to vertices as expander graphs.

Dinur’s proof of the PCP theorem builds heavily on random walks on expander graphs. In the following we study high dimensional analogues of random walks in high dimensional simplicial complexes.

#### On high dimensional random walks.

In this paper we define high order random walks on simplicial complexes. Our definition generalizes the notion of random walks on graphs to high dimensional simplicial complexes.

For a $d$-dimensional simplicial complex, we define the high order random walk on it for any dimension $0 \le i < d$. This walk moves at random between neighboring $i$-faces (e.g., edges) of the complex, where two $i$-faces are considered neighbors if they share a common $(i+1)$-face (e.g., a triangle).

The question we address in this paper is the following.

#### Question.

[*Are there bounded degree high dimensional simplicial complexes in which all of the high order random walks converge rapidly to their stationary distribution*]{}.

In a nutshell, our answer is the following. We provide a local criterion on a complex which implies the rapid convergence of all high order random walks on it. Moreover, we show an explicit family of bounded degree complexes which satisfy that criterion, and hence are bounded 