Ricci curvature of semigroup walks on graphs and generator representations

Abstract

We present an operator-theoretic framework for the Ollivier--Ricci curvature of (one-parameter families of) random walks generated by positive semigroups on graphs. On a metrically complete graph (G,d) that is d-locally finite, we introduce a class of admissible semigroup random walks whose infinitesimal first-order approximation is itself a random walk. We prove that the Ollivier--Ricci curvature of an admissible semigroup random walk is well-defined and completely determined by its first-order approximation. We then establish a variational representation of the curvature of random walks in terms of a notion of the initial-time generator. This representation applies in particular to admissible semigroup random walks. Finally, we use this representation to define an operator-theoretic Ollivier--Ricci curvature for operators acting on spaces of functions over general metric spaces and establish sufficient conditions for well-definedness. This framework contains, in particular, Markovian random walks associated with discrete Laplacians.

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BibTeX RIS

BibTeX

@misc{lakzian2026ricci,
  title = {{Ricci curvature of semigroup walks on graphs and generator representations}},
  author = {Sajjad Lakzian},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.40301},
  eprint = {2610.40301},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.40301},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Ricci curvature of semigroup walks on graphs and generator representations
AU  - Sajjad Lakzian
PY  - 2026
DA  - 2026/10/29
PB  - arXiv
DO  - 10.48550/arxiv.2610.40301
UR  - https://arxiv.org/abs/2610.40301
AB  - We present an operator-theoretic framework for the Ollivier--Ricci curvature of (one-parameter families of) random walks generated by positive semigroups on graphs. On a metrically complete graph (G,d) that is d-locally finite, we introduce a class of admissible semigroup random walks whose infinitesimal first-order approximation is itself a random walk. We prove that the Ollivier--Ricci curvature of an admissible semigroup random walk is well-defined and completely determined by its first-order approximation. We then establish a variational representation of the curvature of random walks in terms of a notion of the initial-time generator. This representation applies in particular to admissible semigroup random walks. Finally, we use this representation to define an operator-theoretic Ollivier--Ricci curvature for operators acting on spaces of functions over general metric spaces and establish sufficient conditions for well-definedness. This framework contains, in particular, Markovian random walks associated with discrete Laplacians.
ER  -

Abstract

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