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In the slow bond problem the rate of a single edge in the Totally AsymmetricSimple Exclusion Process (TASEP) is reduced from 1 to $1-\varepsilon$ for somesmall $\varepsilon>0$. Janowsky and Lebowitz posed the well-known question ofwhether such very small perturbations could affect the macroscopic current.Different groups of physicists, using a range of heuristics and numericalsimulations reached opposing conclusions on whether the critical value of$\varepsilon$ is 0. This was ultimately resolved rigorously inBasu-Sidoravicius-Sly which established that $\varepsilon_c=0$.  Here we study the effect of the current as $\varepsilon$ tends to 0 and indoing so explain why it was so challenging to predict on the basis of numericalsimulations. In particular we show that the current has an infinite order phasetransition at 0, with the effect of the perturbation tending to 0 faster thanany polynomial. Our proof focuses on the Last Passage Percolation formulationof TASEP where a slow bond corresponds to reinforcing the diagonal. We give amultiscale analysis to show that when $\varepsilon$ is small the effect ofreinforcement remains small compared to the difference between optimal and nearoptimal geodesics. Since geodesics can be perturbed on many different scales,we inductively bound the tails of the effect of reinforcement by controllingthe number of near optimal geodesics and giving new tail estimates for thelocal time of (near) geodesics along the diagonal.

Infinite order phase transition in the slow bond TASEP