Winter 2022 STAT 36510

Syllabus for topic course in random growth model

Preliminary


Universality in complex random systems is a striking concept which has played a central role in the direction of research within probability, mathematical physics and statistical mechanics. It turns out that a variety of physical systems and mathematical models, including randomly growing interfaces, certain stochastic PDEs, traffic models, all demonstrate the same universal statistical behaviors in their long-time/large-scale limit. These systems are said to lie in the Kardar-Parisi-Zhang (KPZ) universality class. 

In this course, we are going to focus on one fundamental model in this universality class — the corner growth model, which can also viewed as interacting particle system and percolation model.

Here are some contents that we would cover:

The corner growth model and some of its relatives

Deterministic large scale limits

The last-passage Markov chain

Tracy-Widom distribution

Distributional limit for the last-passage time

Evaluations


The evaluation is based on a written project. The written project provides you with an opportunity to research a topic of personal interest to you (related to the topics of the course). The project should be on the order of 5 – 10 pages in the standard amsart document class in LaTeX (i.e., no modifications to font or margin size, etc). You will have the opportunity to give a short presentation (20 minutes) in the end of the semester.

Random Growth Models

Shape fluctuations and random matrices

Cube root fluctuations for the corner growth model associated to the exclusion process

On randomized sketching algorithms and the Tracy-Widom law

Distribution of the largest eigenvalue for real Wishart and Gaussian random
matrices and a simple approximation for the Tracy-Widom distribution 

Lecture note

We will follow the lecture note by Timo Seppalainen: Lecture notes on Corner Growth Model

Handwritten lecture notes:

Lecture 1; Lecture 2; Lecture 3

Office hour

The office hour is flexible and can be scheduled by appointment.

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