Local to global machinery plays an important role in the study of simplicial complexes, since the seminal work of Garland  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 [2,14], an equivalence with a global random walk definition, generalizing  and a trickling down theorem, generalizing . 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 [18,19], 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 .
|Title of host publication||14th Innovations in Theoretical Computer Science Conference, ITCS 2023|
|Editors||Yael Tauman Kalai|
|Publisher||Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing|
|State||Published - 1 Jan 2023|
|Event||14th Innovations in Theoretical Computer Science Conference, ITCS 2023 - Cambridge, United States|
Duration: 10 Jan 2023 → 13 Jan 2023
|Name||Leibniz International Proceedings in Informatics, LIPIcs|
|Conference||14th Innovations in Theoretical Computer Science Conference, ITCS 2023|
|Period||10/01/23 → 13/01/23|
Bibliographical noteFunding Information:
Funding Tali Kaufman: Research supported by ERC and BSF. Ran J. Tessler: (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.
© Tali Kaufman and Ran J. Tessler; licensed under Creative Commons License CC-BY 4.0.
- Garland Method
- High dimensional Expanders