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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 2-hour lecture provides a valuable overview of current knowledge in this fascinating area of mathematical physics.
