Random matrix theory has a history going back to mathematical statistics in the 1920's and nuclear physics in the 1950's. In the last 20-30 years the subject has expanded considerably and is now a broad research area. It has connections to many parts of mathematics and other fields, e.g. asymptotic analysis, combinatorics, number theory, statistics, statistical physics, numerical analysis, and communications theory. This means that one can study the area of random matrices from many different points of view and there are many possible approaches to solving problems within the subject. The course will cover some basic aspects of random matrices with an emphasis on basic mathematical concepts and techniques.

Lectures

All lecture will be online using zoom. Zoom link for lectures:

Lecture zoom

After each lecture we will upload lecture notes.

Date	Teacher	Topic	Lecture notes
October 6, at 15.15	Johansson	Lecture 1: Introduction to Random Matrices	LectureNotes1.pdf
October 14, at 15.15	Johansson	Lecture 2: Introduction to Discrete Models
October 21, at 15.15	Duits	Lecture 3: Determinantal point processes: Discrete case	F3.pdf
October 28, at 15.15	Duits	Lecture 4: Determinantal point processes: general case	F4.pdf
November 4, at 15.15	Duits	Lecture 5: Biorthogonal ensembles	F5.pdf
November 11, at 15.15	Duits	Lecture 6: GUE: Asymptotics	F6.pdf
November 18, at 15.15	Duits	Lecture 7: Determinantal point processes: Asymptotics
November 25, at 15.15	Johansson	Lecture 8: More on discrete models: Tracy-Widom in random growth	Lecture8.pdf
December 2, at 15.15	Johansson	Lecture 9: Extended processes. Dyson Brownian motion.	Lecture9.pdf
December 9, at 15.15	Schnelli	Lecture 10: Semicircle law, Stieltjes transform, Green function.	Lecture10.pdf
January 20, at 15.15.	Schnelli	Lecture 11: Non-invariant ensembles	Lecture11.pdf
January 27, at 15.15	Schnelli	Lecture 12: Non-invariant ensemble	Lecture12.pdf
February 28		Deadline homework
March 17, at 15.15		Presentations 1
March 24, at 15.15		Presentations 2
March 31		Deadline notes

Examination

Examination will be based on hand-ins and presentation. The precise form of examination will be determined later and depends on the Covid-19 situation.

Homework problems: RMTExercisesv7.pdf

Note new version Februari 19..

The solutions should be handed in no later than February 28, 2021. For pass you should hand in at least seven of the ten problems.

The other part of the examination will consist of giving a short presentation (max. 20 min) and writing an accompanying note on a topic related to the course. Suggestions for topics (pdf). Contact for topic distribution: schnelli@kth.se

Topic:	Student:	Presentation date:
Weyl integration formula	Nils Hemmingsson	March 17
RSK correspondance	Petter Restadh	March 17
Semicircle law via moment method	Aleksa Stankovic	March 17
Selberg integral	Daniel Ahlsen	March 17
Strong Szego theorem	Aryaman Jal	March 24
Haar unitary matrices	Klara Courteaut	March 17
Detection thresholds in spiked random matrix models	Tianfang Zhang	March 17
Weak local semicircle law via Schur complement	Joakim Cronvall	March 24
Helffer-Sjöstrand formula and rigidity of eigenvalues	Björn Martinsson	March 24
Kastelyn theory of dimer models	Scott Mason	March 24
Log-correlated fields and height fluctuations GUE	Wenkui Liu	March 24
	Philippe Moreillon	March 24

Teachers

Maurice Duits, Kurt Johansson and Kevin Schnelli.

Recommended extra literature

A. Borodin, Determinantal point processes, arXiv:0911.1153
K. Johansson, Random Matrices and Determinantal Point Processes arXiv:0510038
K.Johnasson, Edge fluctuations of limit shapes, arXiv:1704.06035
A. Soshnikov, Determinantal Random Point Fields arXiv:0002099

FSF3624 HT20 (50776) Random Matrices