1.INTRODUCTION entries are independent and identically distributed,except for the symmetry con- straints.For xER,let o,denote the Dirac measure at x,i.e the unique proba- bility measure satisfying ffd&r=f(x)for all continuous functions on R.Let LN-N-)denote the empirical measure of the eigenvalues of Xy. Wigner's Theorem(Theorem 2.1.1)asserts that,under appropriate assumptions on the law of the entries,Lw converges(with respect to the weak convergence of measures)towards a deterministic probability measure,the semicircle law.We present in Chapter 2 several proofs of Wigner's Theorem.The first,in Section 2.1,involves a combinatorial machinery,that is also exploited to yield central limit theorems and estimates on the spectral radius of XN.After first introducing in Section 2.3 some useful estimates on the deviation between the empirical mea- sure and its mean,we define in Section 2.4 the Stieltjes transform of measures and use it to give another quick proof of Wigner's theorem. Having discussed techniques valid for entries distributed according to general laws,we turn attention to special situations involving additional symmetry.The simplest of these concerns the Gaussian ensembles,the GOE and GUE.so named because their law is invariant under conjugation by orthogonal(resp.,unitary) matrices.The latter extra symmetry is crucial in deriving in Section 2.5 an explicit joint distribution for the eigenvalues(thus,effectively reducing consideration from a problem involving order of N2 random variables,namely the matrix entries,to ones involving only N variables).(The GSE,or Gaussian symplectic ensemble, also shares this property and is discussed briefly.)A large deviations principle for the empirical distribution,which leads to yet another proof of Wigner's Theorem, follows in Section 2.6. The expression for the joint density of the eigenvalues in the Gaussian ensem- bles is the starting point for obtaining local information on the eigenvalues.This is the topic of Chapter 3.The bulk of the chapter deals with the GUE,because in that situation the eigenvalues form a determinantal process.This allows one to effectively represent the probability that no eigenvalues are present in a set as a Fredholm determinant,a notion that is particularly amenable to asymptotic analysis.Thus,after representing in Section 3.2 the joint density for the GUE in terms of a determinant involving appropriate orthogonal polynomials,the Hermite polynomials,we develop in Section 3.4 in an elementary way some aspects of the theory of Fredholm determinants.We then present in Section 3.5 the asymptotic analysis required in order to study the gap probability at 0,that is the probabil- ity that no eigenvalue is present in an interval around the origin.Relevant tools, such as the Laplace method,are developed along the way.Section 3.7 repeats this analysis for the edge of the spectrum,introducing along the way the method of1. INTRODUCTION 3 entries are independent and identically distributed, except for the symmetry con￾straints. For x ∈ R, let δx denote the Dirac measure at x, i.e the unique proba￾bility measure satisfying R f dδx = f(x) for all continuous functions on R. Let LN = N −1 ∑ N i=1 δλi(XN) denote the empirical measure of the eigenvalues of XN. Wigner’s Theorem (Theorem 2.1.1) asserts that, under appropriate assumptions on the law of the entries, LN converges (with respect to the weak convergence of measures) towards a deterministic probability measure, the semicircle law. We present in Chapter 2 several proofs of Wigner’s Theorem. The first, in Section 2.1, involves a combinatorial machinery, that is also exploited to yield central limit theorems and estimates on the spectral radius of XN. After first introducing in Section 2.3 some useful estimates on the deviation between the empirical mea￾sure and its mean, we define in Section 2.4 the Stieltjes transform of measures and use it to give another quick proof of Wigner’s theorem. Having discussed techniques valid for entries distributed according to general laws, we turn attention to special situations involving additional symmetry. The simplest of these concerns the Gaussian ensembles, the GOE and GUE, so named because their law is invariant under conjugation by orthogonal (resp., unitary) matrices. The latter extra symmetry is crucial in deriving in Section 2.5 an explicit joint distribution for the eigenvalues (thus, effectively reducing consideration from a problem involving order of N 2 random variables, namely the matrix entries, to ones involving only N variables). (The GSE, or Gaussian symplectic ensemble, also shares this property and is discussed briefly.) A large deviations principle for the empirical distribution, which leads to yet another proof of Wigner’s Theorem, follows in Section 2.6. The expression for the joint density of the eigenvalues in the Gaussian ensem￾bles is the starting point for obtaining local information on the eigenvalues. This is the topic of Chapter 3. The bulk of the chapter deals with the GUE, because in that situation the eigenvalues form a determinantal process. This allows one to effectively represent the probability that no eigenvalues are present in a set as a Fredholm determinant, a notion that is particularly amenable to asymptotic analysis. Thus, after representing in Section 3.2 the joint density for the GUE in terms of a determinant involving appropriate orthogonal polynomials, the Hermite polynomials, we develop in Section 3.4 in an elementary way some aspects of the theory of Fredholm determinants. We then present in Section 3.5 the asymptotic analysis required in order to study the gap probability at 0, that is the probabil￾ity that no eigenvalue is present in an interval around the origin. Relevant tools, such as the Laplace method, are developed along the way. Section 3.7 repeats this analysis for the edge of the spectrum, introducing along the way the method of