Super-clustering of consecutive numbers in $p$-shifted random permutations

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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