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We consider the one-dimensional asymmetric zero-range process starting from astep decreasing profile. In the hydrodynamic limit this initial condition leadsto the rarefaction fan of the associated hydrodynamic equation. Under thisinitial condition and for totally asymmetric jumps, we show that the weightedsum of joint probabilities for second class particles sharing the same site isconvergent and we compute its limit. For partially asymmetric jumps we derivethe Law of Large Numbers for the position of a second class particle under theinitial configuration in which all the positive sites are empty, all thenegative sites are occupied with infinitely many first class particles and witha single second class particle at the origin. Moreover, we prove that among theinfinite characteristics emanating from the position of the second classparticle, this particle chooses randomly one of them. The randomness is givenin terms of the weak solution of the hydrodynamic equation through some sort ofrenormalization function. By coupling the zero-range with the exclusion processwe derive some limiting laws for more general initial conditions.

On the asymmetric zero-range in the rarefaction fan