Beyond universality in random matrix theory

In order to have a better understanding of finite random matrices with non-Gaussian entries, we study the 1/N expansion of local eigenvalue statistics in both the bulk and at the hard edge of the spectrum of random matrices. This gives valuable information about the smallest singular value not seen in universality laws. In particular, we show the dependence on the fourth moment (or the kurtosis) of the entries. This work makes use of the so-called complex Gaussian divisible ensembles for both Wigner and sample covariance matrices. 1 Beyond Universality The desire to assess the applicability of universality results in random matrix theory has pressed the need to go beyond universality, in particular the need to understand the influence of finite n and what happens if the matrix deviates from Gaussian normality. In this note, we provide exact asymptotic correction formulas for the smallest singular value of complex matrices and bulk statistics for complex Wigner matrices. “Universality,” a term encountered in statistical mechanics, is widely found in the field of random matrix theory. The universality principle loosely states that eigenvalue statistics of interest will behave asymptotically as if the matrix elements were Gaussian. The spirit of the term is that the eigenvalue statistics will not care about the details of the matrix elements. It is important to extend our knowledge of random matrices beyond universality. In particular, we should understand the role played by • finite n and • non Gaussian random variables. From an application viewpoint, it is very valuable to have an estimate for the departure from universality. Real problems require that n be finite, not infinite, and it has long been observed computationally that ∞ comes very fast in random matrix theory. The applications beg to know how fast. From a theoretical viewpoint, there is much to be gained in searching for proofs that closely follow