In exponential directed last-passage percolation, each vertex in Z^2 is assigned an i.i.d. exponential weight, and the geodesic between a pair of vertices refers to the up-right path connecting them, with the maximum total weight along the path. It is a natural question to ask what a geodesic looks like locally, and how weights on and nearby the geodesic behave. In this talk, I will present some new results on this. We show convergence of the distribution of the 'environment' as seen from a typical point along the geodesic, and convergence of the corresponding empirical measure, as the geodesic length goes to infinity. In addition, we obtain an explicit description of the limiting environment, which depends on the direction of the geodesic. This in principle enables one to compute all the local statistics of the geodesic, and I will talk about some surprising and interesting examples.
This is based on joint work with James Martin and Allan Sly.