**Advanced Topics in Probability - Percolation (0366-4926-01)**

Spring 2013, Tel Aviv University

Location: Dan David 204, Tuesdays 10-13

Instructor: Ron Peled

The course is on Percolation Theory, with a focus on percolation on Euclidean lattices such as Z^d.

Required prerequisites: The courses Probability for Mathematicians or Probability for Sciences or Advanced Probability Theory.

Recommended prerequisite: Functions of a Real Variable.

Tasks: There will be several homework assignments. There may be a final exam according to the number of participants.

Syllabus:
Galton-Watson trees and percolation on trees.

Basic facts on percolation on Z^d: Existence of a non-trivial critical probability. FKG, BK and Reimer inequalities. Russo's formula. Lower bounds on the connected component size in critical percolation. Uniqueness of the infinite cluster in super-critical percolation.

Comparison of the critical probabilities of similar percolation models. A lower bound on the percolation probability of barely super-critical models.

The sub-critical phase of percolation: Exponential decay of the cluster-size distribution - Aizenman-Barsky / Menshikov theorem.

Percolation in two dimensions: Use of duality techniques. A proof that the critical probability for bond percolation on Z^2 is 1/2. Cardy's formula and conformal invariance.

The super-critical phase of percolation in 3-dimensions and more: Properties of the percolation when p is close to 1. The Liggett-Schonmann-Stacey theorem on domination of a dependent percolation by an independent one. Slab percolation - the Grimmett-Marstrand theorem.

Further possible topics as time permits: Percolation on groups. Percolation in very high dimensions. Introduction to SLE.

Main relevant books: Grimmett / Percolation. Grimmett / Probability on graphs. Bollobás and Riordan / Percolation. Ariel Yadin's lecture notes. Hugo Duminil-Copin's lecture notes (in French). Lyons with Peres / Probability on trees and networks. Some of Gábor Pete's lecture notes may also be relevant.

__Announcements__

There will be no class on May 7 since I will be away. A makeup class is scheduled on Friday, May 17, from 10:10 to 13:00 in Schreiber 008.

__Requirements__

Several mandatory homework assignments will be given during the course.

Exercise 1. The exercise needs to be handed in by
March 5'th in class.

Exercise 2. The exercise needs to be handed in by
March 12'th in class.

Exercise 3. The exercise needs to be handed in by
March 19'th in class.

Exercise 4. The exercise needs to be handed in by
April 9'th in class.

Exercise 5. The exercise needs to be handed in by
April 23'rd in class.

Exercise 6. The exercise needs to be handed in by
April 30'th in class.

Exercise 7. The exercise needs to be handed in by
May 21'st in class.

Exercise 8. The exercise needs to be handed in by
May 28'st in class.

Exercise 9. The exercise needs to be handed in by
June 11'th in class.

Exercise 10. The exercise needs to be handed in by
June 18'th in class.

__Lectures so far__

__Lecture 1 (26.2)__: Introduction, Galton-Watson trees (survival has positive probability if and only if mean offspring number is at least 1). See the book of Harris on Branching processes, or David Williaws' Probability with Martingales, or, for a more advanced treatment, the book of Lyons with Peres.

__Lecture 2 (5.3)__: Continuation of Galton-Watson trees under assumption of finite variance for offspring distribution. Probability to survive n generations in sub-critical and critical cases. Growth of the tree in super-critical case. Yaglom limit law for number of offsprings in generation n in critical case. Basic percolation on Z^d: Peierls argument to show p_c(d)>0 for all d>1.

Material drawn from Harris' book on branching processes, paper of Kesten-Ney-Spitzer, the book of Lyons with Peres and Grimmett's percolation book.

__Lecture 3 (12.3, 1.5 hour class due to strike)__: Proof that p_c(d)<1 for all d≥2. Probability of increasing events increases with the percolation parameter. Beginning of the proof of Harris inequality (FKG for percolation).

Material based on Grimmett's percolation book.

__Lecture 4 (19.3)__: Description of general FKG inequality. Statement and proof of the Van den Berg-Kesten inequality. Description of Reimer's inequality. Application of the BK inequality show that the mean cluster size at p_c is infinite. Application to show positive probability at p_c of "easy-way" crossing of boxes in Z^d.

Material based on Grimmett's percolation and probability on graphs books and discussions with Gady Kozma.

__Lecture 5 (9.4)__: Comparison of p_c for the square and triangular lattices using the Aizenman-Grimmett method. Hammersley's theorem on the exponential decay of the radius distribution when the expected cluster size is finite. Beginning of the proof of Aizenman-Barsky to the Aizenman-Barsky / Menshikov theorem.

Material based on Grimmett's percolation and probability on graphs book.

__Lecture 6 (23.4)__: The Aizenman-Barsky proof of the Menshikov / Aizenman-Barsky theorem that the expected cluster size is finite when p<p_c.

Material based on Grimmett's percolation book.

__Lecture 7 (30.4)__: Exponential decay of tail of cluster size. Supercritical phase: uniqueness of the infinite cluster. Right continuity of theta(p) on [0,1].

Material based on Grimmett's percolation book.

__Lecture 8 (17.5)__: The percolation critical point for Z^2 is 1/2 - Zhang's argument and use of the Menshikov / Aizenman-Barsky theorem. Russo-Seymour-Welsh for the triangular lattice. Introduction to conformal invariance.

Material based on Grimmett's probability on graphs book.

__Lecture 9 (21.5 - only two hours)__: Cardy's formula and its history. Beginning of the proof of Smirnov's theorem.

Material based on Grimmett's probability on graphs book and the book by Bollobás and Riordan.

__Lecture 10 (28.5)__: End of proof of Cardy-Smirnov theorem. Introduction to SLE, arm exponents and scaling relations.

Material based on Grimmett's probability on graphs book and Hugo Duminil-Copin's lecture notes.

__Lecture 11 (4.6)__: Super-critical percolation in dimensions 3 and higher: Some statements without proof - p_c(half space) = p_c(Z^d), P(0<->distance n, but not to infinity) decays exponentially, P(n≤|C_0|<infinity) decays exponentially in n^((d-1)/d). Proof of statements for large enough p. Definition of slab critical point. Use of it to start proving the statements for p>p_c.

Material based on Grimmett's percolation book and discussions with Gady Kozma.

__Lecture 12 (11.6)__: Super-critical percolation in dimensions 3 and higher: Proof that P(n≤|C_0|<infinity) decays exponentially in n^((d-1)/d) for all p>p_c(Z^d) using static renormalization results. Static renormalization statement and proof. Introduction to the Grimmett-Marstrand theorem and its proof.

Material based on Grimmett's percolation book.

__Lecture 13 (18.6)__: (Most of the) proof of the Grimmett-Marstrand theorem. Survey of some related topics not treated in our course: Percolation on Cayley graphs of groups, percolation on finite graphs (e.g., the hypercube or the complete graph), long-range percolation on Z, the triangle condition and its uses in high-dimensional percolation, the random cluster model.

Material based on Grimmett's percolation book and discussions with Gady Kozma.