Garland’s technique for posets and high-dimensional Grassmannian expanders

Abstract 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 more 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 trickling down theorem, generalizing [O]. In particular, we show that some posets, such as the Grassmannian poset, exhibit a qualitatively stronger trickling down effect than simplicial complexes. We use these methods, and a novel idea of posetification to the Ramanujan complexes [LSV1, LSV2], to construct a constant degree expanding Grassmannian poset, and analyze its expansion. This is the first construction of such an object, whose existence was conjectured in [DDFH].