Random Walks
Central Question
What structure appears when simple independent choices are accumulated into a path?
Why This Matters
Random walks link combinatorics and probability with recurrence, hitting times and the beginnings of diffusion.
Learning Prompt
Build a simple symmetric random walk on from independent steps. Let me derive its mean, variance and reachable positions, then ask about hitting a level, returning to the origin and escaping forever. Connect path counts with binomial coefficients and use a small finite-interval experiment to investigate ruin probabilities. Make me predict how displacement grows with time and compare one-dimensional behaviour with higher dimensions. Explain which questions concern a single time and which concern an entire path, preparing the scaling-limit story rather than jumping immediately to Brownian motion.
Ideas
Examples
Questions
Connections
- Central Limit Theorem
- Combinatorics Behind the Gaussian
- From Random Walk to Brownian Motion
- Markov Chains
- Week overview