  Recently, a superdiffusion exhibiting the Kardar-Parisi-Zhang (KPZ) scaling
in late-time correlators and autocorrelators of certain interacting many-body
systems has been reported. Inspired by these results, we explore the KPZ
scaling in correlation functions using their realization in the Krylov operator
basis. We focus on the Heisenberg time scale, which approximately corresponds
to the ramp--plateau transition for the Krylov complexity in systems with a
large but finite number degrees of freedom. Two frameworks are under
consideration: i) the system with growing Lanczos coefficients and an
artificial cut-off, and ii) the system with the finite Hilbert space. In both
cases via numerical analysis, we observe the transition from Gaussian to
KPZ-like scaling at the critical Euclidean time $t_{E}^*=c_{cr}K$, for the
Krylov chain of finite length $K$, and $c_{cr}=O(1)$. In particular, we find a
scaling $\sim K^{1/3}$ for fluctuations in the one-point correlation function
and a dynamical scaling $\sim K^{-2/3}$ associated with the return probability
(Loschmidt echo) corresponding to autocorrelators in physical space. In the
first case, the transition is of the 3rd order and can be considered as an
example of dynamical quantum phase transition (DQPT), while in the second, it
is a crossover. For case ii), utilizing the relationship between the spectrum
of tridiagonal matrices at the spectral edge and the spectrum of the stochastic
Airy operator, we demonstrate analytically the origin of the KPZ scaling for
the particular Krylov chain using the results of the probability theory. We
argue that there is some outcome of our study for the double scaling limit of
matrix models. For the case of topological gravity, the white noise
$O(\frac{1}{N})$ term is identified, which should be taken into account in the
controversial issue of ensemble averaging in 2D/1D holography.
