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Stationary diffusion on nonnegatively curved graphs

Primarily AI-generated textHuman understanding: some partsmath.PR — Probabilitymath.CO — Combinatorics

Contributed by Matěj Kripner ↗, Václav Rozhoň ↗, Robert Šámal ↗, Martin Šindelář, Adrián Zámečník

Version 1 / Nov 04, 2026 / CC BY 4.0

Abstract

Lazy random walk on a bounded-degree graph of nonnegative Ollivier–Ricci curvature has diffusive second moment and Gaussian upper tails when started in stationarity. If the maximum degree is DD, then Ed(X0,Xn)2≤4Dn\mathbb E d(X_0,X_n)^2\le4Dn, with optimal order of dependence on DD. The results also hold for stationary random rooted infinite graphs. The proof combines known heat-kernel total-variation decay with a reversible energy identity for the conditional mean displacement. Its exponential form controls variation between starting vertices; martingale concentration then controls the walk itself.

Provenance statement

This work was produced within the Bolzano project (https://bolzano.app), using AI systems for mathematical research and manuscript preparation. The listed authors contributed to the project and the preparation of this release. The manuscript received only limited human checking.

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Bolzano team
Bolzano

References

  1. C. Brena, T. Hutchcroft and F. Münch, Transitive graphs of non-negative Ollivier--Ricci curvature have polynomial growth , arXiv:2512.03968v2 (2026), https://arxiv.org/abs/2512.03968v2.arXiv
  2. F. Münch, Non-negative Ollivier curvature on graphs, reverse Poincaré inequality, Buser inequality, Liouville property, Harnack inequality and eigenvalue estimates , J. Math. Pures Appl. 170 (2023), 231--257, https://doi.org/10.1016/j.matpur.2022.12.007; preprint https://arxiv.org/abs/1907.13514v1.DOI
  3. J. Salez, Sparse expanders have negative curvature , Geom. Funct. Anal. 32 (2022), 1486--1513; preprint numbering from https://arxiv.org/abs/2101.08242v2.link
  4. I. Benjamini and N. Curien, Ergodic theory on stationary random graphs , Electron. J. Probab. 17 (2012), no. 93, 1--20, https://doi.org/10.1214/EJP.v17-2401.DOI
  5. F. Münch and J. Salez, Mixing time and expansion of non-negatively curved Markov chains , J. Éc. polytech. Math. 10 (2023), 575--590, https://doi.org/10.5802/jep.226.DOI
  6. Y. Ollivier, Ricci curvature of Markov chains on metric spaces , J. Funct. Anal. 256 (2009), 810--864, https://doi.org/10.1016/j.jfa.2008.11.001.DOI
  7. Y. Ollivier, A survey of Ricci curvature for metric spaces and Markov chains , in Probabilistic Approach to Geometry, Adv. Stud. Pure Math. 57 , Math. Soc. Japan, Tokyo, 2010, 343--381, https://doi.org/10.2969/aspm/05710343.DOI
  8. Y. Lin, L. Lu and S.-T. Yau, Ricci curvature of graphs , Tohoku Math. J. 63 (2011), 605--627, https://doi.org/10.2748/tmj/1325886283.DOI
  9. R. Bubley and M. Dyer, Path coupling: a technique for proving rapid mixing in Markov chains , in Proc. 38th Annual Symposium on Foundations of Computer Science (FOCS), IEEE, 1997, 223--231, https://doi.org/10.1109/SFCS.1997.646111.DOI
  10. T. K. Carne, A transmutation formula for Markov chains , Bull. Sci. Math. (2) 109 (1985), 399--405.
Show all 14 referencesShow fewer references
  1. J. R. Lee and Y. Peres, Harmonic maps on amenable groups and a diffusive lower bound for random walks , Ann. Probab. 41 (2013), 3392--3419, https://doi.org/10.1214/12-AOP779.DOI
  2. T. Hutchcroft and I. M. Lopez, A relation between isoperimetry and total variation decay with applications to graphs of non-negative Ollivier--Ricci curvature , arXiv:2411.04988v1 (2024), https://arxiv.org/abs/2411.04988v1.arXiv
  3. C. Zhou, Pointwise subexponential growth and near-diffusive displacement on bounded-degree graphs with non-negative Ollivier--Ricci curvature , arXiv:2607.27162v1 (2026), https://arxiv.org/abs/2607.27162v1.arXiv
  4. C. Brena and E. Bruè, Ollivier--Ricci curvature on groups of polynomial growth , arXiv:2608.14232v1 (2026), https://arxiv.org/abs/2608.14232v1.arXiv

Version history

  1. v1Submitted by Václav RozhoňInitial depositCurrentNov 03, 2026