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A random walk is a mathematical formalization of a path consisting of a succession of random steps. It's a fundamental concept in probability theory with applications ranging from physics to finance.
Try the simulation below. Watch how a particle moves randomly on a 2D grid, taking steps in one of four directions (up, down, left, right) with equal probability.
In this 2D random walk simulation, the particle takes discrete steps on a grid. At each time step, it randomly chooses one of four directions with equal probability (1/4 each):
The blue line tracks the actual distance from the origin over time, while the gray dashed line shows the expected value E[r] ≈ √t. Notice how the actual distance fluctuates around this theoretical prediction - sometimes wandering far, sometimes returning closer to home.
This shows the probability distribution of distances at the current time step. The curve represents the theoretical Rayleigh distribution, and the red line marks where your walker currently is. Most walks end up near the peak of the distribution, but outliers are always possible!
After t steps, the root-mean-square distance from the origin is √t. This sublinear growth means:
The walker explores space much more slowly than linear motion would suggest!
While individual walks are unpredictable, the ensemble behavior follows precise statistical laws. The distribution plot shows this beautifully - your single walk is just one sample from this probability distribution.
The beauty of random walks lies in their simplicity - just random steps - yet they exhibit rich mathematical properties and model countless real-world phenomena.