Wigner Matrices and Semicircular Law

Abstract

A Wigner matrix is a symmetric (or Hermitian in the complex case) random matrix. Wigner matrices play an important role in nuclear physics and mathematical physics. The reader is referred to Mehta [212] for applications of Wigner matrices to these areas. Here we mention that they also have a strong statistical meaning. Consider the limit of a normalizedWishart matrix. Suppose that x ₁, …, x _n are iid samples drawn from a p-dimensional multivariate normal population N(μ, I _p ). Then, the sample covariance matrix is defined as

$$ S_n = \frac{1}{{n - 1}}\sum\limits_{i = 1}^n {(x_i - \bar x)} (x_i - \bar x)',$$

where $\overline{x}=\frac{1}{n}\sum\nolimits_{i=1}^{n}{x_i}.$ When n tends to infinity $S_n \rightarrow I_p$ and $\sqrt {n} (S_n - I_p) \rightarrow \sqrt {p} {W_p}$ It can be seen that the entries above the main diagonal of $\sqrt {p} {W_p}$ are iid N(0, 1) and the entries on the diagonal are iid N(0, 2). This matrix is called the (standard) Gaussian matrix or Wigner matrix.