  In a system made up of independent random walks, fluctuations of order
$n^{1/4}$ from the hydrodynamic limit come from particle current across
characteristics. We show that a two-parameter space-time particle current
process converges to a two-parameter Gaussian process. These Gaussian processes
also appear as the limit for the one-dimensional random average process. The
final section of this paper looks at large deviations of the current process.
