Suppose we're selecting points uniformly at random from the $N$-simplex

$S_N = \{x \in \mathbb R^{N+1}: $ all $ x_i \ge 0$ and $x_1 + \ldots x_N = 1\}$.

One way to do this in practice is choose $N-1$ points $a_1, \ldots , a_{N-1}$ uniformly and independently from the unit interval $[0,1]$. Then for $a_0=0$ and $a_1=1$ construct the point $x \in S_N$ with each $x_i = a_i - a_{i-1}$.

One would expect for large $N$ the points $a_i$ to be *evenly spaced* across the interval, and so the the average point looks of $S_n$ looks pretty much like the constant $1/N$ vector. That is to say it's unlikely for any collection of entries to be small. Is anything known about the exact distribution of such collections?

Formally put suppose $X: \Omega \to S_N$ is a uniformly distributed random variable from some probability space onto the $N$-simplex. Define each $X_k : \Omega \to [0,1]$ by $X_k(x) = $ the $k$th largest coordinate of $X(x)$.

I've drawn some samples for $k = n/2$ which is the case I'm mostly interested in. It seems that, after you normalise the variable by multiplying by $N$, the mean tends to about $0.7$ (marked with a vertical line) from above. The distributions are also increasingly tighter bell-curves. The below is with 100,000 samples per curve.

Is there anything like a closed form known for the distribution of $X_k$? If not are there any useful bounds for probabilities like $P(X_k > 1/N \pm \epsilon)$ or $P(X_k < 1/N \pm \epsilon)$ that give answers similar to the behaviour above?

I am also interested in the variables $Y_k = X_1 + \ldots X_k$ if they are any easier to understand analytically, again primarily in the case $k = n/2$. Again the plots look like ever tighter bells and the mean tends to $0.15$ (the black line) from above.