which implies colorful expansion. For any face $\sigma$ in the complex, its [*link*]{} is the local view of $\sigma$, which is obtained by taking all the faces containing $\sigma$ and removing $\sigma$ from all of them. We prove that if the $1$-dimensional skeleton (i.e., the underlying graph) of every link in the complex is a spectral expander graph, then the entire complex is a colorful expander.

We conclude by showing that Ramanujan complexes satisfy this criterion, and hence are colorful expanders. We then use the explicit construction of Ramanujan complexes from [@LSV05.2] in order to achieve an explicit family of bounded degree complexes in which all of the high order random walks converge rapidly to their stationary distribution.

Expander graphs and random walks {#expanders-and-random-walks-section}
--------------------------------

Let $G=(V,E)$ be an undirected graph. Denote by $A$ the adjacency matrix of $G$, where $A(u,v)$ equals the number of edges between $u$ and $v$ (we allow multiple edges). The [*degree*]{} of a vertex is the number of edges containing it. Throughout this section assume for simplicity that $G$ is $k$-regular, i.e., the degree of any vertex is $k$ (we deal with the non-regular case at Section \[non-regular-section\]). The [*Cheeger constant*]{} of $G$ is defined 