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A non-trivial dynamics on the directed landscape

  • Manan Bhatia
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Abstract

We prove the existence of a non-trivial continuous stationary reversible dynamics on the directed landscape as a subsequential scaling limit of Brownian last passage percolation under the Ornstein--Uhlenbeck dynamics. Strong passage-time stability estimates from the companion paper Bhatia '26, together with static endpoint regularity, give tightness at the critical dynamical scale n−1/3. To establish non-triviality of the subsequential limiting dynamics, we show that, in dynamical BLPP, the fractional overlap between geodesics at times zero and a n−1/3 tends to one as first n→∞ and then a↓0. The proof refines the excursion argument of Ganguly-Hammond '24, using strong passage-time stability and the fact that a fixed dynamical time is increasingly subcritical at finer spatial scales.

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Cite

BibTeXRIS
BibTeX
@misc{bhatia2026nontrivial,
  title = {{A non-trivial dynamics on the directed landscape}},
  author = {Manan Bhatia},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.35124},
  eprint = {2609.35124},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.35124},
}
RIS
TY  - UNPB
TI  - A non-trivial dynamics on the directed landscape
AU  - Manan Bhatia
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.35124
UR  - https://arxiv.org/abs/2609.35124
AB  - We prove the existence of a non-trivial continuous stationary reversible dynamics on the directed landscape as a subsequential scaling limit of Brownian last passage percolation under the Ornstein--Uhlenbeck dynamics. Strong passage-time stability estimates from the companion paper Bhatia '26, together with static endpoint regularity, give tightness at the critical dynamical scale n^−1/3. To establish non-triviality of the subsequential limiting dynamics, we show that, in dynamical BLPP, the fractional overlap between geodesics at times zero and a n^−1/3 tends to one as first n→∞ and then a↓0. The proof refines the excursion argument of Ganguly-Hammond '24, using strong passage-time stability and the fact that a fixed dynamical time is increasingly subcritical at finer spatial scales.
ER  -