It was a map showing every detail of the Hogwarts castle and grounds. But the truly remarkable thing was the tiny ink dots moving around it, each labelled with a name in minuscule writing. Astounded, Harry bent over it. A labelled dot in the top left corner showed that Professor Dumbledore was pacing his study; the caretaker's cat, Mrs Norris, was prowling the second floor, and Peeves the poltergeist was currently bouncing around the trophy room.
I don't know what "pacing his study" means here. I looked up the word 'pace', but found none of them seem to fit.
v.t.
to set or regulate the pace for, as in racing.
to traverse with slow, regular steps.
to measure by paces.
to train to a certain pace: to pace a horse.
(of a horse) to run (a distance) at a pace.