  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].
