Universality in Random Matrix Theory

which is the Central Limit Theorem. In principle, all the random variables X1, X2, · · · , XN can be of order 1, hence SN ∼ 1 as well, but the probability of having such a rare event is incredibly small. We can even estimate the bound on the probability for the rare event from the large deviation principle. A similar phenomenon happens when we form a large matrix from i.i.d. random variables and consider the distribution of its eigenvalues. The simplest example for the random matrix model is the Gaussian Orthogonal Ensemble (GOE). Definition 1.1 (Gaussian Orthogonal Ensemble (GOE)). An N × N random matrix H = (hij) is said to be the Gaussian Orthogonal Ensemble if its entries hij are real Gaussian random variables, independent up to the symmetry constraint hij = hji, satisfying 1. Ehij = 0, 2. E|hij | = N−1 for i 6= j, and 3. E|hii| = 2N−1. Similarly, we can define a model for complex Hermitian matrices, which is known as Gaussian Unitary Ensemble (GUE). Definition 1.2 (Gaussian Unitary Ensemble (GUE)). An N × N random matrix H = (hij) is said to be the Gaussian Unitary Ensemble if its entries hij are complex Gaussian random variables, independent up to the symmetry constraint hij = hji, satisfying 1. Ehij = 0, 2. E|hij | = N−1, and 3. E(hij) = 0 if i 6= j. In order to introduce results on GOE or GUE analogous to CLT, we consider the empirical spectral distribution of a random matrix. Definition 1.3. Let λ1 ≤ λ2 ≤ · · · ≤ λN be the eigenvalues of H. Then, the empirical (spectral) measure of H is defined by