In exponential Last-Passage Percolation, each vertex in the 2D lattice 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. This model was first introduced to model fluid flow through a random medium. It is also a central model in the KPZ universality class and related to various natural processes.

A classical question asks what a geodesic looks like locally, and how weights on and nearby the geodesic behave. In this talk, I will present new results on the convergence of the ‘environment’ as seen from a typical point along the geodesic, and convergence of the corresponding empirical measure. In addition, we obtain an explicit description of the limiting ‘environment’. 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.