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In this paper by making use of the "Complexity=Action" proposal, we study the complexity
growth after shock waves in holographic ï¬eld theories. We consider both double black hole-Vaidya
and AdS-Vaidya spacetimes and also a geometry with multiple shocks. We ï¬nd that the Lloyd's
bound is respected during the thermalization process in each of these geometries and at the late
time the complexity growth saturates to its ï¬nal value which is proportional to the energy of the
ï¬nal state. We conclude that the saturation value of complexity growth rate is independent of the
initial temperature and in the case of thermal initial state, it is always less than the value for the
vacuum initial state such that considering multiple shocks it gets more smaller. Our results indicate
that by increasing the temperature of the initial state, the corresponding rate of complexity growth
starts far from ï¬nal saturation rate value.
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