Independence of permutation limits at infinitely many scales

Abstract

We introduce a new natural notion of convergence for permutations at any specified scale, in terms of the density of patterns of restricted width, along with a stricter notion of scalable convergence in which the choice of scale is immaterial. Using these, we prove that asymptotic limits may be chosen independently at a countably infinite number of scales.We illustrate our result with two examples. Firstly, we exhibit a sequence of permutations (ζ j ) such that, for each irreducible p/q ∈ Q ∩ (0, 1], a fixed-length subpermutation of ζ j of width at most |ζ j | p/q is a.a.s. increasing if q is odd, and is a.a.s. decreasing if q is even. In the second, we construct a sequence of permutations (η j ) such that, for every skinny monotone grid class Grid(v), there is a function f v such that any fixed-length subpermutation of η j of width at most f v (|η j |) is a.a.s. in Grid(v).