You can check out a new lecture recording on Spectral Collective YouTube channel on Percolation: a Mathematical Phase Transition:

**Percolation: a Mathematical Phase Transition**

Caio Alves – writing, 3D animation

Aranka Hrušková – writing, clarinet

Vilas Winstein – writing, 2D animation, editing, voice-over

Special thanks to Anisah Awad, Gábor Pete, Jyotsna Sreenivasan, Angie Zavala

Note at 10:42 – The uniqueness of the infinite cluster is known for the d-dimenional lattice since the works of Aizenman, Kesten and Newman - [Uniqueness of the infinite cluster and continuity of connectivity functions for short and long range percolation (1987)] and Burton and Keane - [Density and uniqueness in percolation (1989)]. It does not hold in general: when the graph in question is a regular tree for example, there are always infinitely many clusters during the supercritical phase.

The two last results shown here are only known for site percolation (in which vertices are open or closed instead of edges) in the triangular lattice, where a scaling limit for the boundaries of critical clusters was proved to exist (more on that in the third note). It is believed that these results are universal, that is, valid in great generality for planar percolation processes near criticality.

The third result is from an appendix by Gabor Pete in the paper [Scaling limits for the threshold window: When does a monotone Boolean function flip its outcome? (2017)] by Ahlberg and Steif. Consider an n by n box, and the event where there exists a left-right crossing of said box. Recall the uniform coupling from the video: intuitively, the result is saying that the point at which this crossing emerges in the uniform coupling is with high probability inside an interval of size n^{-3/4} around 1/2.

The fourth result is saying that the average size of the cluster of the origin (or any other given point) goes to infinity as we let p approach the critical parameter like a specific power of the distance between p and p_c. This power is called a critical exponent. The existence of these exponents was proved by Smirnov and Werner in the paper [Critical exponents for two-dimensional percolation (2001)].

Note at 10:52 – Hugo Duminil-Copin has several major contributions to the study of processes arising in statistical physics, including Bernoulli percolation. Among his works on Ising and Ising-like processes we can cite [Random Currents and Continuity of Ising Model’s Spontaneous Magnetization (2015)] with Aizenman and Sidoravicius and [Sharp phase transition for the random-cluster and Potts models via decision trees (2019)] with Raoufi and Tassion.

Note at 12:38 – In the triangular lattice site percolation, Stanislav Smirnov proved the conformal invariance of crossing probabilities at criticality (see https://www.unige.ch/~smirnov/papers/... for an overview), which led to the proof of the existence of scaling limits of exploration curves as Schramm–Loewner evolution processes. See [Critical percolation in the plane (2009)] by Smirnov. This provided a deep understanding of the critical phase in the triangular lattice site percolation, which to this day is not extended to the square lattice.

Note at 17:52 – It is not at all obvious that the probability of being connected to infinity is continuous above criticality. This result can be proved in the d-dimenional hypercubic lattices using the uniqueness of the infinite cluster, and more generally it was proved for transitive graphs (intuitively, graphs in which all vertices look the same) by Häggström, Peres and Schonmann in [Percolation on transitive graphs as a coalescent process: Relentless merging followed by simultaneous uniqueness (1999)].