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

Both the race against Dampé the ghost gravekeeper and the Deku Butler are fully fixed. There were initially rumors that the latter changed depending on what in-game day you did it on, but a quick look at a map of area will show that's impossible. In fact, neither area even has the possibility for alternate routes.