  The symmetric simple exclusion process (SEP) is a paradigmatic model of
transport, both in and out-of-equilibrium. In this model, the study of currents
and their fluctuations has attracted a lot of attention. In finite systems of
arbitrary dimension, both the integrated current through a bond (or a fixed
surface), and its fluctuations, grow linearly with time. Conversely, for
infinite systems, the integrated current displays different behaviours with
time, depending on the geometry of the lattice. For instance, in 1D the
integrated current fluctuations are known to grow sublinearly with time, as
$\sqrt{t}$. Here we study the fluctuations of the integrated current through a
given lattice bond beyond the 1D case by considering a SEP on
higher-dimensional lattices, and on a comb lattice which describes an
intermediate situation between 1D and 2D. We show that the different behaviours
of the current originate from qualitatively and quantitatively different
current-density correlations in the systems, which we compute explicitly.
