Here’s a standard(ish) calculus question, known as Fermat’s principle: what is the quickest way to get from one corner of a field to the opposite one, if one side has been mowed so you can move speedily across it, but the other side is growing wild and slowing you down?
If you set off straight towards the opposite corner, you’ll take the shortest path—but not necessarily the quickest one. The fastest path will involve travelling a bit further on the nice ground, and a bit less far on the overgrown side; it might look a bit like this:
You could sit down and work it out (it’s not too tricky), using a bit of stationary point analysis. Or, you can get ants to solve it for you! Just put a colony of hungry ants in one corner of the field and some food on the other. The foragers will want to collect the food and bring it back as quickly as possible. Some researchers from Regensburg in Germany actually tried it out, and the ants found the quickest route between the two corners, taking into account the different speeds they can move on different surfaces. Thanks, ants. Thants.
We can make the picture a bit more complicated: instead of the field being mown on one side and overgrown on the other, we could assign a different grass height to each point in space. (Mathematicians call this a… field.)
The ants will still want to take the fastest path from one side to the other, but now the calculus problem is starting to get complicated. It can help if we split the field up into sections, and draw a graph representing all the different routes the ants could take.
Once we know how long it will take the ants to move along each edge, we just have to check all the routes and figure out which one is fastest.
Prob-ant-bility
Probability theorists like to ask a very similar question, but about random fields. Instead of fixed timings on each edge, they make each one random. Now that the shortest path is also random, it’s natural to wonder what the path is typically like: how long is it on average? Does it start by going upwards or rightwards? Is it mostly straight or very wiggly?
There are many types of grass, and so many different choices for the random amount of time it takes to cross an edge. You could choose your favourite distribution. For simplicity though, let’s say that the time is a uniformly random amount between one and two minutes.
If our graph is one-dimensional (in other words, a line), there’s only one possible route from left to right.
When there are $n$ segments, the average time to get across the field is close to $n$ times the average length of time per segment—so $3n/2$. This is the law of large numbers.
But when we have an $n \times n$ grid, we can definitely get from corner to corner in time $4n$, and it must take at least time $n$. We know that there exists a number $\alpha\in [1,4]$ such that the total time is very close to $\alpha n$, but unlike the one-dimensional case, no one knows what $\alpha$ is!
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