  Using an excursion-set approach, we revisit the initial spatial clustering of
Primordial Black Holes (PBHs) originating from the Hubble re-entry of large
Gaussian density fluctuations in the early universe. We derive the two-point
correlation functions of PBHs, properly accounting for the "cloud-in-cloud"
mechanism. Our expressions naturally and intrinsically correlate the formation
of pairs of PBHs, which is a key difference with the Poisson model of
clustering. Our approach effectively includes short-range exclusion effects and
clarifies the clustering behaviors at small scale: PBHs are anti-correlated at
short distances. Using a scale-independent collapse threshold, we derive
explicit expressions for the excess probability to find pairs of PBHs separated
by a distance $r$, as well as the excess probability to find pairs with
asymmetric mass ratio. Our framework is model-independent by construction and
we discuss possible other applications.
