In a context where I try to estimate some combinatorial sums, I'm faced with vector random variables $(x_1,...,x_n)$ uniformly distributed with $n \rightarrow \infty$. I want to know if the components have to behave in a wellknown fashion. I recently read the following. (Sourav Chatterjee math summer school 2012)

" Classical example: Uniform distribution on the simplex $\{ (x_1,...,x_n) \ | \ \sum\limits_{k=1}^{n} x_k =1 \}$

In this example, it is known that for n large, the coordinates behave like i.i.d. $Exp(1)$ random variables "

Where can I find a proof of that result or related?

Thanks for any answer/pointer