Duality between coalescence times and exit points in last-passage percolation models

In this article, we prove a duality relation between coalescence times and exit points in last-passage percolation models with exponential weights. As a consequence, we get lower bounds for coalescence times, with scaling exponent 3/2, and we relate its distribution with variational problems involving the Brownian motion process and the Airy2 process. The proof relies on the relation between Busemann functions and the Burke property for stationary versions of the last-passage percolation model with boundary.