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

Let $A_{l;k}^{(n)}\subset S_n$ 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 $p=\{p_j{\}}_{j=1}^{\mathrm{\infty }}$ 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 $\{p_j{\}}_{j=1}^{\mathrm{\infty }}$ is non-increasing, is that $$\begin{array}{rl}{\displaystyle }& {\displaystyle \underset{l\to \mathrm{\infty }}{\lim }\underset{n\to \mathrm{\infty }}{\lim }{P}_n(A_{l,k}^{(n)})=\left(\prod _{j=1}^{k-1}\sum _{i=1}^j{p}_i\right)\left(\prod _{j=1}^{\mathrm{\infty }}\sum _{i=1}^j{p}_i\right),}\end{array}$$ and that if ${\lim }_{n\to \mathrm{\infty }}\min (k_n,n-k_n)=\mathrm{\infty }$, then $$\begin{array}{rl}{\displaystyle }& {\displaystyle \underset{l\to \mathrm{\infty }}{\lim }\underset{n\to \mathrm{\infty }}{\lim }{P}_n(A_{l,k_n}^{(n)})=(\prod _{j=1}^{\mathrm{\infty }}\sum _{i=1}^j{p}_i{)}^2\mathrm{.}}\end{array}$$ In particular these limits are positive if and only if ${\sum }_{j=1}^{\mathrm{\infty }}j{p}_j<\mathrm{\infty }$. 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 $S_{\mathrm{\infty }}$.