Geometry of geodesics through Busemann measures in directed last-passage percolation

Abstract

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. Our results are further connected to the ergodic program for and stability properties of random Hamilton-Jacobi equations. In the exactly solvable exponential model, our results specialize to 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. Furthermore, we compute statistics of locations of instability, where we discover an unexpected connection to simple symmetric random walk.

arXiv commentFixed a few typos. Page layout changed. 59 pages, 13 figures

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• Invariant measures for stochastic conservation laws on the line
• Jointly invariant measures for the Kardar-Parisi-Zhang Equation
• Coalescence estimates for the corner growth model with exponential weights
• Jointly stationary solutions of periodic Burgers flow
• Intertwining the Busemann process of the directed polymer model