of dimension $d$, then $X_\sigma$ is a complex of dimension $d-|\sigma|$. We denote by ${\left\lVert \,\cdot\, \right\rVert}_\sigma$ the norm associated with the link of $\sigma$.

Let us demonstrate the above definitions with a simple example.

Let $X$ be a $2$-dimensional simplicial complex. The complex contains faces of dimensions $0,1$ and $2$, which can be viewed as vertices, edges and triangles. $X(0)$ denotes the vertices, $X(1)$ the edges, and $X(2)$ the triangles. The degree of any vertex $u \in X(0)$ is the number of triangles containing it, i.e., $\deg(u) = |\{t \in X(2) \;|\; u \in t\}|$. The link of a vertex $u \in X(0)$ is a $1$-dimensional simplicial complex defined as follows. Any edge of the form $\{u, v\} \in X(1)$ becomes a vertex in the link, i.e., $v \in X_u(0)$, and any triangle of the form $\{u, v, w\} \in X(2)$ becomes an edge in the link, i.e., $\{v, w\} \in X_u(1)$. Thus, the link of $u$ contains vertices and edges, which correspond to edges and triangles of the complex, respectively.

Our contribution
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In this work we introduce a new notion of random walks on high dimensional simplicial complexes. Our main result is a local criterion on a 