  We explore several aspects of the current fluctuations and correlations in
the box-ball system (BBS), an integrable cellular automaton in one space
dimension. The state we consider is an ensemble of microscopic configurations
where the box occupancies are independent random variables (i.i.d. state), with
a given mean ball density. We compute several quantities exactly in such
homogeneous stationary state: the mean value and the variance of the number of
balls $N_t$ crossing the origin during time $t$, and the scaled cumulants
generating function associated to $N_t$. We also compute two spatially
integrated current-current correlations. The first one, involving the long-time
limit of the current-current correlations, is the so-called Drude weight and is
obtained with thermodynamic Bethe Ansatz (TBA). The second one, involving equal
time current-current correlations is calculated using a transfer matrix
approach. A family of generalized currents, associated to the conserved charges
and to the different time evolutions of the models are constructed. The
long-time limits of their correlations generalize the Drude weight and the
second cumulant of $N_t$ and are found to obey nontrivial symmetry relations.
They are computed using TBA and the results are found to be in good agreement
with microscopic simulations of the model. TBA is also used to compute
explicitly the whole family of flux Jacobian matrices. Finally, some of these
results are extended to a (non-i.i.d.) two-temperatures generalized Gibbs state
(with one parameter coupled to the total number of balls, and another one
coupled to the total number of solitons).
