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A well-known question in planar first-passage percolation concerns the convergence of the empirical distribution of weights as seen along geodesics. We demonstrate this convergence for an explicit model, directed last-passage percolation on\mathbb{Z}^{2}with i.i.d. exponential weights, and provide explicit formulae for the limiting distributions, which depend on the asymptotic direction. For example, for geodesics in the direction of the diagonal, the limiting weight distribution has density(1/4+x/2+x^{2}/8)e^{-x}, and so is a mixture of Gamma(1,1), Gamma(2,1), and Gamma(3,1) distributions with weights1/4,1/2, and1/4respectively. More generally, we study the local environment as seen from vertices along geodesics (including information about the shape of the path and about the weights on and off the path in a local neighborhood). We consider finite geodesics from(0,0)ton\boldsymbol{\rho}for some vector\boldsymbol{\rho}in the first quadrant, in the limit asn\to\infty, as well as semi-infinite geodesics in direction\boldsymbol{\rho}. We show almost sure convergence of the empirical distributions of the environments along these geodesics, as well as convergence of the distributions of the environment around a typical point in these geodesics, to the same limiting distribution, for which we give an explicit description.We make extensive use of a correspondence with TASEP as seen from an isolated second-class particle for which we prove new results concerning ergodicity and convergence to equilibrium. Our analysis relies on geometric arguments involving estimates for last-passage times, available from the integrable probability literature. 

We study maximal length collections of disjoint paths, or ‘disjoint optimizers’, in the directed landscape. We show that disjoint optimizers always exist, and that their lengths can be used to construct an extended directed landscape. The extended directed landscape can be built from an independent collection of extended Airy sheets, which we define from the parabolic Airy line ensemble. We show that the extended directed landscape and disjoint optimizers are scaling limits of the corresponding objects in Brownian last passage percolation (LPP). As two consequences of this work, we show that one direction of the Robinson-Schensted-Knuth bijection passes to the KPZ limit, and we find a criterion for geodesic disjointness in the directed landscape that uses only a single parabolic Airy line ensemble. The proofs rely on a new notion of multi-point LPP across the parabolic Airy line ensemble, combinatorial properties of multi-point LPP, and probabilistic resampling ideas. 

Abstract In the slow bond problem the rate of a single edge in the Totally Asymmetric Simple Exclusion Process (TASEP) is reduced from 1 to for some small . Janowsky and Lebowitz  posed the well‐known question of whether such very small perturbations could affect the macroscopic current. Different groups of physicists, using a range of heuristics and numerical simulations reached opposing conclusions on whether the critical value of is 0. This was ultimately resolved rigorously in Basu‐Sidoravicius‐Sly which established that . Here we study the effect of the current as 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 the current has an infinite order phase transition at 0, with the effect of the perturbation tending to 0 faster than any polynomial. Our proof focuses on the Last Passage Percolation formulation of TASEP where a slow bond corresponds to reinforcing the diagonal. We give a multiscale analysis to show that when is small the effect of reinforcement remains small compared to the difference between optimal and near optimal geodesics. Since geodesics can be perturbed on many different scales, we inductively bound the tails of the effect of reinforcement by controlling the number of near optimal geodesics and giving new tail estimates for the local time of (near) geodesics along the diagonal. 

Abstract We study uniformly random lozenge tilings of general simply connected polygons. Under a technical assumption that is presumably generic with respect to polygon shapes, we show that the local statistics around a cusp point of the arctic curve converge to the Pearcey process. This verifies the widely predicted universality of edge statistics in the cusp case. Together with the smooth and tangent cases proved by Aggarwal‐Huang and Aggarwal‐Gorin, these are believed to be the three types of edge statistics that can arise in a generic polygon. Our proof is via a local coupling of the random tiling with nonintersecting Bernoulli random walks (NBRW). To leverage this coupling, we establish an optimal concentration estimate for the tiling height function around the cusp. As another step and also a result of potential independent interest, we show that the local statistics of NBRW around a cusp converge to the Pearcey process when the initial configuration consists of two parts with proper density growth, via careful asymptotic analysis of the determinantal formulas.