Dimer Models in Statistical Mechanics

This is a short introduction to dimer models in statistical mechanics and the combinatorics that is involve is their solution. This is a short course aimed at undergraduates that is given online at the Summer School at Sanya, Hainan.

Here is how to connect to the lectures:

https://zoom.us/j/4552601552?pwd=cWxBUjlIN3dxclgrZWFEOC9jcmlwUT09

Zoom Meeting ID: 4552601552

Passcode: YMSC

Dimer models appeared in statistical mechanics in earlier 1960’s. These are models in equilibrium statistical mechanics on a two dimensional lattice where the interaction is determined by certain combinatorial exclusion rules. Dimer configurations on plane (and surface) graphs have many combinatorially equivalent interpretations.

As in any model of equilibrium statistical mechanics the main challenge is to find the characteristics of the model, including its correlation functions in the limit when the “size of the system” goes to infinity.

The lectures will be given jointly with Emily Bain who will focus on numerical aspects of Markov simulations for dimer models. These numerical methods are known for physicists as Monte-Carlo or Metropolis-Hastings algorithms.

Tentative plan of the mini-course is as follows:

Lecture 1.(NR) Short introduction to equilibrium statistical mechanics of lattice models. Dimer models on a graph.Correlation functions. Examples. The thermodynamics limit.

Lecture 2.(NR) Dimer models in statistical mechanics. Equivalent combinatorial objects: tilings, lattice path. Kasteleyn solution to a dimer model on a plane graph.

Lecture 3.(NR) Dimer models on large domains. The thermodynamic limit for dimer models on lattices.

Lecture 4. (NR) The limit shape phenomenon in dimer models on bipartite lattices. 

Lecture 5. (EB) How to study dimer models numerically? Introduction to Markov processes (special type of stochastic processes).

Lecture 6. (EB) Sampling dimer models using Markov processes.

Lectures, video files are here. First part of the first lecture is missing (I forgot to start recording):

Notes that I made during the lectures are here:

My notes that I prepared for lectures are here:

The proof of the Kasteleyn theorem for plane graphs is here:

Emily Bain slides:

Here is the code for sampling dimer configurations:

https://math.berkeley.edu/~ebain/code_setup.html

You can also find materials for lectures 5-6 at Emily’s website:

https://math.berkeley.edu/~ebain/.

Prerequisites. Elements of probability theory. Linear algebra.Some knowledge of statistical mechanics will be helpful, but it is not necessary.

References:

1) Kerson Huang, Statistical Mechanics (classical textbook on statistical mechanics)

2) Richard Kenyon, An introduction to the dimer model, arxiv: math/0310326

3) Barry McCoy, Tai Tsun Wu, The two-dimensional Ising model, 2nd edition, 2014, Dover Publications.

4) James Norris, Markov Chains, 1998 (Chapter 1).

5) David Keating and Ananth Sridhar, Random tilings with the gpu, 2018 (can be found on the arxiv)

6) Notes for the class.