  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, so we take the needed estimate as a hypothesis. The
process is 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 n^{2/3}(log n)^{1/3}, where n is the ratio of the macroscopic
and microscopic space scales. The conjectured correct order should be n^{2/3}.
Landim et al. have proved a lower bound of order n^{5/8} 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.
