\sigma_{ij}^2$ be the variance for the probability measure $\nu_{ij}$ with the normalization property that $\sum_{i} \sigma^2_{ij} = 1$ for all $j$. Under essentially the only condition that $c\le N \sigma_{ij}^2 \le c^{-1}$ for some constant $c>0$, we prove that, in the limit $N \to \infty$, the eigenvalue spacing statistics of $H$ in the bulk of the spectrum coincide with those of the Gaussian unitary or orthogonal ensemble (GUE or GOE). We also show that for band matrices with bandwidth $M$ the local semicircle law holds to the energy scale $M^{-1}$.

• The paper establishes bulk universality by showing that local eigenvalue statistics of generalized Wigner matrices conform to classical ensembles.
• It employs the local semicircle law and Green’s function comparison method to link local eigenvalue density with moment matching conditions.
• The findings extend the universality concept to matrices with independent, non-identical entries, impacting various fields such as physics and communications.

Overview of "Bulk Universality for Generalized Wigner Matrices"

The paper "Bulk Universality for Generalized Wigner Matrices" by Erdos, Yau, and Yin explores the concept of bulk universality in the context of random matrix theory. Specifically, the authors focus on extending the theory of bulk universality to generalized Wigner matrices, which have entries with independent but not necessarily identical distributions.

The concept of universality in random matrices is rooted in the observation that certain statistical properties of eigenvalues do not depend on the detailed distribution of the matrix elements but rather only on some general conditions like symmetry class. The authors address both the familiar Gaussian Unitary Ensemble (GUE) and Gaussian Orthogonal Ensemble (GOE) and span their examination to matrices with more generalized structures.

Methodology and Results

A key contribution of this paper is the proof of the bulk universality for generalized Wigner matrices. The authors achieve this by demonstrating that the local eigenvalue density and correlation functions of these matrices conform to the semicircle law and sine kernel typically associated with GUE/GOE.

They leverage the local semicircle law, a powerful result that estimates the local eigenvalue density, to as small a scale as essentially $N^{-1}$ (with logarithmic factors). Their results are stronger in the context of particular matrix classes, such as band matrices where variance decays beyond the band.

To extend these results to the broadest class of matrices, the authors use the Green's function comparison method. They establish that as long as the first few moments of two matrix ensembles match, their local statistics would also match to a significant approximation.

Theorems and Proofs:

1. Local Semicircle Law: The paper provides a rigorous derivation of the local semicircle law, stating that the eigenvalue distribution of high-dimensional Hermitian random matrices follows the semicircle law even at very small scales.
2. Green's Function Comparison: A critical theorem presented is the Green’s function comparison theorem, which aids in demonstrating that eigenvalue correlation functions of certain random matrix ensembles align closely under mild moment matching conditions.
3. Universality for Generalized Wigner Matrices: The universality results are shown to extend beyond classical Wigner matrices to generalized forms, utilizing distribution tail decay conditions instead of stricter moment matching.

Implications and Future Work

The implications of this work are profound, providing a unifying framework for understanding eigenvalue distributions across a vast array of random matrices encountered in physics, statistics, and wireless communication theory. This generalization to broader matrix ensembles could inspire further exploration into non-Hermitian and sparse matrices, expanding the horizons of random matrix theory even further. Additionally, the underlying techniques, such as the use of Green’s function comparisons, might inform new methods in other areas of mathematical physics.

While the paper solidifies the universality concept for generalized Wigner matrices, open questions remain about its applicability to more complex systems, new forms of universality in random structures, and stronger bounds without extensive moment matching. Future developments could involve applying similar analyses to quantum systems and other non-standard models in complex random environments.

In conclusion, the paper establishes a detailed framework for understanding the statistical behavior of generalized Wigner matrices, opening avenues for application in various scientific fields, while setting a precedent for further theoretical developments.