  While the notion of quantum chaos is tied to random matrix spectral
correlations, also eigenstate properties in chaotic systems are often assumed
to be described by random matrix theory. Analytic insights into eigenstate
correlations can be obtained by the recently introduced partial spectral form
factor. Here, we study the partial spectral form factor in chaotic dual-unitary
quantum circuits in the thermodynamic limit. We compute the latter for a finite
subsystem in a brickwork circuit coupled to an infinite complement. For initial
times, shorter than the subsystem's size, spatial locality and (dual) unitarity
implies a constant partial spectral form factor, clearly deviating from the
linear ramp of the random matrix prediction. In contrast, for larger times we
prove, that the partial spectral form factor follows the random matrix result
up to exponentially suppressed corrections. We supplement our exact analytical
results by semi-analytic computations performed in the thermodynamic limit as
well as with numerics for finite-size systems.
