Garland's Technique for Posets and High Dimensional Grassmannian Expanders [article]

Tali Kaufman, Ran J. Tessler
2022 arXiv   pre-print
Local to global machinery plays an important role in the study of simplicial complexes, since the seminal work of Garland [G] 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 [KO,AL], an equivalence with a global random walk definition, generalizing [DDFH] and a
more » ... ing down theorem, generalizing [O]. 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 [LSV1,LSV2], 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 [DDFH].
arXiv:2101.12621v3 fatcat:xru7b4ytunhobmaa7mfmvwc6dq