  We consider the time evolution of a state in an isolated quantum spin lattice
system with energy cumulants proportional to the number of the sites $L^d$. We
compute the distribution of the eigenvalues of the time averaged state over a
time window $[t_0,t_0+t]$ in the limit of large $L$. This allows us to infer
the size of a subspace that captures time evolution in $[t_0,t_0+t]$ with an
accuracy $1-\epsilon$. We estimate the size to be $
\frac{\sqrt{2\mathfrak{e}_2}}{\pi}\mathrm{erf}^{-1}(1-\epsilon)
L^{\frac{d}{2}}t$, where $\mathfrak{e}_2$ is the energy variance per site, and
$\mathrm{erf}^{-1}$ is the inverse error function.
