Walking Wild I

Though physicists began looking at random processes involving large numbers of molecules near the start of the twentieth century (see "Quivering Particles II"), it wasn't until 1920 that mathematicians began to develop a convincing mathematical model of Brownian motion, starting with the work of Norbert Wiener (1894-1964).

At the heart of the mathematics was the difficult problem of making precise mathematical sense of the notion of a particle moving at random.

A Brownian particle suspended in a liquid knows neither when the next shove will occur nor in which direction and how forcefully it will be propelled. Its displacement at any given moment is independent of its past history. These characteristics put Brownian motion in the category of a Markov process, named for mathematician Andrey A. Markov (see "Climbing and Sliding").

One of the simplest examples of such a process is a one-dimensional random walk, in which a "walker" is confined to a long, narrow path and moves forward or backward according to the results of repeatedly tossing a coin. The walker takes a step in one direction if the outcome is heads and in the opposite direction if the outcome is tails.

This graph shows the results of a one-dimensional random walk. The horizontal axis represents the number of steps taken, and the vertical axis shows how many steps you are away from your starting point if you start of 0. Steps in the forward direction are positive (upward) and steps backward are negative (downward).

For a walk along an infinite track, you can calculate a walker's long-term behavior. The resulting trail wanders back and forth along the track, and the probability of the wanderer's being a certain distance away from the starting point after taking a given number of steps is defined by a bell-shaped curve known as a Gaussian (or normal) distribution. The larger the number of steps, the wider would be the curve.

Indeed, the expected average distance of the walker from the starting point after a certain number of equal steps is simply the length of the step times the square root of the number of steps. For infinitely many coin tosses, a random walk confined to a line corresponds to one-dimensional Brownian motion.

One consequence of this type of erratic movement back and forth along a line is that a random walker is certain to return to the origin (or to any particular position on the track)—eventually.

This might sound like a good strategy in the admittedly unlikely situation of someone who's lost on a tightrope: Just take steps at random and you'll end up anywhere you want to be. But it might take longer than a lifetime to get there.

It's straightforward to extend the random-walk model to two dimensions. Take steps to the east, west, north, or south, randomly choosing each direction with equal probability (perhaps by using a tetrahedral die). You can imagine this walk going from vertex to vertex on an infinite checkerboard lattice.

If such a walk continues for an arbitrarily long time, the walker is certain to touch every vertex, including a return visit to its starting point.

In this illustration of a random walk in two dimensions, a walker starts off from a point on the left-hand side of the checkerboard grid, taking random steps to the east, north, west, or south.

The fact that returning to the origin is guaranteed in one and two dimensions suggests that there will be infinitely many returns. Once a walk gets back to the origin, it's like starting from scratch, and there will be a second return, then a third, and so on.

Similarly, such a random walk will visit every point infinitely many times.

Things are a little different in three dimensions. A walker can go up and down as well as in each of the four compass directions, so a standard cubic die serves as a suitable randomizer to determine the movements.

This time, however, even if a walker takes infinitely many steps, the probability of returning to the origin is only about .34. There's so much room available in three dimensions that it becomes considerably harder for a walker to find the way back to the starting point by chance.

In this computer-generated representation of a random walk in three-dimensional space, a walker has an equal probability of moving forward, backward, right, left, up, or down. This particular walk goes for 2,100 steps, beginning with blue, then continuing with magenta, red, yellow, green, cyan, and white segments. Extended indefinitely, the walk has only a 34 percent chance of ever returning to its starting point. G.M. Viswanathan

Indeed, this mathematical result affords an important lesson for anyone who is lost in space. Unless you make it home again within your first few steps, you're likely to end up lost forever (see "Wandering in Space"). No amount of aimless wandering will get you back after such a start. There are simply too many ways to go wrong.

Next: Walking Wild II