2024 Fall
MATH C223A 001 - LEC 001
Advanced Topics in Probability and Stochastic Process
Geometric aspects of high dimensional Gibbs measures
Shirshendu Ganguly
Class #:33574
Units:3
Instruction Mode:
In-Person Instruction
Offered through
Mathematics
Current Enrollment
Total Open Seats:
1
Enrolled: 4
Waitlisted: 0
Capacity: 5
Waitlist Max: 5
No Reserved Seats
Hours & Workload
6 hours of outside work hours, and 3 hours of instructor presentation of course materials.
Other classes by Shirshendu Ganguly
Course Catalog Description
The topics of this course change each semester, and multiple sections may be offered. Advanced topics in probability offered according to students demand and faculty availability.
Class Description
Description: High dimensional Gibbs measures are ubiquitous in probability theory, statistical mechanics and theoretical computer science. A generic example, say on the hypercube Q=\{-1,1\}^n, is given by a Hamitonian H: Q o \mathbb{R}, with the measure \mu at \sigma \in Q being proportional to \exp\left(-eta H(\sigma)
ight) where eta>0 is the inverse temperature parameter. Z, the normalizing constant, is known as the partition function. Central examples include the Ising model, spin glass models, exponential random graphs, the hardcore model and so on. Such Gibbs measures also admit natural Glauber dynamics which keep \mu stationary.
The energy landscape induced by the Hamiltonian can often be quite complicated admitting exponentially many near ground states (the ground state is the state with the minimal energy).
In this course we will learn various techniques to estimate the partition function as well as establish structure theorems for a variety of Gibbs measures and the behavior of the associated Glauber dynamics. Phenomena such as super-concentration, chaos, multiple valleys and peaks, replica symmetry/breaking and metastability will be explored.
We will assume familiarity with the basics of graduate probability equivalent to Stat C205A.
Rules & Requirements
Repeat Rules
Reserved Seats
Current Enrollment
No Reserved Seats
Textbooks & Materials
See class syllabus or https://calstudentstore.berkeley.edu/textbooks for the most current information.
Guide to Open, Free, & Affordable Course Materials
Associated Sections
None