Percolation by Spectral Collective wins SoME2
After last year’s inaugural success of Summer of Math Exposition (SoME), a contest organised by one of the biggest mathematical YouTube channels 3Blue1Brown, SoME2 was held this year, attracting several hundred entries. The submission of a trio of our own employees, who collaborate under the name Spectral Collective, was chosen among the five winners in the broadly defined competition that accepts “videos, blogs, video games, interpretive dances or anything else”, as long as it is about math. The winning video that 3Blue1Brown described as masterfully done and praised for its level of clarity explains the phase transition that occurs in Bernoulli bond percolation on the square grid: if we fix a parameter p between 0 and 1 and keep every edge of the grid independently of the other edges with probability p, erasing it with probability 1-p, then there is a critical value p_c such that for p>p_c, the percolated grid contains an infinite connected component, while for p
Apart from rigorous proofs, the viewer can enjoy many visualisations of the uniform coupling which show how with increasing p, the connected components, also known as clusters, grow and merge.
Spectral Collective was founded in 2021 by Caio Alves, a postdoc in the Noise Sensitivity group, Aranka Hrušková, a Young Researcher currently visiting Simons Institute for the semester, and Vilas Winstein, who was a Fulbright scholar in Budapest and started his PhD at UC Berkeley in August.
Caio Alves, Aranka Hrušková and Vilas Winstein - from our institute - made a video introduction to the mathematical concept of Benjamini-Schramm convergence, which is a type of graph limit theory which works well for sparse graphs. We hope that most of it is understandable by a wide audience with some mathematical background (including some prior exposure to graph theory), but to get the most out of the video it is helpful to know some probability theory and general topology.
They entered it in a competition run by 3Blue1Brown, and they ended up being selected as one of the top 30 videos, out of around 1300.
They are planning on making some more videos; in fact they’re working on some now.
In the second video, they compute the Benjamini-Schramm graph limit of the sequence of complete finite binary trees. If you don't already know what Benjamini-Schramm convergence is, please watch the previous video (top), which gives an introduction to the concept.