Consider the following subset of the unit cube in $\mathbb R^n$: $$ \mathcal D = \{ p = (p_1,p_2,\dots,p_n) \in [0,1]^n:\; p_1 \le p_2 \le \cdots \le p_n\}. $$ We would like to construct a probability measure supported on $\mathcal D$, continuous w.r.t. to the $n$-dimensional Lebesgue measure and "somehow" diffuse for large $n$. By this last statement, I roughly mean that the variances of the components of the vector $p$ stay bounded away from 0 under that measure. The problem is quite open-ended, but I will try to make this a bit more precise towards the end.

To see the difficulty, consider the general approach of starting with a probability distribution on $[0,1]^n$ and conditioning it to lie in $\mathcal D$. If we start with the uniform distribution on $[0,1]^n$ and perform this conditioning, we are effectively looking at the *order statistics* of a sample of size $n$ from the uniform distribution on $[0,1]$. The resulting marginal distributions of the components of $p$ are well-known: $p_k \sim \text{Beta}(k, n+1-k)$ whose variance rapidly goes to zero as $n \to \infty$.

In general, there seems to be a repulsion among the coordinates of $p$, induced by the order constraint, that forces their variances to shrink. The open-ended question is whether we can fight this repulsion and come up with a distribution whose marginal coordinates have more or less constant variances as $n\to \infty$?

We can also make the problem more precise by asking this (though I am afraid it might be too much to ask):

Is there a distribution on $[0,1]^n$ ($n \ge 2$) supported on $\mathcal D$ and absolutely continuous w.r.t. to the Lebesgue measure whose marginal distributions are all uniform on $[0,1]$?

A second more relaxed question could be something like this: For a distribution $Q$ on $[0,1]^n$, let $\text{var}_i(Q)$ be the variance of its $i$ coordinate (i.e., the variance of $p_i$ where $p \sim Q$).

Is there a sequence of distributions $Q_{n}$ on $[0,1]^n$ ($n \ge 2$) supported on $\mathcal D$ and absolutely continuous w.r.t. to the Lebesgue measure, with the propery that the sequence $n \mapsto \min_{1 \le i\le n} \text{var}_i(Q_n)$ is bounded away from zero as $n\to \infty$?

We can also relax the minimum to, say, the arithmetic average if the above is still too stringent.

A bit more generally, I am interested in the answer to the higher-order versions of this problem, for example, the case where we have a collection of points $(p_{ij})_{i,j=1}^{n_1,n_2}$ arranged in the $2$-dimensional lattice $[n_1] \times [n_2]$ with the constraints being imposed by the natural partial order, $p_{i,j} \le p_{i',j'}$ if $i \le i'$ and $j \le j'$. You can then imagine the 3-dimensional version and so on.