STAT 36510 Random Growth model and KPZ equation (2023 Winter)

Time: TR 11:00 am- 12:20 pm Room: Jones 303

Textbook: Directed Polymers in Random Environments (by Francis Comets) to an external site.

Course overview:

The model of directed polymers in a random environment is a simplified model for stretched elastic chains pinned by random impurities. The main question is:

What does a random walk path look like if rewards and penalties are randomly distributed in the space?

In large generality, it experiences a phase transition from diffusive behavior to localized behavior. In this course, we place a particular emphasis on the localization phenomenon. The model can be mapped or related to many others, including interacting particle systems, percolation, queueing systems, randomly growing surfaces, and biological population dynamics. Even though it attracts considerable research activity in stochastic processes and statistical physics, it still keeps many secrets, especially in space dimensions two and larger. It has non-Gaussian scaling limits and belongs to the so-called KPZ universality class when the space dimension is one.

We consider the simplest discrete setup throughout the lecture. The only exception is the KPZ equation — a stochastic partial differential equation — for the decisive reason that it is the scaling limit of the discrete model in one dimension with vanishing interaction, and therefore, it finds a natural place.

In this course, I will also review some standard tools in probability theory such as stochastic calculus.

Course evaluation: based on class participation and a final presentation on literature reading.


01/05Polymer measure and Partition function
01/10Monotonicity and Phase transition
01/12A Martingale associated to the Partition function
01/17The second moment method and L^2 region
01/19Diffusive behavior in the L^2 region
01/24The Log-Gamma Polymer
01/26no class
01/31Stochastic analysis for the space-time white noise
02/02The KPZ equation and the Stochastic Heat Equation
02/07The Continuum Directed Polymer (CDRP)
02/09Convergence from Discrete Polymer models to the CDRP
02/14Localization and Delocalization
02/16Bounds on the Critical Temperature by Size-Biasing
02/21Path localization
02/23Localization in low dimensions
02/28Localization in the Log-Gamma Polymer
03/02Student Presentation