Super-clustering of consecutive numbers in $p$-shifted random permutations
Ross G. Pinsky
备注：
The material in this paper was incorporated into another existing paper on the arXiv, and the revised version of this other paper was uploaded to the arXiv under the revised title, "Clustering of consecutive numbers in permutations under Mallows distributions and super-clustering under general $p$-shifted distributions" (arXiv:2008.07215v2)
Let $A^{(n)}_{l;k}\subset S_n$ denote the event that the set of $l$ consecutive numbers $\{k,k+1,\cdots, k+l-1\}$ appear in a set of $l$ consecutive positions. Let $p=\{p_j\}_{j=1}^\infty$ be a distribution on $\mathbb{N}$ with $p_j>0$. Let $P_n$ denote the probability measure on $S_n$ corresponding to the $p$-shifted random permutation. Our main result, under the additional assumption that $\{p_j\}_{j=1}^\infty$ is non-increasing, is that $$ \begin{aligned} &\lim_{l\to\infty}\lim_{n\to\infty}P_n(A^{(n )}_{l,k})=\big(\prod_{j=1}^{k-1}\sum_{i=1}^jp_i\big) \big(\prod_{j=1}^\infty\sum_{i=1}^jp_i\big), \end{aligned} $$ and that if $\lim_{n\to\infty}\min(k_n,n-k_n)=\infty$, then $$ \begin{aligned} &\lim_{l\to\infty}\lim_{n\to\infty}P_n(A^{(n )}_{l,k_n})= \big(\prod_{j=1}^\infty\sum_{i=1}^jp_i\big)^2. \end{aligned} $$ In particular these limits are positive if and only if $\sum_{j=1}^\infty jp_j<\infty$. We say that super-clustering occurs when the limits are positive. We also give a new characterization of the class of $p$-shifted probability distributions on $S_\infty$.
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