In this work we focus mostly on random unitary rotations, that is, when the system belongs to the unitary symmetry class. The orthogonal case is only briefly mentioned in Appendixes pp5 and pp6-s2.

These averages of products of matrix elements are the only ones formulated by the full ETH [see Eq. (1a)]; the interest towards those quantities is motivated by the evaluation of thermal correlation functions. Indeed, when calculating thermal moments, they naturally appear: $$⟨A^n{⟩}_{\beta }=\frac{\text{Tr}(A^n{e}^{-\beta H})}{\text{Tr}(e^{-\beta H})}=\frac{1}{Z}{\sum }_{i_1,\dots ,i_n}{A}_{i_1{i}_2}{A}_{i_2{i}_3}\cdots A_{i_n{i}_1}{e}^{-\beta E_{i_1}},$$(42)with $Z={\sum }_i{e}^{-\beta E_i}$ partition function at temperature $T={\beta }^{-1}$.

Those products of matrix elements are usually excluded by ETH discussions, because they give a negligible contribution in the calculation of thermal correlation functions [see Eq. (42)], for entropic reasons. This also explains why free probability, rather than classical probability theory, is relevant for quantum dynamics.

From the derivation in Appendix pp2, it becomes clear that higher-order contributions are considerably more difficult to compute, which goes beyond the scope of this work. Therefore, we do not provide subleading corrections for simple loops.

These interactions, similarly to those considered in [97], were specifically chosen to ensure the system belongs to the unitary symmetry class.

As these differences are also modulo $2\pi$, we restrict to the interval ${\omega }_{\alpha \beta }\in [-\pi ,\pi )$.

This is not the case for the entire symmetric group; indeed, the entire Cayley graph itself does not form a lattice.

This is not true if considering the geodesic interval from the identity to a generic maximal cycle: in this case, it is an isomorphism, but not an embedding.

This can be proven identifying $S_n$ with an orientable surface, and applying Euler’s formula: the only possible solutions to the triangle inequality are $d(e,\beta )+d(\beta ,\alpha )=d(e,\alpha )+2g$, for integer $g\ge 0$ [113].

The subtle point is that in general $\alpha$ is not forced to be in the interval $[e,{\gamma }_n]$, therefore one cannot identify it with a non-crossing partition; however, it is possible to construct another permutation ${\alpha }^{\prime }={\lambda }^{-1}\alpha \lambda$, with $\lambda \in S_n$, so that $\mathrm{\#}\alpha =\mathrm{\#}{\alpha }^{\prime }$ and ${\alpha }^{\prime }\in [e,{\gamma }_n]$. In this case ${\mathcal{K}}_{\alpha }={\mathcal{K}}_{{\alpha }^{\prime }}$: the term $\mathcal{K}$ is the same when taking $\alpha$ or ${\alpha }^{\prime }$ and therefore can still be identified with the free cumulant associated to ${\alpha }^{\prime }$; we will keep the notation $\alpha$ in the text, intending that when $\alpha$ is not in the interval $[e,{\gamma }_n]$, it has to be thought as ${\alpha }^{\prime }$.

The fact that a product of matrix elements can be described uniquely by a partition happens only in the case of ETH-cycles, which partitions the set $\mathbf{i}$ in block of equal indices, since one needs the indices to be on a circle for being in blocks of partitions.

Actually, the only reason why we can study the first order contribution is because of the power $1/D^2$ in Eq. (b6).

For non-Floquet systems one should consider further corrections coming from saddle-point fluctuations.