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Our study of random matrices, to date, has focused on somewhat general ensembles, such as iid random matrices or Wigner random matrices, in which the distribution of the individual entries of the matrices was essentially arbitrary (as long as certain moments, such as the mean and variance, were normalised). In these notes, we now focus on two much more special, and much more symmetric, ensembles:

• The Gaussian Unitary Ensemble (GUE), which is an ensemble of random ${n \times n}$ Hermitian matrices ${M_n}$ in which the upper-triangular entries are iid with distribution ${N(0,1)_{\bf C}}$, and the diagonal entries are iid with distribution ${N(0,1)_{\bf R}}$, and independent of the upper-triangular ones; and
• The Gaussian random matrix ensemble, which is an ensemble of random ${n \times n}$ (non-Hermitian) matrices ${M_n}$ whose entries are iid with distribution ${N(0,1)_{\bf C}}$.

The symmetric nature of these ensembles will allow us to compute the spectral distribution by exact algebraic means, revealing a surprising connection with orthogonal polynomials and with determinantal processes. This will, for instance, recover the semi-circular law for GUE, but will also reveal fine spacing information, such as the distribution of the gap between adjacent eigenvalues, which is largely out of reach of tools such as the Stieltjes transform method and the moment method (although the moment method, with some effort, is able to control the extreme edges of the spectrum).

Similarly, we will see for the first time the circular law for eigenvalues of non-Hermitian matrices.

There are a number of other highly symmetric ensembles which can also be treated by the same methods, most notably the Gaussian Orthogonal Ensemble (GOE) and the Gaussian Symplectic Ensemble (GSE). However, for simplicity we shall focus just on the above two ensembles. For a systematic treatment of these ensembles, see the text by Deift.

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