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This two-hour lecture by Thomas Hutchcroft from the California Institute of Technology explores the dimension dependence of critical phenomena in percolation theory. Dive into Bernoulli bond percolation, where edges in a graph are randomly deleted or retained, and examine how connected components form with different retention parameters. Learn about the phase transition that occurs in lattices of dimension d>1, where an infinite cluster emerges at a critical probability pc(d). Discover the rich, fractal-like geometry of critical percolation that depends heavily on dimension but remains largely independent of lattice choice. The lecture particularly focuses on the qualitative distinctions between low-dimensional cases (d6), and the critical case (d=6), highlighting recent advances in long-range and hierarchical models that have enabled rigorous understanding of intermediate-dimensional critical phenomena. This content is available on CARMIN.tv, a French video platform specializing in mathematics and interdisciplinary scientific content.
