Animated 1D and 2D random walks converging to Brownian motion
Take a step left or right by coin flip, again and again. The path that results has surprising structure, and in the limit it becomes Brownian motion.
Run many walkers stepping +1 or -1 and watch the histogram of final positions. It centers on zero but spreads out like sqrt(n), forming a bell curve.
Each walker independently steps +1 or -1 with equal probability at each time step. The histogram on the right shows the distribution of final positions.
A random walk is a path formed by successive random steps. Despite being built from pure randomness, random walks exhibit deep mathematical structure -- from the square-root scaling of displacement to their convergence to Brownian motion in the continuous limit. They appear throughout physics (diffusion, polymer chains), finance (stock price models), and computer science (randomized algorithms).
In this section, you will explore 1D and 2D random walks, observe how the distribution of positions evolves over time, and see how discrete random walks give rise to continuous Brownian motion as the step size shrinks to zero.
The simplest random walk: at each time step, a particle moves +1 or -1 with equal probability. Run multiple walkers simultaneously and watch the histogram of final positions take shape. After n steps, the expected displacement is zero but the typical distance from the origin grows as sqrt(n).
Key insight: Although the expected position E[X_n] = 0 at every step, the variance grows linearly: Var(X_n) = n. This means the standard deviation is sqrt(n), so walkers drift further from the origin over time despite having no bias. The histogram of final positions approaches a normal distribution by the Central Limit Theorem.
Watch walkers step up, down, left, or right and trace their trails. The dashed circle grows like sqrt(n), and the trails keep doubling back over old ground.
Each walker steps in one of four cardinal directions with equal probability. The dashed amber circle shows the root-mean-square distance from the origin, sqrt(E[r²]) = sqrt(n). (The mean distance E[r] is slightly smaller, about 0.886·sqrt(n).)
Extend the walk to two dimensions: at each step the particle moves up, down, left, or right with equal probability. The trails create intricate fractal-like patterns. A remarkable result by George Polya (1921) states that a 2D random walker will return to the origin with probability 1 -- but in 3D and higher, this is no longer true.
Key insight: The expected distance from the origin after n steps scales as sqrt(n), shown by the dashed circle. The 2D walk is recurrent (returns to the origin infinitely often), which is why the trails appear to revisit areas repeatedly. This property makes 2D random walks fundamentally different from their higher-dimensional counterparts.
Drag the step size slider from large to tiny. Big steps make a jagged staircase; tiny ones make a path that looks smooth but is rough at every scale.
With large step sizes, the walk is visibly discrete. Shrink the step size and the path becomes smoother, converging to a continuous Brownian motion (Wiener process). The dashed lines show the +/- sqrt(t) standard deviation envelope.
Brownian motion (the Wiener process) is the continuous-time limit of a random walk. As the step size shrinks and the number of steps grows, the discrete path converges to a continuous but nowhere-differentiable curve. Adjust the step size slider to see this transition: large steps produce a jagged staircase, while tiny steps yield a smooth-looking path that is actually infinitely rough at every scale.
Key insight: The Donsker invariance principle (functional CLT) states that a properly scaled random walk converges in distribution to Brownian motion. The key scaling: if each step has size delta and takes time delta squared, then in the limit as delta approaches 0, we get a Wiener process with variance equal to t at time t. The dashed envelope shows the +/- sqrt(t) standard deviation boundary.
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