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The talk has two parts. In the first part we speak on the modulus of continuity in Poissonian last passage percolation, a model lying in the KPZ universality class. The geodesics and their energy in this model can be scaled so that transformed geodesics cross unit distance and have fluctuations and scaled energy of unit order, and we refer to such scaled geodesics as polymers and their scaled energies as weights. Polymers may be viewed as random functions of the vertical coordinate and, when they are, we show that they have modulus of continuity whose order is at most $t^{2/3} (\log t^{ −1})^{1/3}$ . Regarding the orthogonal direction, in which growth occurs, we show that, when one endpoint of the polymer is fixed at $(0, 0)$ and the other is varied vertically over $(0, z), z \in [1, 2]$, the resulting random weight profile has sharp modulus of continuity of order $t^{1/3} (\log t^{ −1})^{2/3}$ .

In the second part we speak on the “slow bond” model, where Totally Asymmetric Simple Exclusion Process (TASEP) on $\mathbb{Z}$ (another classical exactly solvable model in the KPZ universality class) is imputed with a slow bond at the origin. Whether or not this effect is detectable in the macroscopic current started from the step initial condition was settled recently in (Basu, Sidoravicius, Sly (2014)) where it was shown that the current is reduced even for arbitrarily small strength of the defect. A conjectural description of properties of invariant measures of TASEP with a slow bond at the origin was provided by Liggett's 1999 book . We establish Liggett’s conjectures and in particular show that TASEP with a slow bond at the origin, starting 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.

The talk is based on joint works with Alan Hammond, Riddhipratim Basu and Allan Sly.