  We calculate the diffusion coefficient of an active tracer in a schematic
crowded environment, represented as a lattice gas of passive particles with
hardcore interactions. Starting from the master equation of the problem, we put
forward a closure approximation that goes beyond trivial mean-field and
provides the diffusion coefficient for an arbitrary density of crowders in the
system. We show that our approximation is accurate for a very wide range of
parameters, and that it correctly captures numerous nonequilibrium effects,
which are the signature of the activity in the system. In addition to the
determination of the diffusion coefficient of the tracer, our approach allows
us to characterize the perturbation of the environment induced by the
displacement of the active tracer. Finally, we consider the asymptotic regimes
of low and high densities, in which the expression of the diffusion coefficient
of the tracer becomes explicit, and which we argue to be exact.
