complex which implies the rapid mixing of all of its associated high order random walks. In order to assert their rapid mixing we introduce a new definition for combinatorial expansion of high dimensional simplicial complexes, which we term colorful expansion. We prove that this local criterion on a complex implies colorful expansion, which in turn implies the rapid convergence of all of the high order random walks on it. We then show an explicit construction of bounded degree complexes which satisfy this criterion, and hence are bounded degree complexes in which the high order random walks converge rapidly to their stationary distribution.

### High order random walks

Let $X$ be a $d$-dimensional simplicial complex. We generalize the notion of random walks on graphs to higher dimensions by letting the walk to move on any dimension of the complex. For any $0 \le i < d$ we define the walk on dimension $i$ of the complex as follows. We start from an initial $i$-face $\sigma_0 \in X(i)$. Then, if after $t$ steps we are at $\sigma_t$, the process of choosing $\sigma_{t+1}$ can be described by the following two steps:

1.  Choose an $(i+1)$-face $\tau \supset \sigma_t$ uniformly at random.

2.  Choose 