PreprintCC BY 4.0arXiv
On the most likely geodesic in last passage percolation
Abstract
We consider the problem of identifying the paths most likely to occur as geodesics in last passage percolation. Heuristics suggest that these modal paths should be the extreme corner paths, going straight between the corners of the cube of accessible vertices. We identify three mechanisms which favour such corner paths, and show that they are more likely to appear than all but a vanishingly small portion of paths. We make more specific comparisons in exponential last passage percolation, where moderate deviation estimates may be used to show that corner paths are nearly modal in a precise sense. Finally, we show a form of monotonicity of geodesic probabilities in a special case and conjecture that this holds in general.
Related
- Geodesic traces in dynamical Brownian last passage percolation
- Invariance in half-space last passage percolation and directed polymers
- Stationary Half-space Geometric Last Passage Percolation without Random Shift
- Half-space last passage percolation with inhomogeneous first row
- Conditional Flat Exponential Directed Last Passage Percolation under a One-Point Upper Large Deviation Event
BibTeX
@misc{mckeown2026most,
title = {{On the most likely geodesic in last passage percolation}},
author = {Sam McKeown and Shuta Nakajima},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.11378},
eprint = {2608.11378},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.11378},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the most likely geodesic in last passage percolation
AU - Sam McKeown
AU - Shuta Nakajima
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2608.11378
UR - https://arxiv.org/abs/2608.11378
AB - We consider the problem of identifying the paths most likely to occur as geodesics in last passage percolation. Heuristics suggest that these modal paths should be the extreme corner paths, going straight between the corners of the cube of accessible vertices. We identify three mechanisms which favour such corner paths, and show that they are more likely to appear than all but a vanishingly small portion of paths. We make more specific comparisons in exponential last passage percolation, where moderate deviation estimates may be used to show that corner paths are nearly modal in a precise sense. Finally, we show a form of monotonicity of geodesic probabilities in a special case and conjecture that this holds in general.
ER -