Recently I have come across a distribution defined on the open ranked simplex $\nabla^{n-1}_+ = \{\vec x \in \mathbb{R}^n:\sum_{k=1}^n x_k =1, x_1 \geq x_2 \geq \cdots \geq x_n > 0\}$, whose density is given by $$ p(\vec x) = B^{-1} \prod_{k=1}^n x_{k}^{\alpha_k -1},\quad \vec x \in \nabla^{n-1}_+, $$ where $B = \int_{\nabla^{n-1}} \prod_{k=1}^n x_{k}^{\alpha_k -1} dx$ is the normalizing factor and $\alpha_k$'s positive. (The only difference from the usual Dirichlet distribution is the usual domain $\Delta^{n-1}_+$ being replaced by the rank-based one $\nabla^{n-1}_+$, and also a different normalizing factor.)
We are interested in under what conditions on $\alpha_k$'s will make the following statement holds true: Assume $X^{n}$ follows the ranked Dirichlet distribution in dimension $n$ with parameters $\{\alpha^n_k\}_{k=1}^n$, for any fixed small $\epsilon >0$, $$ \sup_{n} \left\{\frac{1}{n^2}\sum_{k=1}^{n-1} \sum_{l>k} \mathbb{P} \left( X^n_{k} - X^n_{l} < \frac{\epsilon}{n}\right) \right\} \leq f(\epsilon) $$ for some function $f$ such that $\lim_{\epsilon \rightarrow 0} f(\epsilon) = 0$.
Intuitively, this says the averaging probability of gaps between coordinates being less than a dimension-scaled threshold is small. While it seems a pure calculus problem, the computation of the target probability becomes tricky once involved with this "rank" structure. I greatly appreciate any suggestions on how to bound this probability with some choice of $\alpha$ or any reference where you may have seen similar stuff.
