Krylov complexity in large-$q$ and double-scaled SYK model

Considering the large-$q$ expansion of the Sachdev-Ye-Kitaev (SYK) model in the two-stage limit, we compute the Lanczos coefficients, Krylov complexity, and the higher Krylov cumulants in subleading order, along with the $t\mathrm{/}q$ effects. The Krylov complexity naturally describes the “size” of the distribution, while the higher cumulants encode richer information. We further consider the double-scaled limit of SYK${}_q$ at infinite temperature, where $q\sim \sqrt{N}$. In such a limit, we find that the scrambling time shrinks to zero, and the Lanczos coefficients diverge. The growth of Krylov complexity appears to be “hyperfast”, which is previously conjectured to be associated with scrambling in de Sitter space.