Prove RMT equality for higher moments of the spectral form factor in dual-unitary circuits

Prove that, for brickwork dual-unitary Floquet circuits with independently distributed one-site random unitaries (and a fixed two-site dual-unitary gate not equal to SWAP), the thermodynamic limit of the higher moments of the spectral form factor K_n(t)=E[|tr(U^t)|^{2n}] equals the random-matrix-theory prediction K_n(t)=n! t^n up to corrections of order t/𝒩, for all fixed n and t as the system size L→∞.

Background

The spectral form factor (SFF) provides a standard probe of spectral correlations and quantum chaos. For dual-unitary circuits with i.i.d. one-site randomness, the two-point SFF K(t) was determined exactly in the thermodynamic limit and shown to reproduce the RMT linear ramp.

The natural extension to higher moments K_n(t) involves analysing a space transfer matrix whose leading eigenvectors have been characterised; this yields a lower bound consistent with the RMT result, but the full equality has not been established.