31.1. Introduction http://www.ck12.org
Answer
Two dimensional walks can be used to model the spread of insects — mosquitoes, for instance — over an
area, while three-dimensional random walks accurately describe the behavior of gas atoms under a wide range
of conditions.Finally, let’s try to answer the last question: what exactly are we trying to find here? Remember, the usefulness of
any model is measured by its applications: if you model the weather, hopefully you will be able to predict it using
your model.
Let’s look at the applications we listed above, and think of relevant questions. In terms of stock prices, one might
ask: "what is the probability that a given stock will bexdollars above or below its starting point after a given period
of time? In the random walk model, this translates to: What is the probability that a walker will be a distanceXfrom
the origin after a given number of steps?
In terms of gas molecules, we might ask: if you break a beaker of some gas, how quickly will the gas spread through
a room or area? In random walk terms, this becomes: How quickly does the walker tend to move away from the
origin. This is a model of diffusion.
Finally, in our insect model, we might ask: how long will it take, on average, for some infectious insects to reach an
area some distance away from their starting point? In our model, this becomes: How long will it take, on average,
for the walker to reach some distance from the origin?
To recap, some important questions we might try to answer about the random walk model are:
- What is the probability that a walker will be a distanceDfrom the origin after a given number of steps?
- How quickly does the walker tend to move away from the origin?
- How long will it take, on average, for the walker to reach some distance from the origin?
Note: The underlying idea behind all these questions can be summarized in the following manner:What is the net
effect, on average, of the canceling out of steps in opposite directions in a random walk?Understanding this
will not only help us with the mathematics that follows, but is also key to generalizing the model to two and three
dimensions.
We will only consider the first question theoretically, but the other two can be explored using the computational
models of the next chapter.
What are we looking for?
Let’s look at the first of the questions above in more mathematical detail — we would like to find is the probability
that afterNflips, or steps, the walker is D steps to the right (or left) of the origin (starting point)? Remember, at any
given step the walker steps to the right if the coin lands on heads (probabilityP, which is now known) and left if the
coin lands on tails (probability 1−P).
Fair Coin Three-Step Case
In physics derivations, it’s often possible to obtain an intuition about the right way to find a general result or formula
by considering simple specific cases first. We will use this method here — let’s look at the possible outcomes (that
is, step sequences) of a three step random walk whereP= 1 /2 (it’s a fair coin, the walker is equally likely to step
left or right at every step). The various possibilities for this case are illustrated below: