eigenvalue of the normalized adjacency matrix is bounded away from $-1$. In any case, combinatorial expansion is the crucial condition for rapid mixing of random walks.

High dimensional simplicial complexes
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A [*simplicial complex*]{} $X$ with a set of vertices $V$ is a collection of subsets of $V$ with a closure property, namely, for any subset $\sigma \in X$, all of its subsets $\tau \subset \sigma$ are also in $X$. Each such subset is called a [*face*]{} of the complex. The [*dimension*]{} of a face $\sigma \in X$ is one less than the number of vertices in it, i.e., $\dim(\sigma)=|\sigma|-1$. We also define the empty set as a face of the complex, which is the unique face of dimension $-1$. The dimension of the entire complex is defined as the maximal dimension of a face in it. The complex is said to be [*pure*]{} if for any face $\sigma \in X$ with $\dim(\sigma) < \dim(X)$ there is a face of maximal dimension $\tau \in X$, $\dim(\tau)=\dim(X)$, such that $\sigma \subset \tau$. We only deal with pure simplicial complexes. We denote by $X(i)$ the faces of the complex of dimension $i$, which are called in short the $i$-faces of the complex. 