  We calculate the Emptiness Formation Probability (EFP) in the spin-Calogero
Model (sCM) and Haldane-Shastry Model (HSM) using their hydrodynamic
description. The EFP is the probability that a region of space is completely
void of particles in the ground state of a quantum many body system. We
calculate this probability in an instanton approach, by considering the more
general problem of an arbitrary depletion of particles (DFP). In the limit of
large size of depletion region the probability is dominated by a classical
configuration in imaginary time that satisfies a set of boundary conditions and
the action calculated on such solution gives the EFP/DFP with exponential
accuracy. We show that the calculation for sCM can be elegantly performed by
representing the gradientless hydrodynamics of spin particles as a sum of two
spin-less Calogero collective field theories in auxiliary variables.
Interestingly, the result we find for the EFP can be casted in a form
reminiscing of spin-charge separation, which should be violated for a
non-linear effect such as this. We also highlight the connections between sCM,
HSM and $\lambda=2$ spin-less Calogero model from a EFP/DFP perspective.
