Loop Challenge: Random Path

So I've made a filter that uses a loop to simulate random "brownian" motion. Basically, you take a dot, and you move it some small random distance in some random direction, and then you do it again and again. Watch the dot eventually meander off the screen. (actually my filter does single-distance diagonal motion only, but it's still random)

The challenge:
()Trace the path this dot takes, or at least show all the different positions of the dot (all the iterations) at the same time.
()If you have a different way of drawing a random path/scribble (with loops or not), that would be interesting, too.
()Try a guided (not random) path (like a river through the terrain)
()Can you do multiple dots/particles at the same time?
()What else can you do with paths?

Random Motion.ffxml

Damn, I like that solution, gotta try it on my alphabet problem!

Great, thanks. I'm sure this will be way faster than what I've tried so far.

On the original subject, here's my basic Scribble filter, nowhere near as fast as CFandM's, but it's another way to draw a random path natively and with a loop. Perhaps more interestingly it features a line drawing snippet.

The Scribble.ffxml

Thanks Indigo...hmm could also be the missle paths in that old game "missle command"