  Previous work has explored the connections between three concepts -- operator
size, complexity, and the bulk radial momentum of an infalling object -- in the
context of JT gravity and the SYK model. In this paper we investigate the
higher dimensional generalizations of these connections. We use a toy model to
study the growth of an operator when perturbing the vacuum of a CFT. From
circuit analysis we relate the operator growth to the rate of increase of
complexity and check it by complexity-volume duality. We further give an
empirical formula relating complexity and the bulk radial momentum that works
from the time that the perturbation just comes in from the cutoff boundary, to
after the scrambling time.
