Permutations with few inversions are locally uniform

TL;DR

This paper investigates the local and global properties of permutations with few inversions, revealing thresholds for local uniformity and inversion counts, but notes that some proofs are flawed and results are unproven.

Contribution

It identifies thresholds for local uniformity in permutations with few inversions and explores the local-global dichotomy, though some proofs remain unverified.

Findings

Threshold for local uniformity at rom m/n

Global permutation structure differs from local intervals

Results are not fully established due to proof flaws

Abstract

We prove that permutations with few inversions exhibit a local-global dichotomy in the following sense. Suppose ${\boldsymbol\sigma}$ is a permutation chosen uniformly at random from the set of all permutations of $[n]$ with exactly $m=m(n)\ll n^2$ inversions. If $i<j$ are chosen uniformly at random from $[n]$, then ${\boldsymbol\sigma}(i)<{\boldsymbol\sigma}(j)$ asymptotically almost surely. However, if $i$ and $j$ are chosen so that $j-i\ll m/n$, and $m \ll n^2/\log^2 n$, then $\lim_{n\to\infty}\mathbb{P}\big[{\boldsymbol\sigma}(i)<{\boldsymbol\sigma}(j)\big]=\frac{1}{2}$. Moreover, if $k=k(n)\ll \sqrt{m/n}$, then the restriction of ${\boldsymbol\sigma}$ to a random $k$-point interval is asymptotically uniformly distributed over $\mathcal{S}_k$. Thus, knowledge of the local structure of ${\boldsymbol\sigma}$ reveals nothing about its global form. We establish that $\sqrt{m/n}$ is the threshold for local uniformity and $m/n$ the threshold for inversions, and determine the behaviour in the critical windows. As pointed out by a referee, there are flaws in the proofs that do not seem easily rectifiable (see comments on pages 9 and 15). So the results stated above have not been established.