## Empirical measures, geodesic lengths, and a variational formula in first-passage percolation

### With Erik Bates, University of Wisconsin-Madison

# Empirical measures, geodesic lengths, and a variational formula in first-passage percolation

We consider the standard first-passage percolation model on Z^d, in which each edge is assigned an i.i.d. nonnegative weight, and the passage time between any two points is the smallest total weight of a nearest-neighbor path between them. This induces a random “disordered” geometry on the lattice. Our primary interest is in the empirical measures of edge-weights observed along geodesics in this geometry, say from 0 to [nxi], where xi is a fixed unit vector. For various dense families of edge-weight distributions, we prove that these measures converge weakly to a deterministic limit as n tends to infinity. The key tool is a new variational formula for the time constant. In this talk, I will derive this formula and discuss its implications for the convergence of both empirical measures and lengths of geodesics.

- Speaker: Erik Bates, University of Wisconsin-Madison
- Tuesday 02 November 2021, 14:00–15:00
- Venue: MR12 Centre for Mathematical Sciences.
- Series: Probability; organiser: Sourav Sarkar.