Stronger Trickling Down in Posets Implies Expanding Sparsification of the Grassmanian Complex

In simplicial complexes it is well known that many of the global properties of the complex, can be deduced from expansion properties of its links. This phenomenon was first discovered by Garland [G]. In this work we develop a local to global machinery for general posets. We first show that the basic localization principle of Garland generalizes to more general posets. We then show that notable local to global theorems for simplicial complexes arise from general principles for general posets with expanding links. Specifically, we prove the following theorems for general posets satisfying some assumptions: Expanding links (one sided expansion) imply fast convergence of high dimensional random walks (generalization [KO,AL]); Expanding links imply Trickling down theorem (generalizing [O]); and a poset has expanding links (with two sided expansion) iff it satisfies a global random walk convergence property (generalization [DDFH]). We axiomatize general conditions on posets that imply local to global theorems. By developing this local to global machinery for general posets we discover that some posets behave better than simplicial complexes with respect to local to global implications. Specifically, we get a trickling down theorem for some posets (e.g. the Grassmanian poset) which is better behaved than the trickling down theorem known for simplicial complexes. In addition to this machinery, we also present a method to construct a new poset out of a pair of an initial poset and an auxiliary simplicial complex. By applying this procedure to the case where the pair is the Grassmanian poset and a bounded degree high dimensional expander, we obtain a bounded degree Grassmanian poset. We prove, using the tools described above, that this poset is a bounded degree expanding Grassmanian poset, partially proving a conjecture of [DDFH].