Does anyone know anything about the determinant of a random $n\times n$ row stochastic matrix? What I have in mind is that the rows are independently selected from the uniform distribution on the unit $(n-1)$-dimensional simplex: $x_1+\cdots+x_n=1$. I'm interested in upper (and lower) bounds on the expected absolute value of the determinant as a function of $n$.

Thanks for any references! I found something due to Nguyen for the random doubly stochastic matrices, but didn't see anything for the easier (?) singly stochastic case.

**EDIT** thanks to Igor's answer below, I have an answer to the original question that seems likely. The paper referenced by Igor gives the empirical spectral distribution of $A_n$ to be uniform on the disk of radius $1/\sqrt n$. This suggests that the determinant should be something like $ (ne)^{-n/2}$. A result somewhat like this was proved for matrices with iid entries by Nguyen and Vu.