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This lecture explores the fascinating world of Last Passage Percolation (LPP), a model of random geometry focusing on directed paths in random environments. Discover how the Kardar-Parisi-Zhang (KPZ) universality theory explains random fluctuations in LPP when environment distributions have light tails and fast correlation decay, but fails in critical environments with hierarchical, fractal-like structures. Learn about groundbreaking research by Ganguly, Ginsburg, and Nam that developed a multi-scale analysis framework for studying LPP in hierarchical environments, establishing bounds on critical exponents for two canonical examples: i.i.d. environments with critical power-law tails and hierarchical approximations of two-dimensional Gaussian Free Fields. The 58-minute presentation also touches on connections to fractal percolation and related polymer models from ongoing research with Ginsburg and Jing.
