2017/04/25 by Riddhipratim Basu, Basu, Riddhipratim, Sourav Sarkar +3
Mathematics · #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Statistical Mechanics (cond-mat.stat-mech) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1704.07799
openalex publication_date 2017/04/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Totally Asymmetric Simple Exclusion Process (TASEP) on ℤ is one of the classical exactly solvable models in the KPZ universality class. We study the "slow bond" model, where TASEP on ℤ is imputed with a slow bond at the origin. The slow bond increases the particle density immediately to its left and decreases the particle density immediately to its right. Whether or not this effect is detectable in the macroscopic current started from the step initial condition has attracted much interest over the years and this question was settled recently (Basu, Sidoravicius, Sly (2014)) by showing that the current is reduced even for arbitrarily small strength of the defect. Following non-rigorous physics arguments (Janowsky, Lebowitz (1992, 1994)) and some unpublished works by Bramson, a conjectural description of properties of invariant measures of TASEP with a slow bond at the origin was provided in Liggett's 1999 book. We establish Liggett's conjectures and in particular show that TASEP with a slow bond at the origin, started from step initial condition, converges in law to an invariant measure that is asymptotically close to product measures with different densities far away from the origin towards left and right. Our proof exploits the correspondence between TASEP and the last passage percolation on ℤ2 with exponential weights and uses the understanding of geometry of maximal paths in those models.