Where Do You End Up After Many Random Steps? Further Than You Think and Not Far
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Taking steps in random directions produces a path that drifts outward at a rate nobody guesses correctly. The same mathematics describes diffusing molecules, spreading heat, share prices and the shape of a polymer.
How far you get
Consider steps of equal length taken in random directions, with each step independent of the last. The average position after many steps is the starting point, since the steps cancel on average, which is correct and unhelpful because it says nothing about how far the walker has wandered. The useful quantity is the typical distance from the start regardless of direction, and that grows as the square root of the number of steps rather than in proportion to it. A walker taking a hundred steps is typically about ten steps away, and taking ten thousand steps puts them about a hundred away. The square root is the single most important fact about the subject and it explains why diffusion is fast over short distances and impossibly slow over long ones.
Where the same mathematics applies
A remarkable range of processes are described by the same equations:
- •Molecules diffusing through a gas or liquid, which is the original physical instance
- •Heat spreading through a solid, which follows the same form
- •The visible jiggling of small particles suspended in fluid, explained by Einstein in 1905
- •The shape of a long flexible polymer molecule, where each link points randomly relative to the last
- •Share prices under the standard financial model, where returns rather than prices take the steps
- •Search and foraging behaviour in animals, and algorithms that sample large spaces
The dimension that decides everything
A striking result concerns whether a wanderer returns to the start. In one dimension, moving left or right at random, return is certain given enough time, and the same holds in two dimensions on a grid. In three dimensions it is not, with the probability of ever returning being roughly a third. The result was established by Polya in 1921 and is summarised in a well-known remark that a drunk man will find his way home while a drunk bird may not. The reason is that the space available grows faster with distance in higher dimensions, so the walker has more room to get lost in. The consequence matters physically, since it determines whether a diffusing particle will eventually find a target, which bears on chemical reaction rates and on how molecules find binding sites.
The particle that proved atoms
The physical application that mattered most historically concerns the visible jiggling of pollen grains and dust suspended in water, observed under a microscope and unexplained for most of a century. Einstein showed in 1905 that if matter is made of molecules in constant motion, a suspended particle is struck unevenly from all sides and performs exactly such a walk, and he derived a quantitative prediction relating the spreading to temperature, viscosity, particle size and the number of molecules in a given amount of substance. Perrin measured it carefully over the following years, confirmed the prediction and extracted a value for that number, which was among the most direct demonstrations that molecules exist and are not merely a useful bookkeeping device. The work was recognised with a Nobel Prize in 1926 and effectively ended serious scientific doubt about atoms.
Where the simple model fails
Real processes depart from the assumptions in ways that matter. Steps are frequently not independent, since a moving particle or a price has momentum or memory, which produces correlations and changes how the distance grows. Step lengths may follow a heavy-tailed distribution, producing occasional enormous jumps and a walk that spreads far faster than the square root, which describes some animal foraging and some financial movements better than the standard model. Boundaries and obstacles change everything, since a walk confined to a region behaves quite differently from a free one. And the financial application is the most contested, since the standard model assumes independent normally distributed steps and actual markets show both correlation and heavy tails, which is why models built on the simple version underestimate extreme moves.
The takeaway
Typical distance from the start grows as the square root of the number of steps, so ten thousand steps leaves a walker about a hundred away, which is why diffusion is quick over short distances and hopeless over long ones. Return to the start is certain in one and two dimensions and only about one chance in three in three. Correlated steps and heavy-tailed jumps both break the simple model.