I have $N$ independent random unit vectors $\{v_i\}$ in $\mathbb{R}^n$, where N > n. I need a concentration inequality of the form $$\text{P}(|v_i \cdot v_j| > \epsilon \,\,\,\, \forall i, j = 1, \dots, N: i \neq j)\leq \psi(\epsilon)$$ where hopefully $\psi(\epsilon)$ is something small.

I think that I can use Johnson-Lindenstrauss to do this for isotropic vectors (e.g. by choosing orthogonal basis for $\mathbb{R}^N$ and projecting into $\mathbb{R}^n$ with a random subgaussian matrix).

Are there results of this form that hold when the $\{v_i\}$ are not distributed isotropically, for instance Gaussian with covariance $\Sigma$? For instance, when there is some weak correlation/dependence between the components of each of the $v$ --- maybe $|\Sigma_{ij}| \leq \alpha$ when $i\neq j$?

(Any seemingly related results in this area are much appreciated!)