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. 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

Diffusive scaling limit of the Busemann process in Last Passage Percolation

“…A path on which the total weight is attained is called a geodesic from x to y. Lattice LPP belongs to a large family of LPP models which seem to share the same limiting behaviour of the fluctuations of their observables. Since the seminal work of Rost [36] where the first order of L(0, y) was established for y large in exponential LPP, much progress has been made in the study of these models [1,13,26,28,35].…”

“…An infinite geodesic is an up-right path on the random environment such that restricted to any two points on it is a geodesic. Questions such as the existence and uniqueness of such geodesics as well as other finer results have been studied in first passage percolation (FPP) by C. Newman and co-authors [24,25,27,29], while results of that flavour in LPP can be found in [5,7,8,19,26,32,38]. One of the main tools of studying infinite geodesics in random (directed) metrics models is the Busemann function -a random stationary real valued function defined on Z 2 × Z 2 which satisfies the cocycle property.…”

“…Originating in hyperbolic geometry, the Busemann functions were introduced first to FPP by Newman [29] and Hoffman [23] and later to LPP in [20]. They underlie many of the techniques used to study infinite geodesics [5,38] and have deep connections to the set of exceptional directions at which uniqueness of the geodesics fail [26]. Busemann functions are also important in the study of point-to-point geodesics [3], their coalescence [12] and the regularity of the passage time profile around a point [3,12,[32][33][34].…”

Invariant measures for stochastic conservation laws on the line

“…There is no shear invariance when the Hamiltonian is non-quadratic or when the diffusivity is non-constant. Similar problems in the discrete setting lacking shear-invariance were studied in [15,16]. There, special noise distributions yield an exactly-integrable structure that determines the shape function, compensating for the lack of shear-invariance.…”

“…There, special noise distributions yield an exactly-integrable structure that determines the shape function, compensating for the lack of shear-invariance. In addition, [15,16] proved various results about analogues of invariant measures conditional on certain understanding of the shape function.…”