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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.

We investigate large deviations of the free energy in the O'Connell-Yor polymer

We consider planar directed last-passage percolation on the square lattice with general i.i.d. weights and study the geometry of the full set of semi-infinite geodesics in a typical realization of the random environment. The structure of the geodesics is studied through the properties of the Busemann functions viewed as a stochastic process indexed by the asymptotic direction. In the exactly solvable exponential model we give the first complete characterization of the uniqueness and coalescence structure of the entire family of semi-infinite geodesics for any model of this type. Our results are further connected to the ergodic program for random Hamilton-Jacobi equations and in particular to infinite shocks. In the exponential model we compute some statistics of shocks, where we discover an unexpected connection to simple symmetric random walk. Contents 1. Introduction 1.1. Random growth models 1.2. Geodesics 1.3. Busemann measures 1.4. Shocks 1.5. Organization of the paper 1.6. Setting and notation 2. Preliminaries on last-passage percolation 2.1. The shape function of directed last-passage percolation 2.2. The Busemann process 2.3. Semi-infinite geodesics 2.4. Non-uniqueness of directed semi-infinite geodesics 3. Busemann measures, exceptional directions, and coalescence points 3.1. Coalescence and the Busemann measures 3.2. Exponential case 4. Last-passage percolation as a dynamical system 4.1. Discrete Hamilton-Jacobi equations 4.2. Webs of shocks 4.3. Flow of Busemann measure 5. Shock statistics in the exponential model 6. Busemann measures: proofs 7. Webs of shocks: proofs 7.1. Shock points and graphs 7.2. Density of shocks on the lattice 7.3. Flow of Busemann measures 8. Shocks in the exponential model: proofs 8.1. Palm kernels 8.2. Statistics of shocks

No abstract