Lectures' time and place: Fridays 10:15-11:45 am, H306.
This will be a slow and gentle introduction to some probabilistic aspects of statistical mechanics, with many exercises. Prerequisites are just the basics of probability theory.
There will be an exercise sheet every two weeks, or so; before every second class, students will say which exercises they have solved, and each solution will be presented by a student at the blackboard. One half of the grade will come from these exercise solutions. For the other half of the grade, there will be a written midterm.
Here is the first exercise set. And the second exercise set. Please record which of them you are prepared to present.
Topics:
Percolation theory: definitions and their equivalence. Examples using the Peierls contour method, first and second moment method. The Harris-FKG correlation inequality.
The Ising model on finite graphs: definition, spatial Markov property, basic properties of the partition function, definition of long range order.
Glauber dynamics and other Markov chains. Holley's proof of the FKG-inequality. Infinite volume Gibbs measures.
The FK random cluster model, Edwards-Sokal coupling, the Potts models, Uniform Spanning Tree.
Mean field models: Erdős-Rényi random graph and the Curie-Weiss phase transition.
Pólya’s theorem on recurrence versus transience of simple random walk on Z^d. Green's function.
The Discrete Gaussian Free Field (DGFF) on graphs. Relation to other models, such as Ising.
Intuitive glances at more advanced topics:
Critical point for planar percolation: the Harris-Kesten theorem (1980).
Scaling limits: Brownian motion. Continuum GFF. The conformal invariance of critical planar percolation and other FK models. (Fields medals to W. Werner 2006 and S. Smirnov 2010.)
Mermin-Wagner and Kosterlitz-Thouless on the XY model in the plane: what is a topological phase transition? (Nobel prize in physics 2016)
What is a spin glass? (Nobel prize in physics to Parisi 2021)
Self-organized criticality, such as sandpiles (Bak-Tang-Wiesenfeld 1987, Nobel prize in physics 2033).
Feb 20: We introduced bond and site percolation, defined transitivity of a graph, defined p_c in two different ways, dwelled a little bit on measurability issues (including the statement of the Banach-Tarski paradox), proved p_c(\Z)=1 for the integer line, guessed p_c(\T_d)=1/(d-1) for the d-regular tree, and stated (but have not proved) that \theta(p) is increasing, right-continuous, and left-continuous except possibly at p_c. We will prove most of these and 1/3 \leq p_c(\Z^2) \leq 2/3 next time.
You can watch this award-winning video or start reading Section 12.1 of my PGG lecture notes.
If you are curious about the Banach-Tarski paradox, I have not watched this, but it should be good.
Feb 27: We introduced the notion and proved that P_q stochastically dominates P_p for q>p, and concluded that \theta(p) is monotone increasing. Proved that p_c(\Z)=1. Proved using the probability generating function f(s):=E[s^\xi] of the offspring distribution \xi that a Galton-Watson branching process tree is infinite with positive probability if and only of either P[\xi=1]=1, or \E\xi>1.
See the relevant parts of PGG Section 12.1, or the Lyons-Peres book Section 5.1.
March 6: Discussing a second method to prove the phase transition for GW trees at \E\xi=1; see Exercise 1/5. Kolmogorov's 0/1 law. Proved 1/3 \leq p_c(\Z^2,bond) \leq 2/3.
March 13: Discussed Exercises 1/8 and 9. Percolation on \Z^2 at p=1/2: self-duality. Stated the Harris-FKG inequality and the Russo-Seymour-Welsh box-crossing estimates.
March 20: Discussed Exercises 1/1, 2, and 5. The difference between the Weak and Strong Laws of Large Numbers, and between convergence in probability and almost surely, in general. The Borel-Cantelli lemmas.
March 27: The Ising model of magnetization.
Rick Durrett. Probability: theory and examples. 5th edition. Cambridge University Press, 2019. https://services.math.duke.edu/~rtd/PTE/PTE5_011119.pdf.
Rick Durrett. Random graph dynamics. Cambridge University Press, 2007. https://www.math.duke.edu/~rtd/RGD/RGD.pdf.
Geoffrey Grimmett. Probability on graphs. Cambridge University Press, 2010. http://www.statslab.cam.ac.uk/~grg/books/pgs.html.
Remco van der Hofstad. Random graphs and complex networks, Vol. I. Cambridge University Press, 2017. http://www.win.tue.nl/~rhofstad/NotesRGCN.pdf
Olle Haggstrom: Markov chains and mixing times. American Mathematical Society, 2008. http://pages.uoregon.edu/dlevin/MARKOV/.
David Levin, Yuval Peres, Elizabeth Wilmer. Markov chains and mixing times. American Mathematical Society, 2008. http://pages.uoregon.edu/dlevin/MARKOV/.
Russ Lyons and Yuval Peres. Probability on trees and networks. Cambridge University Press, 2016. http://mypage.iu.edu/%7Erdlyons/prbtree/prbtree.html
Gábor Pete. Probability and geometry on groups. Book in preparation. PGG.pdf