an $i$-face $\sigma_{t+1} \subset \tau$, $\sigma_{t+1} \ne \sigma_t$, uniformly at random and move to it.

We say that $\sigma, \sigma' \in X(i)$ are neighbors, denoted by $\sigma \sim \sigma'$, if they share a common $(i+1)$-face, i.e., $\sigma \cup \sigma' \in X(i+1)$. In order to analyze this walk, we define the following auxiliary graphs.

Let $X$ be a $d$-dimensional simplicial complex. For any $0 \le i < d$ the [*$i$-graph*]{} $G_i = (V_i,E_i)$ (with adjacency matrix $A_i$) is defined as follows.

-   The vertices of $G_i$ are the $i$-faces of $X$, i.e., $V_i = X(i)$.

-   For any two $i$-faces $\sigma,\sigma' \in X(i)$ such that $\sigma \sim \sigma'$, there is an edge between the corresponding vertices in $G_i$, i.e., $E_i=\left\{\!\{\sigma, \sigma'\} \in \binom{X(i)}{2} \mid \sigma \sim \sigma'\right\}$.

Denote by $\pi_0 \in {\mathbb{R}}^{X(i)}$ the probability distribution from which $\sigma_0$ is chosen. As in graphs, the random walk on dimension $i$ of $X$ yields a sequence of probability distributions $\pi_0, \pi_1, \dotsc, \in {\mathbb{R}}^{X(i)}$. Note that the transition from $\pi_t$ to $\pi_{t+1}$ is given by $\pi_{t+1} = \pi_t \widetilde A_i$, where $\widetilde A_i$ is the normalized adjacency matrix of $G_i$. In other words, the random walk on dimension $i$ of the complex is 