31.1. Introduction http://www.ck12.org
- Step size (L).
- Number of steps (N).
We can now describe our model in the terms above. The person flips a coin, which lands on heads or tails according
to the probabilityp. After each flip, he takes a step of lengthLin the direction decided by the coin. AfterNtotal
flips, he stops and records his position.
In general , the valueLcan vary from step to step, but in this chapter, we’re going to focus on random walks with
constant step size, which we can just set toL=1 without losing any generality.
Question
Why can we do this?Answer
Because as long as step sizes are constant, our results would only be off by a proportionality constant for other
random walks.Bookkeeping
Question
Consider a two step random walk (that is,N=2). What are the possible outcomes of such a random walk?
How can we keep track of these?Answer
In a two step random walk, either the steps are in the same direction, or in opposite directions. If they are
in the same direction, the walker will be either two to the left, or two to the right of her original position.
Alternatively, if they are in the same direction, the walker will remain at her starting position at the end of the
walk. We can represent this on a graph where we count the number of steps on thex-axis, and the position on
they-axis (it might seem more natural to use the horizontal axis for position — due to the left/right dichotomy
— but since the number of steps is essentially a ’time’ variable and time is generally the independent variable,
we will follow convention). All the possibilities are represented below: