Modern aspects of Markov chains: entropy, curvature and the cutoff phenomenon

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Markov Chains and the Mysterious Cutoff Phenomenon

This lecture note explores the "cutoff phenomenon" in Markov chains, where some processes abruptly transition to equilibrium instead of gradually converging. It introduces key concepts like mixing times, curvature, and varentropy, applying them to various models like card shuffling and random walks. The author presents a novel criterion based on varentropy for predicting cutoff behavior.

Possible Conflicts of Interest

The author acknowledges funding from an ERC Consolidator Grant, but no other potential conflicts of interest were readily apparent.

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Highly technical and specialized content

This lecture note is intended for a specialized audience familiar with advanced probability theory and stochastic processes. Its technical nature and reliance on specialized mathematical background limit accessibility for broader audiences.

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This lecture note offers a high-quality and detailed exposition of the cutoff phenomenon in Markov chains. It covers important models, introduces advanced concepts like curvature and functional inequalities and clearly illustrates theoretical results with examples. While highly technical, it excels in clarity and organization within its target audience.

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