On pattern-avoiding permutons

Abstract

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 $k$ 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 $(k-1)$ many, and this bound is sharp. We use this to give a simple proof of the `permutation removal lemma'.

17 pages, 4 figures. Reorganization of the previous version of the paper by incorporating the appendix into the main body. Addition of relevant citations and contextualization

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