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Explore the mathematical foundations of polymer models in critical environments through this 49-minute conference talk delivered at IPAM's New Interactions Between Probability and Geometry Workshop. Delve into disordered systems such as directed polymers and spin glasses, which serve as crucial testbeds for understanding how additional randomness affects well-established statistical mechanics models. Examine directed polymers as random distortions of simple random walks, representing one of the simplest yet most challenging examples in this field since their introduction by Huse and Henley in 1985. Discover the central role these models play in understanding protein sequences, random growing interfaces, and stochastic PDEs. Learn about the Kardar-Parisi-Zhang (KPZ) universality theory's predictions for random fluctuations in polymers when underlying disorder has light tails and fast correlation decay. Investigate critical environments that exhibit hierarchical, fractal-like structures where KPZ theory doesn't apply, leading to fluctuation theories with logarithmic corrections and novel critical exponents. Survey recent mathematical developments in studying polymers within critical settings through multiscale analysis, while exploring connections to fractal percolation and Gaussian multiplicative chaos in areas where even physics literature lacks predictions for critical exponents.
