The Drunkard's Walk
The random walk — sometimes called the drunkard's walk — is one of the most fundamental models in probability theory. Imagine a person taking steps in random directions: after many steps, how far are they from where they started? Surprisingly, the answer is not proportional to the number of steps but to its square root. Ten thousand steps take you only 100 step-lengths from the origin on average.
Einstein and the Atom
In 1905, Albert Einstein published his theory of Brownian motion, explaining the erratic jittering of pollen grains in water as the result of random molecular bombardment. His prediction that displacement scales as sqrt(t) was confirmed experimentally by Jean Perrin, providing the first definitive evidence for the existence of atoms and earning Perrin the Nobel Prize.
Diffusion: Random Walks at Scale
The macroscopic consequence of molecular random walks is diffusion — the process by which ink spreads in water, heat conducts through metal, and scents travel through air. The diffusion equation, a partial differential equation derived from random walk statistics, is one of the fundamental equations of physics. Its solutions are Gaussian distributions that spread with sqrt(t).
From Physics to Finance
In 1900, five years before Einstein, Louis Bachelier modeled stock prices as random walks in his doctoral thesis. This insight — that price changes are fundamentally unpredictable — became the foundation of modern quantitative finance. The Black-Scholes option pricing model, Markowitz portfolio theory, and the efficient market hypothesis all rest on random walk assumptions.