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We extend recent results on the exact hydrodynamics of a system of diffusiveactive particles displaying a motility-induced phase separation to account fortypical fluctuations of the dynamical fields. By calculating correlationfunctions exactly in the homogeneous phase, we find that two macroscopic lengthscales develop in the system. The first is related to the diffusive length ofthe particles and the other to the collective behavior of the particles. Thelatter diverges as the critical point is approached. Our results show that thecritical behavior of the model in one dimension belongs to the universalityclass of a mean-field Ising model, both for static and dynamic properties, whenthe thermodynamic limit is taken in a specified manner. The results arecompared to the critical behavior exhibited by the ABC model. In particular, wefind that in contrast to the ABC model the density large deviation function, atits Gaussian approximation, does not contain algebraically decayinginteractions but is of a finite, macroscopic, extent which is dictated by thediverging correlation length.

Exact fluctuating hydrodynamics of active lattice gases -- Typical fluctuations