Geometry of geodesics through Busemann measures in directed last-passage percolation
Geometry of geodesics through Busemann measures in directed last-passage percolation
复制标题
DOI：
10.4171/jems/1246
复制
发表时间：
2019-08
影响因子：
2.6
通讯作者：
Christopher Janjigian;F. Rassoul-Agha;T. Seppalainen
中科院分区：
文献类型：
--
作者：
Christopher Janjigian;F. Rassoul-Agha;T. Seppalainen
我们考虑了具有一般I.I.D.的正方形格子上的平面有向最后通道渗流。并研究了半无限测地线的全集几何在典型随机环境下的实现。通过将Busemann函数视为以渐近方向为指标的随机过程的性质，研究了测地线的结构。在精确可解的指数模型中，我们首次完整地刻画了这类模型的整个半无限测地线族的唯一性和聚结结构。我们的结果进一步与随机Hamilton-Jacobi方程的遍历程序和稳定性有关。在指数模型中，我们计算了一些不稳定位置的统计量，其中我们发现了与简单对称随机游走的意外联系。
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 and stability properties of random Hamilton-Jacobi equations. In the exponential model we compute some statistics of locations of instability, where we discover an unexpected connection to simple symmetric random walk.