关于分部复形的改进型滴流定理
An Improved Trickle-Down Theorem for Partite Complexes
AI总结：
本研究提出分部复形滴流定理的强化版本，证明余维2面链环的平均谱扩张性质可推导至更高余维面的链环，并将其应用于有界度高维扩张子的谱扩张界分析。
摘要：
我们证明了分部复形滴流定理的一个强化版本。给定一个$(d+1)$-分部的$d$维单纯复形，我们证明：若余维数为2的面的链环“平均而言”是$\frac{1-δ}{d}$-（单侧）谱扩张子，则对所有满足$3\leq k\leq d+1$的$k$，任意余维数为$k$的面的链环都是$O(\frac{1-δ}{kδ})$-（单侧）谱扩张子。作为应用，我们将该定理作为黑盒使用，证明在近期构造的有界度高维扩张子中，余维数为$k$的面的链环的谱扩张至多为余维数2的最差面的链环谱扩张的$O(1/k)$比例。
英文摘要：
We prove a strengthening of the trickle down theorem for partite complexes. Given a $(d+1)$-partite $d$-dimensional simplicial complex, we show that if "on average" the links of faces of co-dimension 2 are $\frac{1-δ}{d}$-(one-sided) spectral expanders, then the link of any face of co-dimension $k$ is an $O(\frac{1-δ}{kδ})$-(one-sided) spectral expander, for all $3\leq k\leq d+1$. 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 spectral expansion at most $O(1/k)$ fraction of the spectral expansion of the links of the worst faces of co-dimension $2$.
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