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Garland’s Technique for Posets and High Dimensional Grassmannian Expanders

Authors Tali Kaufman, Ran J. Tessler

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Author Details

Tali Kaufman
  • Department of Computer Science, Bar-Ilan University, Ramat-Gan, Israel
Ran J. Tessler
  • Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel

Funding

  • Kaufman, Tali: Research supported by ERC and BSF.
  • Tessler, Ran J.: (incumbent of the Lillian and George Lyttle Career Development Chair) Research was supported by the ISF grant No. 335/19 and by a research grant from the Center for New Scientists of Weizmann Institute.

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Tali Kaufman and Ran J. Tessler. Garland’s Technique for Posets and High Dimensional Grassmannian Expanders. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 78:1-78:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ITCS.2023.78

Abstract

Local to global machinery plays an important role in the study of simplicial complexes, since the seminal work of Garland [Garland, 1973] to our days. In this work we develop a local to global machinery for general posets. We show that the high dimensional expansion notions and many recent expansion results have a generalization to posets. Examples are fast convergence of high dimensional random walks generalizing [Kaufman et al., 2020], [Alev and Lau, 2020], an equivalence with a global random walk definition, generalizing [Dikstein et al., 2018] and a trickling down theorem, generalizing [Oppenheim, 2018]. In particular, we show that some posets, such as the Grassmannian poset, exhibit qualitatively stronger trickling down effect than simplicial complexes. Using these methods, and the novel idea of posetification to Ramanujan complexes [Lubotzky et al., 2005a], [Lubotzky et al., 2005b], we construct a constant degree expanding Grassmannian poset, and analyze its expansion. This it the first construction of such object, whose existence was conjectured in [Dikstein et al., 2018].

Subject Classification

ACM Subject Classification
  • Theory of computation → Randomness, geometry and discrete structures
Keywords
  • High dimensional Expanders
  • Posets
  • Grassmannian
  • Garland Method

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References

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