Topics in Probability Theory - Topical (MTH 303.5)

Course webpage: https://www.trishen.co.uk/teaching/topics-in-probability-icts

Venue: Feynman Lecture Hall

Class Timings: Wednesdays from 2:00 PM - 5:00 PM

First Meeting: 13 Aug 2025,

Course Type: Topics Course for PhD program in Mathematics

Syllabus : This course covers advanced or specialized topics in probability theory. Themes vary by offering and are selected by the instructor. Possible areas include (but are not limited to) stochastic processes, limit theorems, random structures, statistical mechanics, and probabilistic methods in other fields.

This Semester (Aug-Dec 2025) 

Critical planar statistical physics

Planar statistical physics models exhibit a wide range of beautiful phenomena. The study of two-dimensional percolation models has been at the forefront of many exciting developments in probability theory in the past few decades. In this course, we will study Fortuin-Kastelyn (FK) percolation models, which are a family of percolation models parametrized by $q\geq 0$. The case $q=1$ corresponds to Bernoulli bond percolation. These models are also intimately connected with Ising and Potts lattice spin models. We will study the critical phenomena of these percolation models on $\mathbb Z^2$ for $q \geq 1$.  The goal is to prove that the phase transition is continuous for $1 \leq q \leq 4$ and discontinuous for $q>4$.

Syllabus:

  • The FK model and fundamental properties.
  • Edwards-Sokal coupling with Ising and Potts models.
  • Baxter-Kelland-Wu coupling with 6V.
  • Discontinuity when $q>4$.
  • Russo-Seymour-Welsh theory.
  • Renormalization of crossing probabilities.
  • Parafermionic observables.
  • Continuity when $1 \leq q \leq 4$.

Assessment is by a short presentation at the end of the semester.

Course Outcomes:

  • Understand advanced concepts in one or more areas of modern probability.

  • Develop proficiency in key probabilistic techniques and their applications.

  • Engage with current research-level material through problems and readings.


Course Evaluation:  Short blackboard presentations (c. 15-20 mins) on a selected topic from the course at the end of the semester.




Credit Score: 2