What Is Reachable, and What Traps You
Last timeWhere You Are After Many Hops
Some situations a chain keeps coming back to and some it abandons for good. Sorting them out is the step that has to happen before any long run question has an answer at all.
The previous lesson said that the long run behaviour of a chain is a question
about high powers of its table. It did not say that the question always has an
answer, and it does not.
Before any of that can be settled, the situations have to be sorted. Some of
them the chain keeps coming back to forever. Some it passes through and
abandons. Some trap it completely. Which is which is decided by nothing more
than which arrows exist, and the sorting is the whole content of this lesson.
Reaching
One situation reaches another if there is any route from the first to the second
whose every hop has a chance above zero. The length of the route does not
matter and neither does how small the chance is.
That last point is worth insisting on. A route of six hops each with chance one
in a thousand has a chance of one in a million million million, and it counts
exactly as much as a single hop of chance nine in ten. The question being asked
is whether the thing is possible, and the answers to the long run questions turn
out to depend only on the possibilities and not on their sizes.
Two situations that reach each other are said to communicate, and that relation
sorts the situations into classes: within a class everything reaches everything,
and between classes the reaching runs one way or not at all.
The lesson stops here
6 more paragraphs to go
You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.
The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.
See the planThe contentsThis is the reading half
Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.
The contents