On the asymmetric zero-range in the rarefaction fan

We consider one-dimensional asymmetric zero-range processes starting from a step decreasing profile leading, in the hydrodynamic limit, to the rarefaction fan of the associated hydrodynamic equation. Under that initial condition, and for {\em{ totally asymmetric jumps}}, we show that the weighted sum of joint probabilities for second class particles sharing the same site is convergent and we compute its limit. For {\em{partially asymmetric jumps}}, we derive the Law of Large Numbers for a second class particle, under the initial configuration in which all positive sites are empty, all negative sites are occupied with infinitely many first class particles and there is a single second class particle at the origin. Moreover, we prove that among the infinite characteristics emanating from the position of the second class particle it picks randomly one of them. The randomness is given in terms of the weak solution of the hydrodynamic equation, through some sort of renormalization function. By coupling the {\em{constant-rate totally asymmetric zero-range}} with the totally asymmetric simple exclusion, we derive limiting laws for more general initial conditions.FC