only full $(k-1)$-faces. This implies that the fraction of full $(i+1)$-faces is about the same as the fraction of the full $(j+1)$-faces. This actually finishes the proof, since we get many $(i+1)$-faces that are not full, i.e., contain at least one $i$-face which is not in $W$.

### Explicit construction

As proven in [@EK16], Ramanujan complexes are excellent skeleton expanders. Since our definition of skeleton expansion is slightly different than the one appearing in [@EK16], we modify their proof so the requirement of our definition follows. We show that Ramanujan complexes with thickness large enough are $\alpha$-skeleton expanders with $\alpha$ as small as we want. Then by the explicit construction of Ramanujan complexes from [@LSV05.2] we achieve the following theorem.

\[main-thm-4-informal\] There exists an explicit family of bounded degree high dimensional simplicial complexes in which all of the high order random walks converge rapidly to their stationary distribution.

Related work
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In a recent work Oppenheim [@Opp15] has shown that if all the links of a $d$-dimensional simplicial complex are good spectral expanders and the complex is connected, then the $1$-dimensional skeleton of the complex is an expander graph, or in other words, that the graph $G_0$ is a spectral 