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

Grading policy

The final grade will be based on several homework assignments.

Homework

Homework 1

Homework 2

Classes (collapsible sections contain board pictures)

Class 1 (Sep 2, 2025): Introduction to the course. Percolation on a regular tree

Class 2 (Sep 4, 2025): Percolation on the complete graph (Erdős–Rényi phase transition). Proof follows the paper of Krivelevich-Sudakov

Class 3 (Sep 9, 2025): End of Erdős–Rényi phase transition proof. Tool 1: Monotone coupling

Class 4 (Sep 11, 2025): Bond percolation on Z^d. Proof of phase transition in dimension d>=2

Class 5 (Sep 16): Correlation inequalities: Harris' theorem and the four functions theorem of Ahlswede and Daykin

Class 6 (Sep 18): Proof of Harris' inequality. Application to two-point function lower bound in super-critical phase. The van den Berg--Kesten inequality and application to a Simon-Lieb type inequality

Class 7 (Sep 23): More on the Simon-Lieb type inequality. Finite-size criterion for exponential decay of connectivities. Ergodicity and application to the number of infinite open clusters

Class 8 (Sep 25): Statement of the ergodic theorem. The Russo-Margulis formula for Boolean functions. Pivotal indices and the 4-arm event in two-dimensional percolation

Class 9 (Sep 30): Sharpness of the phase transition (exponential decay throughout sub-critical regime) and mean-field lower bound

Class 10 (Oct 2): Exponential decay in the volume of the cluster in sub-critical regime. Discussion on more general transitive graphs and the phase transition to a unique infinite open cluster. Uniqueness of the infinite open cluster on Z^d (Burton-Keane proof)

Class 11 (Oct 7): Right-continuity in p of the probability that the origin is an infinite cluster. Left continuity at all but the critical p. Two-dimensional bond percolation: Statement of the Harris-Kesten theorem and discussion of duality

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