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This lecture, part 6 of a series by Thomas Hutchcroft from the California Institute of Technology, explores the dimension dependence of critical phenomena in percolation theory. Delve into the complex geometry of Bernoulli bond percolation, where edges in a graph are randomly deleted or retained with a specific probability parameter. Learn how percolation exhibits phase transitions in lattices of dimension d>1, with infinite clusters emerging at 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 in percolation theory.
