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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

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E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Electron. AbstractIn this article, we study weighted particle representations for a class of stochastic partial differential equations (SPDE) with Dirichlet boundary conditions. The locations and weights of the particles satisfy an infinite system of stochastic differential equations. The locations are given by independent, stationary reflecting diffusions in a bounded domain, and the weights evolve according to an infinite system of stochastic differential equations driven by a common Gaussian white noise W which is the stochastic input for the SPDE. The weights interact through V , the associated weighted empirical measure, which gives the solution of the SPDE. When a particle hits the boundary its weight jumps to a value given by a function of the location of the particle on the boundary. This function determines the boundary condition for the SPDE. We show existence and uniqueness of a solution of the infinite system of stochastic differential equations giving the locations and weights of the particles and derive two weak forms for the corresponding SPDE depending on the choice of test functions. The weighted empirical measure V is the unique solution for each of the nonlinear stochastic partial differential equations. The work is motivated by and applied to the stochastic Allen-Cahn equation and extends the earlier of work of Kurtz and Xiong in [14,15].

We prove a shape theorem and derive a variational formula for the limiting quenched Lyapunov exponent and the Green's function of random walk in a random potential on a square lattice of arbitrary dimension and with an arbitrary finite set of steps. The potential is a function of a stationary environment and the step of the walk. This potential is subject to a moment assumption whose strictness is tied to the mixing of the environment. Our setting includes directed and undirected polymers, random walk in static and dynamic random environment, and, when the temperature is taken to zero, our results also give a shape theorem and a variational formula for the time constant of both site and edge directed last-passage percolation and standard first-passage percolation.