Independence of permutation limits at infinitely many scales

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 $(\zeta_j)$ such that, for each irreducible $p/q\in\mathbb{Q}\cap(0,1]$, a fixed-length subpermutation of $\zeta_j$ of width at most $|\zeta_j|^{p/q}$ 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 $(\eta_j)$ such that, for every skinny monotone grid class $\mathsf{Grid}(\mathbf{v})$, there is a function $f_\mathbf{v}$ such that any fixed-length subpermutation of $\eta_j$ of width at most $f_\mathbf{v}(|\eta_j|)$ is a.a.s. in $\mathsf{Grid}(\mathbf{v})$.