  We consider the particle current in the asymmetric avalanche process on a
ring. It is known to exhibit a transition from the intermittent to continuous
flow at the critical density of particles. The exact expressions for the first
two scaled cumulants of the particle current are obtained in the large time
limit $t\to\infty$ via the Bethe ansatz and a perturbative solution of the
TQ-equation. The results are presented in an integral form suitable for the
asymptotic analysis in the large system size limit $N\to\infty$. In this limit
the first cumulant, the average current per site or the average velocity of the
associated interface, is asymptotically finite below the critical density and
grows linearly and exponentially times power law prefactor at the critical
density and above, respectively.
  The scaled second cumulant per site, i.e. the diffusion coefficient or the
scaled variance of the associated interface height, shows the $O(N^{-1/2})$
decay expected for models in the Kardar-Parisi-Zhang universality class below
the critical density, while it is growing as $O(N^{3/2})$ and exponentially
times power law prefactor at the critical point and above. Also, we identify
the crossover regime and obtain the scaling functions for the uniform
asymptotics unifying the three regimes. These functions are compared to the
scaling functions describing crossover of the cumulants of the avalanche size,
obtained as statistics of the first return area under the time space trajectory
of the Vasicek random process.
