method in asserting the rapid convergence of the high order random walks is as follows. We introduce a new notion of high order combinatorial expansion of complexes which we term colorful expansion. A $d$-dimensional simplicial complex is said to be a colorful expander if for any $0 \le i < d$ and any subset $S$ of $i$-faces there is a large set of [*expanding*]{} $(i+1)$-faces, i.e., faces which are hit by $S$ but are not fully covered by $S$. The term colorful expansion comes from the fact that if we color the faces in $S$ with one color and the faces not in $S$ with another color, then we get many colorful $(i+1)$-faces. This notion is of an independent interest as it is a natural generalization of combinatorial expansion of graphs to higher dimensions, which is not implied by previously studied notions such as coboundary or cosystolic expansion (see  [@EK16]). We show that colorful expansion of a complex implies that [*all*]{} of its adjacency matrices $A_i$, $0 \le i < d$, have a concentrated spectrum, and hence all of the high order random walks on it converge rapidly to their stationary distribution.

We provide a local-to-global criterion on a complex 