# Introduction to Random Matrices

### Lecturer: Johannes Alt

Random matrices have been introduced in the first half of the 20th century motivated by applications in statistics and in physics. Currently, it is a very active research area, both in mathematics and in applications. This course gives an introduction to the basic concepts and results of the mathematical theory of random matrices which lies at the intersection of linear algebra, analysis and probability theory.

## Time and place

 Lecture: Wednesday, 13:15 - 14:45 Room SM623 Exercise class: Wednesday, 15:15 - 16:00 Room SM623

## Contents

• Wigner's semicircle law: moment method, Catalan numbers; resolvent method, Stieltjes transform.
• Local semicircle law.
• Sharp norm bound, eigenvalue rigidity and eigenvector delocalization.

## Homework exercises

The homework exercises are posted weekly on Chamilo.

## Contents of the lectures

 February 20, 2019: Goal of the course: understand the behaviour of the eigenvalues of a typical, large Hermitian random matrices Motivations: Hermitian random matrices were introduced due to applications in quantum mechanics and statistics. Moreover, random matrix statistics appear in the nontrivial zeros of the Riemann zeta-function, random growth models and the longest increasing subsequence of random permutations. Definition Gaussian orthogonal ensemble (GOE) and Gaussian unitary ensemble (GUE). Definition of Wigner matrix; real symmetric and complex Hermitian Wigner matrix Definition of empirical spectral measure (uniform probability measure on the spectrum); Empirical spectral measure is a random measure Formulation of a weak version of Wigner’s semicircle law: The expectation of the kth moment of the empirical spectral measure of a real symmetric Wigner matrix converges to the kth moment of Wigner’s semicircle law. Proof via moment method in the next lecture. References for Definitions: Beginning of Section 2.1 and Section 2.2 of the book by Anderson, Guionnet and Zeitouni. February 27, 2019: Proof that the expectation of the moments of the empirical spectral measure of a real symmetric NxN Wigner matrix H converges to the moments of Wigner's semicircle law: The expectation of the kth moment of the empirical spectral measure equals to the expectation of the trace of the kth power of H divided by N. The trace of the kth power of H is written out in terms of entries of H. Using a graphical language the leading orders of these sums are identified. The odd moments are seen to be of order N^{-1/2}. The even moments equal the number of Dyck paths of length k up to an error term of size N^{-1}. Since the number of Dyck paths of length k equals to the (k/2)th Catalan number we conclude from a homework exercise that the even moments coincide in the limit with the moments of Wigner's semicircle law. Reference: Lemma 2.1.6 and Section 2.1.3 in the book by Anderson, Guionnet and Zeitouni. March 6, 2019: Proof that the variance of M_{N,k}, the kth moment of the empirical spectral measure, is of order 1/N^2. Variance is expressed as sum of covariances of products of k entries of H. These products are categorized by graphs in order to count them and reveal certain cancellations. Since the variance of M_{N,k} converges to zero while the expectation of M_{N,k} converges to m_k, the kth moment of the semicircle law, M_{N,k} converge in probability to m_k. Owing to the strong convergence rate of the variance of M_{N,k}, we show that it converges even almost surely to m_k by the Borel-Cantelli Lemma. Empirical spectral measure converges weakly in probability to semicircle law. Reference: Lemma 2.1.7 and Section 2.1.7 as well as Section 2.1.2 in the book by Anderson, Guionnet and Zeitouni. March 13, 2019: Version of semicircle law for intervals: fraction of eigenvalues in an interval converges in probability to the integral of the semicircle over it. This follows directly from the weak convergence in probability. Summary of the results obtained with the moment method for a polynomial integrated against the empirical spectral measure. Comparison to the case when the eigenvalues would be independent. Conclusion: Eigenvalues fluctuate much less. Overview of other applications of the moment method (without proofs): Wigner's semicircle law for complex Hermitian Wigner matrices, for matrices with finite second moment only; convergence of the largest eigenvalue to 2 in probability. Stieltjes transform: definition, inversion formula; Stieltjes transform of the semicircle distribution Reference: Stieltjes transform: Beginning of Section 2.4 in the book by Anderson, Guionnet and Zeitouni. March 20, 2019: Explanation of the next goal: new proof of Wigner's semicircle law with the help of the Stieltjes transform. First part of the preparations: understanding the relation between convergence of Stieltjes transform and convergence of finite measures. Vague convergence of finite measures (this is a weaker notion of convergence than convergence in probability). Good property of vague convergence: sequence of finite measures with uniformly bounded total measure has convergent subsequence. If the limiting measure of a vaguely convergent sequence of probability measures is a probability measure then the convergence holds in weak sense as well. If Stieltjes transforms of a sequence of probability measures converge pointwise then the measures converge vaguely. If Stieltjes transforms of a sequence of random probability measures converge to the Stieltjes transform of a deterministic probability measure in probability then the random measure converge weakly in probability to the deterministic measure. Reference: Beginning of Section 2.4 in the book by Anderson, Guionnet