论文标题
改进的partite综合体定理
An Improved Trickle-Down Theorem for Partite Complexes
论文作者
论文摘要
我们证明了对partite综合体的trick流的加强。给定一个$（d+1）$ - partite $ d $ - 二维简单综合体，我们表明，如果“平均”“平均” co-dimension 2的面部链接是$ \ frac {1-Δ} {d} {D} $ - （单方面）频谱展开器，那么$ k $ k $ k $ k $的链接是$ o（\ frac {1-δ} {kδ}）$ - （单面）频谱扩展器，所有$ 3 \ leq k \ leq d+1 $。对于应用程序，将我们的定理用作黑框，我们表明，在有限程度的高维扩张器的最新构造中，共同数量$ k $的面孔的链接最多，频谱膨胀最多（1/k）$（1/k）$ spectral spectral spectral lacking links links links links links link of co-dimense co-dimense links links lackspeation lights lake light。
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 $2$.
