We will introduce the basic problem of percolation theory. We will define and prove the non-triviality of the critical probability, introduce the notion of increasing events, and state the FKG and BK inequalities. If time permits we will discuss the proofs or proof sketches of these inequalities.

We will state and outline proofs of the FKG inequality, BK inequality, and Russo's formula. If time permits, we will state a result about the exponential decay of the diameter of the cluster containing the origin in the subcritical regime, and outline the main parts of the proof given in notes by Duminil-Copin.

This week, we will go over the proof of Russo's formula (Propositions 2.6 and 2.7 from the notes) and use this result to understand the rate of decay of radius of the finite component containing 0 when the probabilities of edge occurrences is below the critical threshold. We will explore potential ways to partition our system to understand how the decay occurs.

Today we will finish proving that the probability that there is a path connecting 0 to the boundary decays exponentially with the diameter. We will then move on to prove a Russo-Seymour-Welsh type uniform lower bound for the probability of horizontal crossings of rectangles, that only depends on the ratio of the sides.

We will give a modern proof of Harris’ and Kesten’s classical result that the Bernoulli bond percolation threshold in \(\mathbb Z^2\) is 1/2, before moving on to study finer properties of the infinite cluster. In particular, we will show that it is almost surely unique, which, if time permits, we will use to revisit and simplify Harris’ and Kesten’s arguments.

A central phenomenon in mathematical physics is that scaling limits of many random processes turn out to be conformally invariant. Today we will begin to discuss the notion of conformal invariance by studying the example of the scaling limit of certain crossing probabilities in the triangular/hexagonal lattice at criticality. We'll first review some historical work by Cardy and Carleson, and then we'll give a sketch of the proof of Smirnov's theorem which makes these ideas mathematically precise.

We will start by laying the groundwork for defining Schramm-Loewner evolutions, a family of conformally invariant random curves in the plane. We will mention a few of the areas in which they arise apart from critical percolation, and then review some of the complex analysis that will be needed, paying attention to illustrative examples. I will be following the notes of Jason Miller fairly closely.

We will continue covering some prerequisites from complex analysis, in particular we will prove a distortion estimate for conformal maps, which corresponds to chapter 4 of Jason Miller's notes.

We will continue by analyzing the conformal transformations on the complex half-plane that are needed for the introduction of the SLE curves. On the way we will introduce the notion of half-plane capacity, and discuss the conformal invariance of Brownian motion as an important tool.

We will introduce the chordal Loewner evolution and get an intuitive understanding (and prove the existence) of the driving function. If time permits we will define SLE using a conformal Markov property and by essentially setting the driving function to be a real Brownian motion.

We will first deduce Schramm’s theorem, which will finish the definition of SLE. Next, we will give an overview of some fundamental tools we need from stochastic calculus, including Ito's formula and the Levy characterization of Brownian motion. Time permitting, we will introduce Bessel processes and how they relate to the phases of SLE.

We will introduce the exploration process for percolation and discuss a localization property it enjoys. We will then show that the only value of \(\kappa\) such that \(SLE_\kappa\) has the analogous localization process is \(\kappa = 6\), thus showing that this is the only candidate for the scaling limit of the exploration process.