Abstract

Whether a localized microscopic defect will affect the macroscopic behaviour of a system is a fundamental question in statistical mechanics. For the Totally Asymmetric Simple Exclusion Process (TASEP) on $\mathbb{Z}$, this problem was originally posed by Janowsky and Lebowitz and became famous as the ``slow-bond” problem. If the wait time of jump for a particle at the origin is increased from an exponential with rate $1$ to that with rate $1-\epsilon$, is this effect detectable in the macroscopic current? Different groups of physicists, using a range of heuristics and numerical simulations, reached opposing conclusions on whether the critical value of $\epsilon$ is $0$. This was ultimately resolved rigorously in Basu-Sidoravicius-Sly which established that $\epsilon_c=0$. In this talk, we will study the effect of the current as $\epsilon$ tends to $0$ and in doing so explain why it was so challenging to predict on the basis of numerical simulations. In particular, we show that with the effect of the perturbation tends to 0 faster than any polynomial. Our proof focuses on the Last Passage Percolation formulation of TASEP . The talk is based on joint works with Allan Sly and Lingfu Zhang.