An Improved Trickle-Down Theorem for Partite Complexes

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 $\frac{1-\delta}{d}$-(one-sided) spectral expanders, then 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 local spectral expansion at most $O(1/k)$ fraction of the local expansion of worst faces of co-dimension $2$.