any vertex in the graph. In a same way, the case of having an unbounded degree complex so the high order random walks on it will converge rapidly to their stationary distribution is less interesting. The complete high dimensional analogue of the most useful random walks on expander graphs are [*bounded degree*]{} high dimensional simplicial complexes in which all of the high order random walks converge rapidly to their stationary distribution.

#### Better than random.

In the case of graphs, a random walk on a random bounded degree graph is known to converge rapidly to its stationary distribution. The constructions of explicit expander graphs seem to try and simulate the behavior of these random objects as much as possible, i.e., the philosophy is that a random graph is the best one can get and with explicit constructions we try to get as close as we can to random graphs. In the case of high dimensional simplicial complexes, a high order random walk on a random bounded degree complex [*does not*]{} converge rapidly to its stationary distribution. Thus, this phenomenon we study here is very special as it is something we [*cannot achieve with a random object*]{}.

#### Colorful expansion.

Our 