Markov Chains
Central Question
How can dynamics with no memory generate persistent long-term statistical structure?
Why This Matters
Markov chains organize random evolution through transition rules, connecting local dynamics to stationary distributions and the conditions for equilibrium.
Learning Prompt
Develop finite-state Markov chains from a process whose next state depends only on its current state. Let me build a transition matrix for a small example and decide whether distributions should be row or column vectors before computing. Derive stationary distributions as fixed points, then distinguish stationarity, uniqueness and convergence toward stationarity. Use reducible and periodic counterexamples to show why these are different questions. Connect a random walk on a graph with reversibility and detailed balance. Make me predict long-term behaviour from structure rather than simply raising a matrix to a large power.
Ideas
Examples
Questions
Connections
- What Is Randomness Mathematically?
- Law of Large Numbers
- Random Walks
- Statistical Mechanics Connection
- Week overview