Consider the following $n\times n$ random matrix $V_{n}$ where the $(p,q)$ entry is given by $$ V_{n}(p,q):= \frac{1}{\sqrt{n}}\exp(2\pi i(p-1) x_{q}) $$ where $x_{1},x_{2},\ldots,x_{n}$ are iid random variables with uniform distribution on $[0,1]$.

It is not difficult to prove that $V_{n}^{*}V_{n}$ has the same eigenvalues as $X_{n}$ where $$ X_{n}(p,q)=\frac{\sin(n(x_p-x_q)/2)}{n\sin((x_p-x_q)/2)}. $$ This matrix is positive definite and invertible with probability one. The minimum eigenvalue, $\lambda_{1}(n)$, goes to zero as $n\to\infty$. I'm interested in the rate at which this eigenvalue goes to zero. Simulations suggest that $$ \mathbb{E}(\lambda_1(n))\sim \exp(-\alpha n), $$ the expected value decays exponentially. Using the Cauchy interlacing theorem I can only get the upper bound $O(\frac{1}{n^2}).$

Any idea of what can work here?

Thanks!

--Gabriel