A phase transition with no energy and no temperature: bond and site percolation on a lattice, the critical probability, exact solution on a tree, exponential decay below $p_c$ and uniqueness of the infinite cluster above it, planar duality and Kesten's theorem that $p_c = 1/2$ on the square lattice, cluster-size distributions, critical exponents and finite-size scaling, the fractal critical cluster, the mean-field limit and its connection to random graphs, first-passage and invasion percolation, random resistor networks, directed percolation, continuum percolation, and the conformal invariance behind Cardy's formula and SLE.
Free to start · adaptive placement finds your level · reviews timed to your own forgetting.
Every idea is taught with motivation and a worked example before the drills, and an FSRS spaced-repetition engine schedules each review for the day its model predicts you would forget it. A short placement check finds what you already know, so you start Percolation exactly where it's useful.