An upper bound on the fluctuations of a second class particle

This note proves an upper bound for the fluctuations of a second-class particle in the totally asymmetric simple exclusion process. The proof needs a lower tail estimate for the last-passage growth model associated with the exclusion process. A stronger estimate has been proved for the corresponding discrete time model, but not for the continuous time model we work with. So we take the needed estimate as a hypothesis. The process is assumed to be initially in local equilibrium with a slowly varying macroscopic profile. The macroscopic initial profile is smooth in a neighborhood of the origin where the second-class particle starts off, and the forward characteristic from the origin is not a shock. Given these assumptions, the result is that the typical fluctuation of the second-class particle is not of larger order than n2/3(log n)1/3, where n is the ratio of the macroscopic and microscopic space scales. The conjectured correct order should be n2/3. Landim et al. have proved a lower bound of order n5/8 for more general asymmetric exclusion processes in equilibrium. Fluctuations in the case of shocks and rarefaction fans are covered by earlier results of Ferrari–Fontes and Ferrari–Kipnis. Research partially supported by NSF grant DMS-0126775.