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- W2969893246 abstract "We prove that permutations with few inversions exhibit a local-global dichotomy in the following sense. Suppose ${boldsymbolsigma}$ 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 ${boldsymbolsigma}(i)<{boldsymbolsigma}(j)$ asymptotically almost surely. However, if $i$ and $j$ are chosen so that $j-ill m/n$, and $m ll n^2/log^2 n$, then $lim_{ntoinfty}mathbb{P}big[{boldsymbolsigma}(i)<{boldsymbolsigma}(j)big]=frac{1}{2}$. Moreover, if $k=k(n)ll sqrt{m/n}$, then the restriction of ${boldsymbolsigma}$ to a random $k$-point interval is asymptotically uniformly distributed over $mathcal{S}_k$. Thus, knowledge of the local structure of ${boldsymbolsigma}$ 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." @default.
- W2969893246 created "2019-08-29" @default.
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- W2969893246 date "2019-08-20" @default.
- W2969893246 modified "2023-09-25" @default.
- W2969893246 title "Permutations with few inversions are locally uniform" @default.
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