is of the form where the function is called the potential. The Gaussian ensembles are the only common special cases of these two classes of random matrices. Spectral theory of random matrices The spectral theory of random matrices studies the distribution of the eigenvalues as the size of the matrix goes to infinity. Global regime In the global regime, one is interested in the distribution of linear statistics of the form . Empirical spectral measure The empirical spectral measure of is defined by Usually, the limit of is a deterministic measure; this is a particular case of self-averaging. The cumulative distribution function of the limiting measure is called the integrated density of states and is denoted N(λ). If the integrated density of states is differentiable, its derivative is called the density of states and is denoted ρ(λ). The limit of the empirical spectral measure for Wigner matrices was described by Eugene Wigner; see Wigner semicircle distribution and Wigner surmise. As far as sample covariance matrices are concerned, a theory was developed by Marčenko and Pastur. The limit of the empirical spectral measure of invariant matrix ensembles is described by a certain integral equation which arises from potential theory. Fluctuations For the linear statistics , one is also interested in the fluctuations about ∫ f(λ) dN(λ). For many classes of random matrices, a central limit theorem of the form is known. The variational problem for the unitary ensembles Consider the measure where is the potential of the ensemble and let be the empirical spectral measure. We can rewrite with as the probability measure is now of the form where is the above functional inside the squared brackets. Let now be the space of one-dimensional probability measures and consider the minimizer For there exists a unique euilibrium measure through the Euler-Lagrange variational conditions for some real constant where is the support of the measure and . The equilibrium measure has the following Radon–Nikodym density Local regime In the local regime, one is interested in the spacings between eigenvalues, and, more generally, in the joint distribution of eigenvalues in an interval of length of order 1/n. One distinguishes between bulk statistics, pertaining to intervals inside the support of the limiting spectral measure, and edge statistics, pertaining to intervals near the boundary of the support. Bulk statistics Formally, fix in the interior of the support of . Then consider the point process where are