Search for: All records

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.