Universality and Emergent Classicality in Krylov Space

Krylov complexity grows linearly at late times for many systems. In this regime, we give a general spectral formula for the leading growth coefficients of Krylov complexity and its higher moments, without requiring detailed knowledge of the Lanczos coefficients. For states with a narrow energy distribution compared to the bandwidth, dynamics in the Krylov basis resembles motion along a classical trajectory. We demonstrate our results for the critical XY chain and double-scaled SYK. For the latter, we compute the leading late-time behavior of the chord-number moments through one-loop order.