Disordered systems such as directed polymers and spin glasses are ubiquitous in statistical mechanics forming a testbed to study the effects of extra randomness on well understood models. Directed polymers, which are random distortions of the simple random walk, form one of the simplest---yet most challenging---examples in this class. Since their introduction in the seminal work of Huse and Henley in 1985, directed polymers have been central in the understanding of protein sequences, random growing interfaces and stochastic PDEs. When the underlying disorder has light tails and a fast decay of correlation, the random fluctuations of polymers are predicted to be explained by the Kardar–Parisi–Zhang (KPZ) universality theory. However, the KPZ theory is not expected to apply in many natural settings, such as "critical" environments exhibiting a hierarchical, fractal-like structure which should give rise to a fluctuation theory featuring logarithmic corrections with novel critical exponents. Predictions for these exponents are missing, even from the physics literature. In this talk we will survey some recent developments in the mathematical study of polymers in such critical settings involving multiscale analysis, exploring connections to fractal percolation and Gaussian multiplicative chaos.
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