On the size of the space spanned by a nonequilibrium state in a quantum spin lattice system

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 $1-ϵ$. We estimate the size to be $ \frac{\sqrt{2\mathfrak{e}_2}}{\pi}$\mathrm{e}\mathrm{r}\mathrm{f}$^{-1}(1-\epsilon) L^{\frac{d}{2}}t$,where$\mathfrak{e}_2$istheenergyvariancepersite,and$$\mathrm{e}\mathrm{r}\mathrm{f}$^{-1}$ is the inverse error function.