Course: Introduction to Integrable Probability
Lecturer: Alexey Bufetov
Time: Monday 15:15--16:45 , Thursday 15:15--16:45 First Lecture: 19 April
Main course website: https://sites.google.com/site/alexeybufetov/teaching/s21-introduction-to-integrable-probability
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 lectures will happen at Big Blue Button . The link is https://conf.fmi.uni-leipzig.de/b/ale-npm-x68-95g
Access code: 356613
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 ?
- Trainer/in: Alexey Bufetov