  We continue the analysis of the Krylov complexity in the IP matrix model. In
a previous paper, for a fundamental operator, it was shown that at zero
temperature, the Krylov complexity oscillates and does not grow, but in the
infinite temperature limit, the Krylov complexity grows exponentially in time
as $\sim \exp\left( {\mathcal{O}\left( {\sqrt{t}}\right)} \right)$. We study
how the Krylov complexity changes from a zero-temperature oscillation to an
infinite-temperature exponential growth. At low temperatures, the spectral
density is approximated as collections of infinite Wigner semicircles. We
showed that this infinite collection of branch cuts yields linear growth to the
Lanczos coefficients and gives exponential growth of the Krylov complexity.
Thus the IP model for any nonzero temperature shows exponential growth for the
Krylov complexity even though the Green function decays by a power law in time.
We also study the Lanczos coefficients and the Krylov complexity in the IOP
matrix model taking into account the $1/N^2$ corrections. There, the Lanczos
coefficients are constants and the Krylov complexity does not grow
exponentially as expected.
