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

  • Christopher Janjigian

    Purdue University, West Lafayette, USA
  • Firas Rassoul-Agha

    University of Utah, Salt Lake City, USA
  • Timo Seppäläinen

    University of Wisconsin-Madison, USA
Geometry of geodesics through Busemann measures in directed last-passage percolation cover
Download PDF

This article is published open access under our Subscribe to Open model.

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.

Cite this article

Christopher Janjigian, Firas Rassoul-Agha, Timo Seppäläinen, Geometry of geodesics through Busemann measures in directed last-passage percolation. J. Eur. Math. Soc. 25 (2023), no. 7, pp. 2573–2639

DOI 10.4171/JEMS/1246