  We explore the physics of scrambling in the moving mirror models, in which a
two-dimensional CFT is subjected to a time-dependent boundary condition. It is
well-known that by choosing an appropriate mirror profile, one can model
quantum aspects of black holes in two-dimensions, ranging from Hawking
radiation in an eternal black hole (for an "escaping mirror") to the recent
realization of Page curve in evaporating black holes (for a "kink mirror"). We
explore a class of OTOCs in the presence of such a boundary and explicitly
demonstrate the following primary aspects: First, we show that the dynamical
CFT data directly affect an OTOC and maximally chaotic scrambling occurs for
the escaping mirror for a large-$c$ CFT with identity block dominance. We
further show that the exponential growth of OTOC associated with the physics of
scrambling yields a power-law growth in the model for evaporating black holes
which demonstrates a unitary dynamics in terms of a Page curve. We also
demonstrate that, by tuning a parameter, one can naturally interpolate between
an exponential growth associated to scrambling and a power-law growth in
unitary dynamics. Our work explicitly exhibits the role of higher-point
functions in CFT dynamics as well as the distinction between scrambling and
Page curve. We also discuss several future possibilities based on this class of
models.
