**Course Instructor:** Arnab Sen

Office: 238 Vincent Hall

Email: arnab@math.umn.edu

**Class time:** MW 3:35 P.M. -- 4:50
P.M.

** Location: ** Vincent Hall 207

** Office hours: ** After lecture or by appointment.

**Course Description:** This course is an introduction to random matrix theory. Over the last two decades, the random matrix theory has grown into a huge field with diverse themes and has found connections to various branches of mathematics. In this one-semester course, we will attempt to cover only a small selection of topics. Our main focus will be to understand the spectrum of a large random self-adjoint matrix on both local and global scales. However, we will touch upon random non-self-adjoint matrices as well. We will explain various classical models of random matrices including Wigner matrices, Gaussian unitary ensembles, Gaussian beta ensembles, and Ginibre ensembles. In the last part of the course, we will study some spectral properties of the adjacency matrices of large sparse random graphs.

**Course Content:** We plan to cover the following topics:

- Wigner's semicircle law.
- Tridiagonalization of GUE and GOE matrices.
- Exact computation of joint eigenvalue densities for GUE (and Gaussian beta ensembles).
- Local limits of the GUE: the sine kernel (in the bulk) and the Airy kernel (at the edge).
- Circular law.
- Spectral properties of adjacency matrices of large sparse random graphs.

**Prerequisite:** No prior knowledge in random matrix theory is required but students should be comfortable with linear algebra and basic probability theory.

**Evaluation:** 2 homework assignments + a final report (a short
(no more than 3 pages) summary of one of the selected articles on random matrices - the list to be given later) .

This course will not have an official textbook. But the following
references will be useful:

1. G. Anderson, A. Guionnet and O. Zeitouni: An Introduction to Random
Matrices (available online).

2. A. Guionnet: Large random matrices: lectures on macroscopic
asymptotics. Lectures from the 36th Probability Summer School held in
Saint-Flour, 2006 (available online).

3. Terence Tao's blog posts on random matrices (also avaialble as a book).

4. Charles Bordenave, Djalil Chafaï: Around the
circular law.

5. J. B. Hough, M. Krishnapur, Y. Peres and B. Virag: Zeros of
Gaussian Analytic Functions and Determinantal Point Processes. (available online).

**Topics covered:**

- week 1 and 2 (1/22, 1/29): overview, description of important theorems in random matrices; method of moments, wigner's semicircular law, catalan numbers.
- week 3 (2/3, 2/5): finishing the proof of Wigner's semicircular law via method of moments; rank inequality, Hoffman-Wielandt lemma, truncation argument for Wigner matrix; basic properties of Stieltjes transform.
- week 4 (2/10, 2/12): Wigner's semiciruclar law via Stieltjes transform; Azuma-Hoeffding inequality; concentration of Stieltjes transform; concentration of Chromatic number for the Erdos-Renyi graphs.
- week 5 (2/17, 2/19): Talagrand's concentration inequality for product measures; application to maximum eigenvalue; application to longest increasing subsequence of random permutations.
- week 6 (2/24, 2/26): Gaussian ensembles: GOE, GUE, GSE. Tridiagonalization of Gaussian ensembles. Tridiagonal matrices, probability measures on the real line and orthogonal polynomials.
- week 7 (3/3, 3/5): Eigenvalue density for β-tridiagonal matrices. determinantal structure for GUE eigenvalue; Marginal densities, introduction to determinantal point processes
- week 8 (3/10, 3/12): counting number of points in a subset for a DPP - central limit theorem; non-intersecting random walks and determinantal structure; representation of hole probabilities in terms of Fredholm determinants.
- week 9 (3/17, 3/19): Spring break
- week 10 (3/24, 3/26): Bulk asymptotics of GUE, sine kernel, properties of Hermite polynomials, Laplace method, method of steepest descent
- week 11 (3/31, 4/2): Singularity probability of random Bernoulli matrices; No class on 4/2.
- week 12 (4/7, 4/9): Kahn-Komlos-Szemeredi bound on the singularity probability of random Bernoulli matrices, Fourier analytic methods in estimating small ball probabilities.
- week 13 (4/14, 4/16): Eigenvalues of graphs, two fundamental results in spectral graph theory: matrix-tree theorem and Cheeger's inequality (a nice blog post of Luca Trevsian explains the Cheeger's inequality on manifolds).
- week 14 (4/21, 4/23): Uniform d-regular graphs, configuration model, eigenvalue distribution for uniform d-regular graphs: Kesten-Mckay law, concentration inequality for uniform d-regular graphs via Azuma-Hoeffding inequality.
- week 15 (4/28, 4/30): local weak convergence of graphs, proofs of local weak convergence for uniform d-regular graphs and G(n, λ/n).
- week 16 (5/5, 5/7): Local weak convergence implies convergence of spectra, convergence of atoms (Lück approximation), examples (and counter-examples) of local invariants, existence of a dense set of atoms for G(n, λ/n).

**Suggested Reading:**

- An example of a locally finite infinite graph whose adjacency matrix is not self-adjoint by Vladimir Müller.
- N. Alon, M. Krivelevich, V. Vu. On the concentration of eigenvalues of random symmetric matrices, Israel J. Math. 131 (2002), 259-267. Link.
- L. Erdos, B. Schlein, HT Yau. Local semicircle law and complete delocalization for Wigner random matrices, Communications in Mathematical Physics, 287(2), 2009, 641-655. Link
- M. Ledoux. A remark on hypercontractivity and tail inequalities for the largest eigenvalues of random matrices. In Seminaire de Probabilites XXXVII, vol. 1832 of Lecture Notes in Math. (2003) 360-369. Link
- T. Tao, V. Vu. On random ±1 matrices: Singularity and determinant, Random Structures & Algorithms, 28 (1), 2006, 1-23. Link.
- M. Rudelson, R. Vershynin. The Littlewood-Offord Problem and invertibility of random matrices, Adv. Math. 218 (2008), 600-633. Link.
- M. Krivelevich, B. Sudakov. The phase transition in random graphs - a simple proof, Random Structures & Algorithms, 43(2), 131-138. Link.