Dimension Dependence of Critical Phenomena in Percolation - 2/6

This lecture, part 2 of 6 in a series on dimension dependence of critical phenomena in percolation, features Thomas Hutchcroft from the California Institute of Technology presenting at the Institut des Hautes Etudes Scientifiques (IHES). Explore Bernoulli bond percolation, where edges in a graph are randomly deleted or retained, and discover how the geometry of connected components changes across different dimensions. Learn about the phase transition that occurs in lattices of dimension d>1, where an infinite cluster emerges at a critical probability pc(d). Understand the rich, fractal-like geometry of critical percolation that depends heavily on dimension but less on lattice choice, with particular focus on the qualitative distinctions between low dimensions (d6), and the critical case (d=6). Gain insights into recent advances in long-range and hierarchical models that have enabled rigorous understanding of intermediate-dimensional critical phenomena during this 1 hour and 51 minute presentation.