Percolation Spring 2021

1st class: definition of Bernoulli site and bond percolation processes; notation, \theta(p) and existence of phase transition in \Z^2. slides, simulation Julia code, simulation gif.

2nd class: monotone events, the Harris-FKG inequality; ergodicity and invariance of the product measure, uniqueness of the infinite cluster in \Z^d. slides.

3rd class: continuity of \theta; the Margulis-Russo formula. slides

4th class: differential inequalities, sharp thresholds and the OSSS inequality. slides, simulation Julia code, the simulations of the breadth-first search algorithm determining the cluster of the origin in site percolation with p=0.59, box radius 5, 50, and 500.

5th class: expected size of the open cluster of the origin in the subcritical phase, the susceptibility, exponential decay of the one-arm event probability in the subcritical phase part I. slides.

6th class: exponential decay of the one-arm event probability in the subcritical phase part II. Zhang's argument and critical percolation in \Z^2. slides, simulations determining the cluster intersecting the boundary of a box in site percolation: p=0.59, box radius 5, 50 and 500; p=0.55 and box radius 500.

7th class: planar bond percolation, Russo-Seymour-Welsh theory, obtaining a finite size criterion via mulstiscale renormalization. slides.

8th class: planar bond percolation, sharp thresholds for crossing probabilities. Site percolation on the triangular lattice, conformal invariance of crossing probabilities part I. slides.

9th class: site percolation on the triangular lattice, conformal invariance of crossing probabilities part II. slides.

10th class: site percolation on the triangular lattice, conformal invariance of crossing probabilities part III. slides.

11th class: the supercritical phase of percolation on \Z^d, d >= 3: exponential decay of the radius of the finite cluster; percolation on slabs - the Grimmett-Marstrand result part I. slides.

12th class: the Grimmett-Marstrand result part II. slides.

13th class: the Grimmett-Marstrand result part III. slides.

14th class: Cayley graphs and the mass transport principle. the Benjamini Lyons Peres Schramm 1999 result on absence of percolation at criticality for nonamenable Cayley graphs. slides.

The following two books are standard in a percolation course:

Percolation - Geoffrey Grimmett. Percolation, volume 321 of. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 1999

Percolation - Béla Bollobás and Oliver Riordan. Percolation. Cambridge University Press, 2006.

An introduction to Bernoulli percolation covering subjects we will study in the beginning of the course can be found here:

Introduction to Bernoulli percolation - Hugo Duminil-Copin, https://www.ihes.fr/~duminil/publi/2017percolation.pdf.

The following notes talk about differential inequalities of Boolean functions, and their role in proving sharp threshold results:

Sharp threshold phenomena in statistical physics - Hugo Duminil-Copin, https://arxiv.org/abs/1810.03384.

The book I link below covers subjects we will not have time to get into during the course, more specifically, if covers the theory of sharp thresholds for Boolean functions. It is closely related to percolation though, and a beautiful topic.

Noise sensitivity of Boolean functions and percolation - Christophe Garban and Jeffrey E. Steif, http://www.math.chalmers.se/~steif/book.pdf.

The following series of lectures goes into detail about the OSSS inequality and differential inequalities in Bernoulli percolation, as well as some dependent percolation models. There is also a presentation of the beautiful proof of Menshikov's theorem by Duminil-Copin and Tassion that does not use the OSSS inequality in the second talk, this one we didn't go over during class.

Sharp threshold phenomena in statistical physics - A series of lectures by Hugo Duminil-Copin.

https://www.youtube.com/watch?v=WqljnMFqruI

https://www.youtube.com/watch?v=I0lSoWH4GIM

https://www.youtube.com/watch?v=IMbCRVWwMCs

https://www.youtube.com/watch?v=9cwMzU5hBo8

Apart from Chapter 7 of the Bollobas-Riordan book, the following are good resources to study conformal invariance in planar percolation. Note that these notes and lectures sometimes go into SLE theory, which we will not cover in class.

Cardy’s formula on the triangular lattice, the easy way - Vincent Beffara, https://hal.archives-ouvertes.fr/hal-01693159/document

Conformal invariance of lattice models - series of lectures by Stanislav Smirnov, https://www.youtube.com/watch?v=FpmrD2CTe6o&list=PLR6S6vLGnUbxb8c_AiaWNff2u36OV2B22

Conformally Invariant Models - series of lectures by Vladas Sidoravicius and Wendelin Werner (from 2007), https://www.youtube.com/watch?v=5CvbthOqAKA&list=PLo4jXE-LdDTQNXsOgWudHCvMKd7

A detailed proof of the Grimmett-Marstrand result can be seen in chapter 7.1 of Grimmett's book. If you can log in jstor, you can read the original paper here. Another nice result involving dynamic renormalization is the one from Barsky, Grimmett and Newman, which can be seen either on chapter 7.3 of Grimmett's book, or in the original paper. It proves the absence of percolation at criticality for half-spaces.

Here is a link for the Benjamini, Lyons, Peres, Schramm result about absence of percolation at criticality for nonamenable Cayley graphs.

For students actually enrolled in the course, there is the need for a grading system. I opted for a system in which students give an (online) 1 hour lecture via zoom on a pre-selected topic, in the manner they think best. I am going to list below some topics, and you can email me about which topic you want to cover. Please give me an ordered list of the topics you would prefer to cover. I will give preference to people who contact me first. The lectures should take place in May after our classes are over. I'm available to help you with any difficulties that may appear.

The topics:

Overview of other percolation models

Random cluster model. Overviews can be found on The random cluster model by Grimmett, and Lectures on the Ising and Potts models on the hypercubic lattice by Duminil-Copin. It is a dependent percolation model with deep connections to the Potts model in statistical physics. I'm looking for an overview of the process, definitions, basic properties, connections to statistical mechanics. If more than one student is interested in this topic, we could go further into the recent proof of sharpness of the phase transition by Duminil-Copin, Raoufi and Tassion.

Continuum percolation models, Chapter 8 of the Bollobás-Riordan book. There one can see an overview of two continuum percolation models. Different people can choose as subjects one of the following two models: Gilbert disc model and Voronoi percolation, respectively sections 8.1 and 8.3.

Proofs of important results

Asymptotic tail behavior of the radius of a finite open cluster, Chayes, Chayes, Grimmett, Kesten, Schonmann. Can be found on section 8.4 of Grimmett's book second edition. The paper provides a subadditivity argument to show that the exponent of the one-arm decay of the finite cluster actually exists. I specifically want to see the proof of Theorem 8.18 there present.

Differentiability of $\theta$ and $\chi$, Russo http://math.bme.hu/~balint/oktatas/perkolacio/percolation_papers/russo_1.pdf. I learned it from section 8.7 of Grimmett's book second edition. We showed that $\theta$ was continuous outside of the critical point, the truth is that it is actually analytic (this was recently proved by A. Georgakopoulos & C. Panagiotis). The proof that it is infinitely differentiable is much simpler.

Percolation in ∞ + 1 Dimensions, Grimmett and Newman http://www.statslab.cam.ac.uk/~grg/books/hammfest/10-grg.pdf: The authors show the existence of different phases inside the supercritical phase of percolation in a specific one-ended non-amenable infinite graph: there are parameters for which the infinite cluster is unique, and others for which there are infinitely many clusters. This might be a little bit heavier than the other topics, so two students could collaborate to give two talks about this.