Suppose $h$ is vector of eigenvalues of a random matrix with unit trace and $e$ is a vector in $\mathbb{R}^d$ starting with $e_i=1$ and evolving according the following recurrence:
$$e_i \leftarrow h_i \langle h, e\rangle - 2 h_i e_i $$
Is there a way to express $\|e\|_1$ at time $t$ in terms of limiting spectral density of $h$?
Slides 6-7 of Trogden's RMT tutorial shows how to express it in terms empirical spectral density by using $u=(1,1,1,\ldots,1)$ but I'm curious if this can be applied to the limiting spectral density.
