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We develop a new probabilistic method for deriving deviation estimates indirected planar polymer and percolation models. The key estimates are for exitpoints of geodesics as they cross transversal down-right boundaries. Thesebounds are of optimal cubic-exponential order. We derive them in the context oflast-passage percolation with exponential weights with near-stationary boundaryconditions. As a result, the probabilistic coupling method is empowered totreat a variety of problems optimally, which could previously be achieved onlyvia inputs from integrable probability. As applications in the bulk setting, weobtain upper bounds of cubic-exponential order for transversal fluctuations ofgeodesics, and cube-root upper bounds with a logarithmic correction fordistributional Busemann limits and competition interface limits. Several otherapplications are already in the literature.

Optimal-order exit point bounds in exponential last-passage percolation via the coupling technique