Random Walks

A random walk is a path formed by random steps. Despite its simplicity, it models everything from stock prices to particle diffusion.

One-Dimensional Walk

Start at 0. Flip a coin. Heads: step right (+1). Tails: step left (-1). After N steps, where are you?

The expected position is 0, but the expected distance from 0 grows like $\sqrt{N}$. That square root scaling appears everywhere in nature.

Applications

• Brownian motion (physics)
• Stock market models (finance)
• Animal foraging patterns (biology)

Randomness aggregated looks nothing like randomness at the micro scale.