S21: Introduction to Integrable Probability
Course: Introduction to Integrable Probability
Lecturer: Alexey Bufetov
Time: Monday 15:15--16:45 , Thursday 15:15--16:45
Prerequisites: Basic Probability (e.g., what is a random variable ?), basic Linear
Algebra (e.g. what is a determinant ?). The rest will be defined and explained.
Connection Information: first lecture: 19 April
First lectures will happen at Big Blue Button . The link is https://conf.fmi.uni-leipzig.de/b/ale-npm-x68-95g
Access code: compute 356615 - 2 = ?
If you are interested in the course, please send me an email to alexey.bufetov AT math.uni-leipzig.de . I will send the news during the course via the email list. Most important updates will also appear on this webpage.
What this course is about: Integrable probability is a recently (last 20-30 years) emerged field in probability theory. The main characteristic feature of the field is the prominent role of methods and ideas from other parts of mathematics, such as analysis, algebra, combinatorics, and mathematical physics. Below I briefly describe some questions that we will study during this course.
------ Consider a uniformly random permutation of size $N$. Let $L_N$ be the length of the longest increasing subsequence in this permutation (example: for a permutation $ 3 1 5 2 4$ the length of the longest increasing subsequence is 3). How does the random variable $L_N$ behave as $N \to \infty$ ? In words: what is a typical length of the longest increasing subsequence of a large random permutation ?
----- Consider a large square symmetric matrix with entries on and above diagonal filled by independent identically distributed random variables. What one can say about the (random) eigenvalues of this matrix ? In particular, how do they behave as the size of the matrix tends to infinity ?
----- Consider a uniform domino tiling (covering by dominoes without overlap) of the rectangular Aztec diamond --- the region of the square lattice depicted on the Figure above, left panel. When we take the domain very large (see the right panel), we clealry see some structure. Why does it look like this ?
Preliminary plan of the course:
During the course we will cover most / all of the following topics. The program might be adjusted depending on how the course is going and on the interests of listeners.
• Introduction
• Wigner ensemble: symmetric matrices with independent entries. Law of Large
Numbers for eigenvalues: Semicircle law. Moment method. Wishart ensemble
and Marcenko-Pastur distribution.
Asymptotic behavior of the largest eigenvalue of Wigner ensemble
• Symmetric polynomials. Schur polynomials and their properties.
• Domino tilings of the Aztec diamond. Number of tilings.
• Global limit behavior of a measure determined by a Schur generating function.
• Global limit behavior of domino tilings of the Aztec diamond
• Limit behavior of more complicated lozenge and domino tiling models.
• Schur-Weyl and Plancherel probability measures on Young diagrams. Global limit behavior of random Young diagrams under these measures.
Determinantal point processes : general facts
Gaussian Unitary Ensemble: eigenvalues form a determinantal process.
Bulk and edge asymptotic fluctuations of GUE eigenvalues
Schur measure is a determinantal process.
Bulk and edge asymptotic fluctuations of domino tilings of the Aztec diamond
Suggested literature
• G. Anderson, A. Guionnet and O. Zeitouni: An Introduction to Random Matri-
ces; available from the O. Zeitouni’s webpage http://www.wisdom.weizmann.
• T. Tao: Topics in random matrix theory; available at https://terrytao.
wordpress.com/category/teaching/254a-random-matrices/
• D. Romik: The Surprising Mathematics of Longest Increasing Subsequences.
• A. Borodin, V. Gorin: Lectures on Integrable Probability; https://arxiv.
• A. Bufetov, V. Gorin: Representations of classical Lie groups and quantized
free convolution; https://arxiv.org/abs/1311.5780
(more references will be provided during the course).