On pattern-avoiding permutons

The theory of limits of permutations leads to limit objects called permutons, which are certain Borel measures on the unit square. We prove that permutons avoiding a given permutation of order k$$ k $$ have a particularly simple structure. Namely, almost every fiber of the disintegration of the permuton (say, along the x-axis) consists only of atoms, at most (k-1)$$ \left(k-1\right) $$ many, and this bound is sharp. We use this to give a simple proof of the "permutation removal lemma."