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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.
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This content will become publicly available on December 1, 2026
Anomalous Geodesics in the Inhomogeneous Corner Growth Model
Abstract We study Busemann functions, semi-infinite geodesics, and competition interfaces in the exactly solvable last-passage percolation with inhomogeneous exponential weights. New phenomena concerning geodesics arise due to inhomogeneity. These include novel Busemann functions associated with flat regions of the limit shape and thin rectangles, semi-infinite geodesics with intervals of asymptotic directions, non-trivial axis-directed geodesics, intervals with no geodesic directions, and isolated geodesic directions. We further observe a new dichotomy for competition interfaces and second-class customers in a series of memoryless continuous-time queues with inhomogeneous service rates: a second-class customer either becomes trapped or proceeds through the service stations at strictly positive speed.
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- Award ID(s):
- 2448375
- PAR ID:
- 10675249
- Publisher / Repository:
- Springer
- Date Published:
- Journal Name:
- Communications in Mathematical Physics
- Volume:
- 406
- Issue:
- 12
- ISSN:
- 0010-3616
- Page Range / eLocation ID:
- 1-65
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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