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This lecture explores the dimension dependence of critical phenomena in Bernoulli bond percolation, focusing on how connected components behave when edges in a graph are randomly deleted or retained. 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. Examine the qualitative distinctions between low-dimensional cases (d6), and the critical case (d=6), with special attention to the poorly understood dimensions d=3,4,5,6. Gain insights into recent advances in long-range and hierarchical models that have enabled rigorous understanding of intermediate-dimensional critical phenomena. Presented by Thomas Hutchcroft from the California Institute of Technology at the Institut des Hautes Etudes Scientifiques (IHES), this comprehensive exploration of percolation theory runs for nearly two hours.
