Suppose $H$ is a random positive definite matrix normalized to have norm 1. Can we say anything about behavior of $f(s)$?

$$f(s)=\operatorname{Tr}[H(I-H)^s]$$

Taking $H=A^T A$ with entries of $A$ sampled from standard normal or uniform(-1,1) produces essentially identical curves, which suggests universality. What is the shape of this curve?
 
[![enter image description here][1]][1]

If we represent $i$th eigenvalue of $H$ as $y=h(i)$, we can approximate the sum in terms of integral which reduces to Laplace transform

$$f(s)\approx \int_0^\infty h(i) \exp(-s h(i))=\mathcal{L}[yh^{-1}(y)')\mathbb{I}_{0,1}(y)]$$

However, plugging this into Mathematica for $h(i)$ representing semicircle law gives something unintelligible in terms of Bessel and Struve functions.

[Notebook](https://www.wolframcloud.com/obj/yaroslavvb/nn-linear/mathoverflow-random-descent.nb)


Motivation: $f(2s)$ gives the loss observed after $s$ steps of gradient descent on a quadratic $H$ ([math](https://machine-learning-etc.ghost.io/gradient-descent-linear-update/))

  [1]: https://i.sstatic.net/JKlDG.png
  [2]: https://i.sstatic.net/1nGUz.png