Remarks on operator complexity at very long times

Operator complexity is usually discussed in fast scrambling systems (such as SYK) as approximately equivalent to operator size. This should not hold after the scrambling time, because operator size saturates while complexity should grow linearly for asymptotically long times. We discuss how a different notion of operator complexity, called K-complexity, can behave as operator size for early times and still continue growing linearly after the scrambling time. In showing this we use the ETH hypothesis. We also discuss the linear growth of operator complexity from the point of view of the bulk picture in holographic systems. We claim that the momentum/complexity correspondence naturally explains this asymptotic linear growth if complexity is defined using the extremal-volume prescription.