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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?

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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

Motivation: $f(2s)$ gives the loss observed after $s$ steps of gradient descent on a quadratic $H$ (math)

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    $\begingroup$ Did you think about Tauberian theorems? This could give you a first (yet rough) understanding of the behaviour of your function depending on the spectrum. I do know know what would best suit your needs and am not super fit with these theorems but this might be a reasonable way to go. $\endgroup$
    – Gilles Mordant
    Commented Mar 9, 2023 at 11:35
  • $\begingroup$ Yes, it looks like a promising approach, since then I found this paper which seems to do precisely this approach $\endgroup$
    – Yaroslav Bulatov
    Commented Mar 9, 2023 at 23:51
  • $\begingroup$ Great! Keep me posted if you agree, what would come out is interesting. $\endgroup$
    – Gilles Mordant
    Commented Mar 10, 2023 at 13:37

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