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<p>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].</p><p>In particular, we show that some posets, such as the Grassmannian poset, exhibit a qualitatively stronger trickling down effect than simplicial complexes.</p><p>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].</p>

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

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  • 发表日期：

    2025年9月4日

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  • Tali Kaufman,
  • Ran J. Tessler

摘要

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].