equivalent to a random walk on the $i$-graph as above. Thus, our goal is to construct a complex $X$ such that the random walks on [*all*]{} of its induced $i$-graphs, $0 \le i < d$, will converge rapidly to the uniform distribution.

### Rapid mixing criterion

Recall that the link of any face $\sigma \in X$ is a smaller complex which corresponds to the local view of $\sigma$. We essentially show that if every link is expanding in some sense, then all of the high order random walks on $X$ converge rapidly to their stationary distribution.

For any face $\sigma \in X$ the $1$-dimensional skeleton of $X_\sigma$ is defined as $X_\sigma(0) \cup X_\sigma(1)$, i.e., the underlying graph of $X_\sigma$. We say that $X$ is a [*skeleton expander*]{} if the $1$-dimensional skeleton of every link behaves pseudorandomly.

Let $X$ be a $d$-dimensional simplicial complex. $X$ is called an [*$\alpha$-skeleton expander*]{}, $\alpha > 0$, if for any face $\sigma \in X$ (including $\sigma = \emptyset)$ and any subset of vertices $S \subseteq X_\sigma(0)$ it holds that $${\left\lVert E(S) \right\rVert}_\sigma \le {\left\lVert S \right\rVert}_\sigma({\left\lVert S \right\rVert}_\sigma + \alpha),$$ where $E(S) \subseteq X_\sigma(1)$ denotes the set of edges with both endpoints in $S$.

We 