We extend recent results on the exact hydrodynamics of a system of diffusive active particles displaying a motility-induced phase separation to account for typical fluctuations of the dynamical fields. By calculating correlation functions exactly in the homogeneous phase, we find that two macroscopic length scales develop in the system. The first is related to the diffusive length of the particles and the other to the collective behavior of the particles. The latter diverges as the critical point is approached. Our results show that the critical behavior of the model in one dimension belongs to the universality class of a mean-field Ising model, both for static and dynamic properties, when the thermodynamic limit is taken in a specified manner. The results are compared to the critical behavior exhibited by the ABC model. In particular, we find that in contrast to the ABC model the density large deviation function, at its Gaussian approximation, does not contain algebraically decaying interactions but is of a finite, macroscopic, extent which is dictated by the diverging correlation length.
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