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].
@InProceedings{kaufman_et_al:LIPIcs.ITCS.2023.78, author = {Kaufman, Tali and Tessler, Ran J.}, title = {{Garland’s Technique for Posets and High Dimensional Grassmannian Expanders}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {78:1--78:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.78}, URN = {urn:nbn:de:0030-drops-175819}, doi = {10.4230/LIPIcs.ITCS.2023.78}, annote = {Keywords: High dimensional Expanders, Posets, Grassmannian, Garland Method} }
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