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Explore the mathematical connections between deterministic dynamical systems and random walks through this 51-minute conference talk examining the tubular and planar Lorentz gas models. Discover how these deterministic systems serve as analogs to random walks on Z^d for dimensions d=1 and d=2, where the displacement function exhibits statistical properties identical to random walks with either finite or infinite variance depending on the horizon type. Learn about the fascinating case where infinite variance scenarios feature displacement functions whose first moments "barely" fail to achieve L^2 integrability, yet still produce Gaussian asymptotic limit laws under appropriate scaling. Examine the stochastic properties of displacement functions in both finite and infinite variance regimes, and investigate theoretical possibilities for breaking away from Gaussian behavior. Gain insights into advanced topics in dynamical systems theory, probability theory, and mathematical physics through rigorous mathematical analysis presented at the Centre International de Rencontres Mathématiques during their thematic meeting on random walks applications and interactions.
