degree high dimensional simplicial complexes in which all of the high order random walks converge rapidly to their stationary distribution.

The convergence rate of random walks on graphs is known to be deduced from the spectrum of the graph’s adjacency matrix (for more about random walks on graphs see [@Lov93]). Similarly, for a $d$-dimensional simplicial complex and any $0 \le i < d$ we define the matrix $A_i$ to be the adjacency matrix of dimension $i$ of the complex. This matrix is indexed by the $i$-faces of the complex, where $A_i(\sigma,\tau)$ indicate if $\sigma$ and $\tau$ are neighbors. The question we address here is how one could construct a bounded degree $d$-dimensional simplicial complex (for any $d \in {\mathbb{N}}$) whose [*all*]{} adjacency matrices $A_i$, $0 \le i < d$, have a concentrated spectrum, and thus its associated high order random walks (on any of its dimensions) converge rapidly to their stationary distribution.

#### Bounded versus unbounded.

It is easy to obtain an unbounded degree graph in which the random walk will converge rapidly to its stationary distribution. This case is less useful as in most applications it is crucial to have a constant bound on the number of neighbors of 