and Zeitouni. In particular, Theorem 2.4.4. March 27, 2019: Two important ingredients of the second proof of Wigner's semicircle law: Hoffman-Wielandt theorem, Schur complement formula. Proof sketch of the Hoffman-Wielandt theorem via Birhoff-von Neumann theorem and rearrangement inequalities. Beginning of the proof that the Stieltjes transform of the empirical spectral measure of a real symmetric Wigner matrix converges in probability to the Stieltjes transform of the semicircle distribution: Resolvent. Minors. Schur complement formula implies that normalized trace of resolvent satisfies the quadratic equation of the Stieltjes transform of the semicircle distribution up to some error terms. Proof that the first two error terms converge to zero in probability when N goes to infinity. The remaining two error terms will be estimated next time. Reference: Section 2.4.2 in the book by Anderson, Guionnet and Zeitouni. April 3, 2019: End of the proof that the Stieltjes transform of the empirical spectral measure of a real symmetric Wigner matrix converges in probability to the Stieltjes transform of the semicircle law. This last part used some multilinear large deviation estimates derived from the Marcinkiewicz-Zygmund inequality. Motivation of the local semicircle law: it implies that the number of eigenvalues in a shrinking interval converges to the number expected by the semicircle law with very high probability. Informal formulation in terms of the resolvent and Stieltjes transform of the semicircle. Diagonal entries of resolvent converge to this Stieltjes transform; Off-diagonal entries converge to zero. Definition of events that hold with very high probability. Definition of stochastic domination. Precise formulation of the local semicircle law. Precise statement about the approximation of the number of eigenvalues in any interval by semicircle law. Reference: Beginning of Section 2, Section 2.3 and beginning of Section 2.4 of the lectures notes by Benaych-Georges and Knowles. In particular, Theorem 2.6 and Theorem 2.8. April 10, 2019: Version of the local semicircle law with weaker error bound. Proof idea: bootstrapping. Bootstrapping will start at a large imaginary part of the spectral parameter. Then imaginary part will be reduced successively down to the optimal scale. Preparations of the proof: resolvent identities, Ward identity, large deviation bounds expressed in the sense of stochastic domination. Self-consistent equation for diagonal entries of resolvent which is obtained from the Schur complement. Formulation and proof of the main estimates that control the error terms if the imaginary part is large or there is a weak initial bound. Reference: Section 3 and beginning of Section 5 of the lectures notes by Benaych-Georges and Knowles. May 8, 2019: Stability of the equation defining the Stieltjes transform m of the semicircle law. Initial step of the bootstrapping: proof of the local law when the imaginary part of the spectral parameter z is bigger or equal to one. Proof of the weak local law stated last time for all z in a finite lattice. Idea of the proof: successively decreasing the imaginary part of z by using Lipschitz-continuity in z, main estimates established last time and the stability of the equation for m. Continuity argument yields weak local law everywhere. Reference: Section 5 of the lectures notes by Benaych-Georges and Knowles. May 15, 2019: Fluctuation averaging: If the offdiagonal terms G_{ij} of the resolvent G converge to zero at a certain rate then the average of Q_i 1/G_{ii} converges to zero with the square of this rate. The summands Q_i 1/G_{ii} are of the same order as G_{ij} and a cancellation in the average improves this rate. The proof of the fluctuation averaging is based on tracking this cancellation in high moment estimates. Sketch of the estimate of the second moment. Full proof in Proposition 6.1 of the lectures notes by Benaych-Georges and Knowles. Proof of the local semicircle law, Theorem 3.2, using the fluctuation averaging. Statement and proof of an improved local semicircle law outside of [-2,2] which is the support of the semicircle law. Reference: Section 6 and the beginning of the proof of Proposition 6.1 in Section 7 of the lectures notes by Benaych-Georges and Knowles. May 22, 2019: Proof of the semicircle law on small scales: The normalized number of eigenvalues in any interval is close to the number prescribed by the semicircle law up to an error term of size N^{-1} in the sense of stochastic domination. Proof via the Helffer-Sjöstrand formula (without proof): This formula can be used to express the integral of a function with respect to some measure in terms of an integral of the Stieltjes transform of this measure. We also need a uniform version of the local semicircle law. Norm of a Wigner matrix is bounded by 2 + O(N^{-2/3 + \eps}). Proof by deriving a contradiction to the improved local semicircle law outside of [-2,2], Corollary 3.17. Reference: Theorem 2.8, Section 8 and the beginning of Section 9 in the lectures notes by Benaych-Georges and Knowles. May 29, 2019: Proof of Eigenvalue rigidity: eigenvalues are very close to their typical locations given by the N-quantiles of the semicircle distribution. Proof of Eigenvector delocalization: all components of any eigenvector of a Wigner matrix have essentially the same modulus. Outlook: Universality of eigenvalue statistics on the microscopic scale. Sine kernel. Courses on Random Matrices and Universality during the masterclass 2019/2020 in mathematical physics. Reference: Theorem 2.9, Theorem 2.10, Section 9 and Section 11.2 in the lectures notes by Benaych-Georges and Knowles.

--- Last updated on May 29, 2019 ---