  We develop a new probabilistic method for deriving deviation estimates in
directed planar polymer and percolation models. The key estimates are for exit
points of geodesics as they cross transversal down-right boundaries. These
bounds are of optimal cubic-exponential order. We derive them in the context of
last-passage percolation with exponential weights with near-stationary boundary
conditions. As a result, the probabilistic coupling method is empowered to
treat a variety of problems optimally, which could previously be achieved only
via inputs from integrable probability. As applications in the bulk setting, we
obtain upper bounds of cubic-exponential order for transversal fluctuations of
geodesics, and cube-root upper bounds with a logarithmic correction for
distributional Busemann limits and competition interface limits. Several other
applications are already in the literature.
