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.