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arXiv · 2208.04486

An Improved Trickle-Down Theorem for Partite Complexes

Dorna Abdolazimi·Shayan Oveis Gharan
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Abstract

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 $\frac{1-\delta}{d}$-(one-sided) spectral expanders, then the link of any face of co-dimension $k$ is an $O(\frac{1-\delta}{k\delta})$-(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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Dorna Abdolazimi, Shayan Oveis Gharan. 2022-08-09. An Improved Trickle-Down Theorem for Partite Complexes. https://arxiv.org/abs/2208.04486

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