4
$\begingroup$

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.

Improve this question
$\endgroup$
2
  • $\begingroup$ Did you find any other paper or write one yourself on distribution of such determinants? $\endgroup$
    – Maesumi
    Commented Nov 20, 2017 at 14:17
  • $\begingroup$ Could you please list the title of the paper by Nguyen and Vu you referenced? $\endgroup$
    – Hans
    Commented Jan 30, 2018 at 22:28

1 Answer 1

Reset to default
4
$\begingroup$

The paper to look at is Bordenave, Caputo, Chafai, which is cited in Nguen's paper.

Improve this answer
$\endgroup$
3
  • $\begingroup$ Thanks Igor - It seems that that model is a bit different to the one I'm considering: they are taking rows formed by taking iid non-negative entries and then normalizing (so that the $i$th entry is $X_i/(X_1+\ldots+X_n)$), which seems to be somewhat different (even if the $X$'s are uniform) than uniformly sampling over the simplex (e.g. conditional on $X_1+\ldots+X_n=n-0.5$, all of the $X_i$'s are above 0.5). From the introduction to the paper, the spirit seems to be that small changes to the distribution make small changes to the spectrum so presumably this paper gives the right distribution. $\endgroup$
    – Anthony Quas
    Commented Dec 2, 2016 at 1:00
  • 3
    $\begingroup$ One obtains the uniform distribution on the simplex by taking $X_i$ to be iid exponential variables. $\endgroup$
    – Nick Cook
    Commented Dec 2, 2016 at 1:52
  • $\begingroup$ Oh yes. I see that. $\endgroup$
    – Anthony Quas
    Commented Dec 2, 2016 at 2:20

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .