note that in the above definition we can replace the requirement of $\alpha$-skeleton expansion for $\sigma = \emptyset$ with the weaker requirement that the complex is connected. This could be done since by [@Opp15] if the complex is connected, then the $\alpha$-skeleton expansion of $\sigma =\emptyset$ follows from the $\alpha$-skeleton expansion of all $\sigma \in X$, $\sigma \neq \emptyset$.

For some intuition regrading the above definition, consider a random graph $G = (V,E)$ whose edges are distributed uniformly. Let $S \subseteq V$ be any subset of vertices of size $\gamma|V|$. Then the probability for any edge to have both endpoints in $S$ is $\gamma^2$, and hence the expected size of $E(S)$ is $\gamma^2|E|$. From this point of view, the $\alpha$-skeleton expansion property means that the underlying graph of every link “looks like” a random graph, up to an error of $\alpha$.

The main result of this paper is the following theorem.

\[main-thm-3-informal\] If a complex is an $\alpha$-skeleton expander for small enough $\alpha$, then all of the high order random walks on it converge rapidly to their stationary distribution.

### Colorful expansion

Our new definition for combinatorial expansion of high dimensional simplicial complexes is the main building block in the 