In Ocarina of Time and Majora's Mask, are the "follow-guy maze puzzle" routes randomly picked each time?

In both Ocarina of Time and Majora's Mask, there's one part where you are supposed to follow a ghost or other creature in an underground maze of sorts. Anyone who has played either game knows what I refer to. It strikes me as a golden opportunity to build in some simple randomness, where the creature you are supposed to follow picks not the same route every time, or for every player.

Maybe the route that the character will pick differs from save game to save game, so that if you have to retry it on the same save slot, you each time get the same route (and can memorize it), but it's not safe to write a "walkthrough" where you lay out the exact route, because all players may not get the same one route, but a slightly/very different one?

But on the other hand, I have an odd feeling that game makers, at least back in the day, wanted there to be a reliable single way to beat it, and that randomness was almost frowned upon.

In these two specific games, the latter being a kind of "remix" of the former and very similar in many ways, are those "follow-guy maze puzzle"s:

1. A fixed, single route for every player for every save game?
2. Varying per save-game in one of a few pre-determined routes?
3. Varying per save-game in several places of the maze, meaning numerous possible routes?
4. Varying every single time you try it, even within the same save game? (From my memory, this cannot be it.)
5. Some other possibility which I haven't thought of?