Time: TR 11:00 am- 12:20 pm Room: Jones 303
Textbook: Directed Polymers in Random Environments (by Francis Comets)
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.
|01/05||Polymer measure and Partition function|
|01/10||Monotonicity and Phase transition|
|01/12||A Martingale associated to the Partition function|
|01/17||The second moment method and L^2 region|
|01/19||Diffusive behavior in the L^2 region|
|01/24||The Log-Gamma Polymer|
|01/31||Stochastic analysis for the space-time white noise|
|02/02||The KPZ equation and the Stochastic Heat Equation|
|02/07||The Continuum Directed Polymer (CDRP)|
|02/09||Convergence from Discrete Polymer models to the CDRP|
|02/14||Localization and Delocalization|
|02/16||Bounds on the Critical Temperature by Size-Biasing|
|02/23||Localization in low dimensions|
|02/28||Localization in the Log-Gamma Polymer|