English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We prove that permutations with few inversions exhibit a local-globaldichotomy in the following sense. Suppose ${\boldsymbol\sigma}$ is apermutation 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 atrandom 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 asymptoticallyuniformly distributed over $\mathcal{S}_k$. Thus, knowledge of the localstructure of ${\boldsymbol\sigma}$ reveals nothing about its global form. Weestablish that $\sqrt{m/n}$ is the threshold for local uniformity and $m/n$ thethreshold 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 seemeasily rectifiable (see comments on pages 9 and 15). So the results statedabove have not been established.

Permutations with few inversions are locally uniform