Random Matrices: The circular Law (0708.2895v5)

Abstract: Let $\a$ be a complex random variable with mean zero and bounded variance $\sigma^{2}$. Let $N_{n}$ be a random matrix of order $n$ with entries being i.i.d. copies of $\a$. Let $\lambda_{1}, ..., \lambda_{n}$ be the eigenvalues of $\frac{1}{\sigma \sqrt n}N_{n}$. Define the empirical spectral distribution $\mu_{n}$ of $N_{n}$ by the formula $$ \mu_n(s,t) := \frac{1}{n} # {k \leq n| \Re(\lambda_k) \leq s; \Im(\lambda_k) \leq t }.$$ The Circular law conjecture asserts that $\mu_{n}$ converges to the uniform distribution $\mu_\infty$ over the unit disk as $n$ tends to infinity. We prove this conjecture under the slightly stronger assumption that the $(2+\eta)\th$-moment of $\a$ is bounded, for any $\eta >0$. Our method builds and improves upon earlier work of Girko, Bai, G\"otze-Tikhomirov, and Pan-Zhou, and also applies for sparse random matrices. The new key ingredient in the paper is a general result about the least singular value of random matrices, which was obtained using tools and ideas from additive combinatorics.

• The paper proves the circular law for random matrices under the relaxed condition of bounded (2 + agreek>eta</tagreek>)th moment, using techniques from additive combinatorics.
• This extended proof significantly broadens the circular law's applicability, including distributions like Bernoulli variables and certain sparse matrices, and provides a rate of convergence.
• The generalized circular law has potential impacts on statistical physics, telecommunications, and numerical linear algebra, inspiring new research into other matrix ensembles and computational methods.

Overview of the Circular Law in Random Matrices

The paper "Random Matrices: The Circular Law" authored by Terence Tao and Van Vu offers an exhaustive exploration into the circular law conjecture, a prominent topic in the field of random matrix theory since its proposition in the 1950s. The authors rigorously prove this conjecture under a relatively relaxed condition, thereby expanding the applicability of the circular law.

At its core, the circular law conjecture posits that for a random matrix with i.i.d. entries generated from a complex random variable with zero mean and finite variance, the empirical spectral distribution converges to a uniform distribution over the unit disk in the complex plane as the matrix size grows to infinity. Earlier results and partial proofs were contingent on more stringent assumptions, among them the bounded density of the entries.

Key Contributions and Methodology

The principal contribution of this paper is the proof of the circular law under the condition that the (2 + η)th moment of the random variable is bounded for any η > 0. This result does not rely on the bounded density condition, which has historically restricted the types of distributions to which the law could be applied. The elimination of this requirement marks a significant advancement as it includes distributions like Bernoulli variables, which were previously excluded.

The authors' approach integrates advanced techniques from additive combinatorics to control the least singular value of random matrices. Specifically, they employ an enhanced version of the Inverse Littlewood-Offord theorem, a tool critical to estimating the concentration probabilities of linear combinations of random variables. By leveraging this theorem, Tao and Vu achieve new lower bounds on the least singular value, which is pivotal to demonstrating strong convergence.

Numerical Results and Theoretical Implications

An intriguing aspect of their findings is that the circular law holds more broadly than previously affirmed, encompassing classes of random matrices that were not addressed in earlier literature. For instance, they extend their analysis to sparse matrices, showing the law's validity even when the matrix is perturbed in highly structured yet sparse manners.

This work not only settles longstanding questions regarding the circular law but also sets a precedent for quantifying the rate of convergence. With additional technical considerations, they show that for matrices with bounded (2 + η)th moment, the spectral distribution converges uniformly at a rate proportional to n^-η', where η' is derived from η.

Future Directions

The theoretical framework established in this paper paves the way for further exploration into random matrix theory and its applications. The broader applicability of the circular law implies potential impacts on disciplines such as statistical physics, telecommunications, and numerical linear algebra, where understanding the behavior of eigenvalues is crucial.

Furthermore, these results warrant an examination of whether similar relaxation in conditions can apply to other matrix ensembles or laws of large random matrices. Additionally, this work could inspire a refinement of algorithms in computational mathematics that exploit spectral properties, potentially enhancing performance and accuracy in practice.

Conclusion

In summary, Terence Tao and Van Vu's paper represents a milestone in the field of random matrices. By proving the circular law under less restrictive moment conditions, they significantly broaden the scope of this fundamental result. It is a contribution that not only resolves a historical conjecture but also invites new avenues of research in both theoretical and applied contexts.