Math 878T/Stat 818T (Fall 2025)

Introduction to Statistical Physics

Syllabus

Statistical Physics studies how the macroscopic properties of materials arise from the materials’ microscopic description. It is a branch of physics with a rich mathematical theory, studied by physicists and mathematicians alike (with four fields medals awarded for its study in the last 20 years).

Significant focus is given to phase transitions: dramatic changes in the macroscopic properties occurring as a result of small changes in the microscopic description (or the outside environment). Familiar examples of such transitions include the boiling of water at 100 degrees Celsius or the spontaneous magnetization of metals at low temperature.

The course will introduce the mathematical theory of statistical physics on lattices, based on probability theory (and touching upon combinatorics and analysis). This will be done through consideration of several basic models, including percolation, Ising and the spin O(n) models. We will detail some of the progress made and highlight remaining challenges. Time permitting, we will discuss additional settings such as random surfaces and disordered systems.

Prerequisites

No background in physics is required. We will make use of measure-theoretic probability theory, on the level of Stat 600 (it is possible to take Stat 600 in parallel to the course).

Reference Materials

1. Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction / Friedli and Velenik. https://www.unige.ch/math/folks/velenik/smbook/

2. Lectures on the Ising and Potts models on the hypercubic lattice / Duminil-Copin.
Introduction to Bernoulli percolation / Duminil-Copin.
https://www.ihes.fr/~duminil/teaching.html

3. Lectures on the Spin and Loop O(n) models / Peled and Spinka.
https://arxiv.org/abs/1708.00058

4. Probability on Graphs / Grimmett.
https://www.statslab.cam.ac.uk/~grg/books/pgs.html

5. Ferromagnetic spin systems / Bauerschmidt.
https://cims.nyu.edu/~bauerschmidt/teaching/math253x/spin.pdf