rapid mixing proof. For an $i$-cochain $W \in C^i(X)$, we say that an $(i+1)$-face is [*expanding*]{} if it hits $W$ but not fully covered by $W$.

Let $X$ be a $d$-dimensional simplicial complex. For any $i$-cochain $W \in C^i(X)$, $0 \le i < d$, the [*expanding faces*]{} of $W$ are defined as $$\psi(W) = \{\tau \in X(i+1) \;|\; \exists\, \sigma, \sigma' \subset \tau: \sigma \in W,\, \sigma' \in X(i) \setminus W\}.$$

Then we define colorful expansion as follows.

Let $X$ be a $d$-dimensional simplicial complex. $X$ is called an [*$\epsilon$-colorful expander*]{}, $\epsilon > 0$, if for any $i$-cochain $W \in C^i(X)$, $0 \le i < d$, $0 < {\left\lVert W \right\rVert} \le 1/2$, $$\frac{{\left\lVert \psi(W) \right\rVert}}{{\left\lVert W \right\rVert}} \ge \epsilon.$$

Our main result is then a corollary of the following two theorems.

\[main-thm-1-informal\] If a complex is an $\alpha$-skeleton expander for small enough $\alpha$, then it is a colorful expander.

\[main-thm-2-informal\] If a complex is a colorful expander, then all of the high order random walks on it converge rapidly to their stationary distribution.

Since Theorem \[main-thm-1-informal\] is the main technical part of this paper, we provide here a short overview of the proof.

#### Informal overview of the proof 