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Explore the surprising connections between three fundamental concepts in statistical physics through this research lecture that investigates stretched two-dimensional random walks near impermeable boundaries. Examine how Brownian bridges behave in the vicinity of an impermeable disc using the optimal-fluctuation approach, discovering that the transverse span scales with the Kardar-Parisi-Zhang (KPZ) exponent of 1/3. Learn about the proposed connection between KPZ-like statistics and Lifshitz tails through an analogy with the one-dimensional Balagurov-Vaks trapping problem, revealing how these phenomena emerge in deterministic large-deviation landscapes. Investigate how interpreting the radial component of random walks as diffusion in a conformally invariant 1/r² potential recovers the Efimov-BKT (Berezinskii-Kosterlitz-Thouless) behavior in renormalization-group flow. Understand the large-deviation perspective that explains how typical paths responsible for BKT-like behavior belong to a sub-ensemble of stretched Brownian bridges operating in the large-deviation regime, providing a unified framework for understanding these seemingly disparate physical phenomena.
