2002/05/15 by Timo Seppäläinen, Timo Seppalainen, Seppalainen, Timo
Mathematics · #60K35 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #math.PR #msc:60K35
paper · pdf · doi:10.48550/arxiv.math/0205176
15 pages
arxiv created 2002/05/15 · openalex publication_date 2002/05/15 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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.