  Krylov complexity, as a novel measure of operator complexity under Heisenberg
evolution, exhibits many interesting universal behaviors and also bounds many
other complexity measures. In this work, we study Krylov complexity
$\mathcal{K}(t)$ in Random Matrix Theory (RMT). In large $N$ limit: (1) For
infinite temperature, we analytically show that the Lanczos coefficient
$\{b_n\}$ saturate to constant plateau $\lim\limits_{n\rightarrow\infty}b_n=b$,
rendering a linear growing complexity $\mathcal{K}(t)\sim t$, in contrast to
the exponential-in-time growth in chaotic local systems in thermodynamic limit.
After numerically comparing this plateau value $b$ to a large class of chaotic
local quantum systems, we find that up to small fluctuations, it actually
bounds the $\{b_n\}$ in chaotic local quantum systems. Therefore we conjecture
that in chaotic local quantum systems after scrambling time, the speed of
linear growth of Krylov complexity cannot be larger than that in RMT. (2) For
low temperature, we analytically show that $b_n$ will first exhibit linear
growth with $n$, whose slope saturates the famous chaos bound. After hitting
the same plateau $b$, $b_n$ will then remain constant. This indicates
$\mathcal{K}(t)\sim e^{2\pi t/\beta}$ before scrambling time $t_*\sim
O(\beta\log\beta)$, and after that it will grow linearly in time, with the same
speed as in infinite temperature. We finally remark on the effect of finite $N$
corrections.
