Is there a nice expression for $V(s)$ defined below for $1<p\le 2$ and large $d$?
$$V(s)=\frac{1}{d}\sum_{ij=(1,1)}^d \left(f(i,s)-f(j,s)\right)^2$$ $$f(i,s)=i^{-p}(1-i^{-p})^s$$
Empirically it appears to settle into log-linear growth after initial transient period
Motivation: this gives variance of gradient descent estimate with random starting point after $s/2$ steps. $s=2$ case was solved here