Document Open Access Logo

An Improved Trickle down Theorem for Partite Complexes

Authors Dorna Abdolazimi, Shayan Oveis Gharan

PDF
Thumbnail PDF

File

LIPIcs.CCC.2023.10.pdf
  • Filesize: 0.7 MB
  • 16 pages

Document Identifiers

Author Details

Dorna Abdolazimi
  • University of Washington, Seattle, WA, USA
Shayan Oveis Gharan
  • University of Washington, Seattle, WA, USA

Funding

  • Abdolazimi, Dorna: Research supported by NSF grant CCF-1907845 and Air Force Office of Scientific Research grant FA9550-20-1-0212.
  • Oveis Gharan, Shayan: Research supported by Air Force Office of Scientific Research grant FA9550-20-1-0212, Simons Investigator grant.

Acknowledgements

The discussion that initiated this work took place at the DIMACS Workshop on Entropy and Maximization. We would like to thank the DIMACS center and the workhop organizers for making this happen. In particular, we would like to thank Tali Kaufman for raising the question of an improved trickle down theorem for sparse simplical complexes in that workshop. We also would like to thank Ryan O’Donnell and Kevin Pratt for helpful discussions on high dimensional expanders based on Chevalley groups.

Cite AsGet BibTex

Dorna Abdolazimi and Shayan Oveis Gharan. An Improved Trickle down Theorem for Partite Complexes. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 10:1-10:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.CCC.2023.10

Abstract

We prove a strengthening of the trickle down theorem for partite complexes. Given a (d+1)-partite d-dimensional simplicial complex, we show that if "on average" the links of faces of co-dimension 2 are (1-δ)/d-(one-sided) spectral expanders, then the link of any face of co-dimension k is an O((1-δ)/(kδ))-(one-sided) spectral expander, for all 3 ≤ k ≤ d+1. For an application, using our theorem as a black-box, we show that links of faces of co-dimension k in recent constructions of bounded degree high dimensional expanders have spectral expansion at most O(1/k) fraction of the spectral expansion of the links of the worst faces of co-dimension 2.

Subject Classification

ACM Subject Classification
  • Theory of computation → Expander graphs and randomness extractors
  • Theory of computation → Error-correcting codes
  • Theory of computation → Random walks and Markov chains
Keywords
  • Simplicial complexes
  • High dimensional expanders
  • Trickle down theorem
  • Bounded degree high dimensional expanders
  • Locally testable codes
  • Random walks

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Dorna Abdolazimi, Kuikui Liu, and Shayan Oveis Gharan. A matrix trickle-down theorem on simplicial complexes and applications to sampling colorings. In FOCS, pages 161-172. IEEE, 2021. Google Scholar
  2. Vedat Levi Alev, Fernando Granha Jeronimo, and Madhur Tulsiani. Approximating constraint satisfaction problems on high-dimensional expanders. In David Zuckerman, editor, FOCS, pages 180-201. IEEE Computer Society, 2019. Google Scholar
  3. Nima Anari, Kuikui Liu, and Shayan Oveis Gharan. Spectral independence in high-dimensional expanders and applications to the hardcore model. SIAM Journal on Computing, FOCS20:1-37, 2020. URL: https://doi.org/10.1137/20M1367696.
  4. Nima Anari, Kuikui Liu, Shayan Oveis Gharan, and Cynthia Vinzant. Log-concave polynomials II: high-dimensional walks and an FPRAS for counting bases of a matroid. In Moses Charikar and Edith Cohen, editors, STOC, pages 1-12. ACM, 2019. Google Scholar
  5. Roger Carter. Simple groups of Lie type. John Wiley & Sons, 1989. Google Scholar
  6. Zongchen Chen, Kuikui Liu, and Eric Vigoda. Optimal mixing of glauber dynamics: entropy factorization via high-dimensional expansion. In Samir Khuller and Virginia Vassilevska Williams, editors, STOC, pages 1537-1550. ACM, 2021. Google Scholar
  7. Yotam Dikstein and Irit Dinur. Agreement testing theorems on layered set systems. In David Zuckerman, editor, FOCS, pages 1495-1524. IEEE Computer Society, 2019. URL: https://doi.org/10.1109/FOCS.2019.00088.
  8. I. Dinur and T. Kaufman. High dimensional expanders imply agreement expanders. In FOCS, pages 974-985, 2017. Google Scholar
  9. Irit Dinur, Shai Evra, Ron Livne, Alexander Lubotzky, and Shahar Mozes. Locally testable codes with constant rate, distance, and locality. In Stefano Leonardi and Anupam Gupta, editors, STOC, pages 357-374. ACM, 2022. Google Scholar
  10. Irit Dinur, Yuval Filmus, Prahladh Harsha, and Madhur Tulsiani. Explicit sos lower bounds from high-dimensional expanders. In James R. Lee, editor, 12th Innovations in Theoretical Computer Science Conference, ITCS 2021, January 6-8, 2021, Virtual Conference, volume 185 of LIPIcs, pages 38:1-38:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  11. Irit Dinur, Prahladh Harsha, Tali Kaufman, Inbal Livni Navon, and Amnon Ta-Shma. List-decoding with double samplers. SIAM J. Comput., 50(2):301-349, 2021. Google Scholar
  12. Shai Evra and Tali Kaufman. Bounded degree cosystolic expanders of every dimension. In Proceedings of the forty-eighth annual ACM symposium on Theory of Computing, pages 36-48, 2016. Google Scholar
  13. Tali Kaufman and Izhar Oppenheim. Construction of new local spectral high dimensional expanders. In Ilias Diakonikolas, David Kempe, and Monika Henzinger, editors, STOC, pages 773-786. ACM, 2018. URL: https://doi.org/10.1145/3188745.3188782.
  14. Alexander Lubotzky, Beth Samuels, and Uzi Vishne. Explicit constructions of ramanujan complexes of type ad. European Journal of Combinatorics, 26(6):965-993, 2005. Google Scholar
  15. Alexander Lubotzky, Beth Samuels, and Uzi Vishne. Explicit constructions of ramanujan complexes of type Ãd. European Journal of Combinatorics, 26(6):965-993, 2005. Combinatorics and Representation Theory. URL: https://doi.org/10.1016/j.ejc.2004.06.007.
  16. Alexander Lubotzky, Beth Samuels, and Uzi Vishne. Ramanujan complexes of type a d. Israel journal of Mathematics, 149:267-299, 2005. Google Scholar
  17. Ryan O'Donnell and Kevin Pratt. High-Dimensional Expanders from Chevalley Groups. In Shachar Lovett, editor, 37th Computational Complexity Conference (CCC 2022), volume 234 of Leibniz International Proceedings in Informatics (LIPIcs), pages 18:1-18:26, Dagstuhl, Germany, 2022. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. Google Scholar
  18. Izhar Oppenheim. Local spectral expansion approach to high dimensional expanders part i: Descent of spectral gaps. Discrete and Computational Geometry, 59(2):293-330, 2018. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023 Schloss Dagstuhl - LZI GmbH Imprint Privacy Contact