High-Dimensional Expanders 1
summary
Defines a d-dimensional lambda link expander: every link (neighborhood of a face) has 1-skeleton with spectral gap ≤ lambda, and shows this local property forces global expansion via Oppenheim’s trickling-down theorem (if all links are λ-expanders then the global normalized second eigenvalue ≤ λ/(1−λ)). Constructs and analyzes examples (complete complex, d+1-partite, spherical buildings, LSV Ramanujan complexes whose links equal spherical buildings) and develops linear-algebraic tools—up/down operators and non-lazy M+ walks—to relate link spectra to high-order random-walk mixing.
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