Markov Chains: Convergence and Spectral Gap

This lecture delves into the convergence properties of Markov chains, exploring the concept of spectral gap and how it affects the speed of convergence towards the stationary distribution. The instructor explains the importance of ergodicity, periodicity, and irreducibility in determining the convergence behavior of Markov chains, using examples to illustrate different scenarios where convergence may be fast or slow. Additionally, the concept of lazy chains, where transitions are occasionally skipped, is introduced as a method to potentially accelerate convergence by modifying the chain's structure. The lecture concludes by discussing how spectral gap analysis can provide insights into the convergence rate of Markov chains, highlighting the trade-offs between different chain configurations and their impact on convergence speed.