Garland's Technique for Posets and High Dimensional Grassmannian Expanders

TL;DR

This paper develops a local-to-global framework for posets, extending high-dimensional expansion concepts, and constructs the first constant degree expanding Grassmannian poset, confirming a long-standing conjecture.

Contribution

It generalizes high-dimensional expansion results to posets, introduces Posetification, and constructs the first constant degree expanding Grassmannian poset.

Findings

Posets exhibit stronger trickling down effects than simplicial complexes

High-dimensional random walks can be generalized to posets

Constructed the first constant degree expanding Grassmannian poset

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 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 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].