d\mu=\partial_p\colon C_p^{(2)}({\mathcal{SC}_*},\mu)\to C_{p-1}^{(2)}({\mathcal{SC}_*},\mu),\\
\int^\oplus_{{\mathcal{SC}_*}}\partial_{p,[K,x]}^* d\mu=\partial_p^*\colon C_{p-1}^{(2)}({\mathcal{SC}_*},\mu)\to C_{p}^{(2)}({\mathcal{SC}_*},\mu),\end{aligned}$$ with operator norm equals the $\mu$-essential supremum. Since $\partial_{p-1}\circ\partial_p([K,x])=0$ for all $[K,x]\in{\mathcal{SC}_*}$ and $p\in{{\mathbb N}}$ we get a chain complex $$\begin{aligned}
...\xrightarrow[]{\partial_{p+1}}C_p^{(2)}({\mathcal{SC}_*},\mu)\xrightarrow[]{\partial_p}C_{p-1}^{(2)}({\mathcal{SC}_*},\mu)\xrightarrow[]{\partial_{p-1}}...\end{aligned}$$ and hence, we can define the *p-th reduced homology of ${\mathcal{SC}_*}$ with respect to $\mu$* by $$\overline{H}^{(2)}_{p}({\mathcal{SC}_*},\mu)=\ker \partial_p/\overline{\mathrm{im}\;\partial_{p+1}} .$$ Further, we can define the Laplace operator $\Delta_p=\partial_{p-1}^*\circ\partial_p+\partial_{p+1}\circ\partial_{p+1}^*$ and the direct integral Hilbert space of *harmonic chains* $\mathcal{H}_{p}^{(2)}({\mathcal{SC}_*},\mu)$. There is, as long as $\Delta_p$ is a bounded operator, an Hodge isomorphism (cf. [@eckmann p. 192]): $$\begin{aligned}
\overline{H}^{(2)}_{p}({\mathcal{SC}_*},\mu)\cong\mathcal{H}_{p}^{(2)}({\mathcal{SC}_*},\mu).\end{aligned}$$

For every random rooted simplicial complex $\mu$ of bounded degree $\deg(\mu)=D$ the boundary operator $\partial_p$, its adjoint $\partial_p^*$ and the Laplace operator $\Delta_p$ are bounded operators on $C_p^{(2)}({\mathcal{SC}_*},\mu)$ for all $p\in{{\mathbb N}}$.

First remark that $\|\partial_p\|\le \sqrt{p+1}$ is independent of the vertex degree since every $p$-simplex has $p+1$ faces. Let us look at $\partial_p^*$. First consider the coboundary operator $\partial^*_p$ on a fixed simplicial complex $K$ with vertex degree bounded by $D$. For every $(p-1)$-simplex $\sigma$ of $K$ its image $\partial_p^*(\sigma)$ consists of at most $D-(p-1)$ *p*-simplices since every vertex, which is contained in $\sigma$, must be contained in the $p$-simplices of the image of $\sigma$ and since the degree is bounded, there can only 