An Improved Trickle-Down Theorem for Partite Complexes
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Given a d+1-partite d-dimensional simplicial complex, we prove a generalization of the trickle-down theorem. We show that if "on average" faces of co-dimension 2 are 1-δ/d-(one-sided) spectral expanders, then any face of co-dimension k is an O(1-δ/kδ)-(one-sided) spectral expander, for all 3≤ k≤ d+1. For an application, using our theorem as a black-box, we show that links of faces of co-dimension k in recent constructions of bounded degree high dimensional expanders have local spectral expansion at most O(1/k) fraction of the local expansion of worst faces of co-dimension 2.
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