Localization Schemes - A Framework for Proving Mixing Bounds for Markov Chains

Explore a comprehensive seminar on localization schemes and their application in proving mixing bounds for Markov chains. Delve into the connection between Spectral Independence and Stochastic Localization techniques, unifying and extending these approaches for both discrete and continuous settings. Examine the concept of "localization schemes" and their corresponding Markov chains, learning how to derive mixing bounds through analysis of the localization process. Discover generalizations of Spectral Independence and Entropic Independence, and understand how to recover main theorems using simple martingale arguments. Apply the presented machinery to obtain simple proofs for various mixing bounds, including the first O(n log n) bound for mixing time of the hardcore model in the tree-uniqueness regime and KL-divergence decay bound for log-concave sampling. Gain insights into topics such as global dynamics, grabber dynamics, nonuniqueness, discrete hypercube, sampling independent sets, covariance and correlation matrices, and operator norms.