PreprintCC BY 4.0arXiv
The starting point of a directed geodesic is special
Abstract
For any t∈ [0,1/2), consider the process ηt:[0,1/2]↦ℝ defined by ηt(s)=Π(t+s)−Π(t), where Π is the directed geodesic from (0,0) to (0,1) in the directed landscape. Let 0≤ t<u<1/2. We show that the laws of ηt and ηu are mutually absolutely continuous if and only if t>0. This proves Conjecture 14.5 in Dauvergne, Ortmann, and Virág, 2022 in the affirmative.
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BibTeX
@misc{tassopoulos2026starting,
title = {{The starting point of a directed geodesic is special}},
author = {Pantelis Tassopoulos and Sourav Sarkar},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39675},
eprint = {2609.39675},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39675},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The starting point of a directed geodesic is special
AU - Pantelis Tassopoulos
AU - Sourav Sarkar
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39675
UR - https://arxiv.org/abs/2609.39675
AB - For any t∈ [0,1/2), consider the process η_t:[0,1/2]↦ℝ defined by η_t(s)=Π(t+s)−Π(t), where Π is the directed geodesic from (0,0) to (0,1) in the directed landscape. Let 0≤ t<u<1/2. We show that the laws of η_t and η_u are mutually absolutely continuous if and only if t>0. This proves Conjecture 14.5 in Dauvergne, Ortmann, and Virág, 2022 in the affirmative.
ER -