Ollivier-Ricci curvature for causal sets

@misc{barton2026ollivierricci,
  title = {{Ollivier-Ricci curvature for causal sets}},
  author = {Joe Barton and Samuël Borza and Jona Röhrig},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.04910},
  eprint = {2606.04910},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.04910},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Ollivier-Ricci curvature for causal sets
AU  - Joe Barton
AU  - Samuël Borza
AU  - Jona Röhrig
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2606.04910
UR  - https://arxiv.org/abs/2606.04910
AB  - We introduce a novel notion of Ollivier-Ricci curvature for causal sets using Lorentzian optimal transport. The construction is motivated by a new Lorentzian asymptotic formula of independent interest, which recovers timelike Ricci curvature, up to higher-order terms, from the transport distance between probability measures on nearby causal diamonds. Passing to the discrete setting, this leads to a mesoscopic notion of Ricci curvature defined along maximal chains and built from probability measures on causal diamonds. We study several variants, including idle and Lin-Lu-Yau type curvatures, prove local-to-global propagation results and timelike Bonnet-Myers theorems, and compute the curvature for a range of explicit causal sets. We design high-density Poisson sprinkling numerical experiments recovering the expected constant-curvature signatures of Minkowski, de Sitter, and anti-de Sitter space. These results provide evidence that the construction captures timelike Ricci curvature from order-theoretic data.
ER  -