a compact subspace of ${\mathcal{SC}_*}$ [@bowen].\
We call a probability measure on ${\mathcal{SC}_*}$ a *random rooted simplicial complex* ([*rrsc*]{}).\
Let ${\mathcal{SC}_{**}}$ be the space of isomorphism classes of connected locally finite doubly rooted simplicial complexes. An element $[K,x,y]$ of ${\mathcal{SC}_{**}}$ consists of a connected simplicial complex $K$ and an ordered pair of vertices $x,y\in V(K)$. The topology is given by $$\begin{aligned}
U_r([K,x,y])\coloneqq\left\{[L,p,q]\in{\mathcal{SC}_{**}}\;\middle|\begin{array}{ll}
(B_{(K,x)}(r),x)\cong (B_{(L,p)}(r),p) \text{ and}\\
(B_{(K,y)}(r),y)\cong (B_{(L,q)}(r),q)\end{array}\right\}.\end{aligned}$$ An important class of random rooted simplicial complexes are the *unimodular* ones:

\[definition unimodular\] A probability measure $\mu$ on ${\mathcal{SC}_*}$ is called *unimodular* if $$\int_{{\mathcal{SC}_*}}\sum_{y\in V(K)}f([K,x,y])d\mu([K,x])=\int_{{\mathcal{SC}_*}}\sum_{x\in V(K)}f([K,x,y])d\mu([K,y])$$ for all Borel measurable functions $f\colon{\mathcal{SC}_{**}}\to \mathbb{R}_{\ge 0}$.

We define the *maximum degree* of an element $[K,x]\in{\mathcal{SC}_*}$ by $$\mathrm{deg}_{max}([K,x])=\sup_{x\in V(K)}\{\deg(x)\},$$ where $\deg(x)$ denotes the vertex degree, i.e. the number of 1-simplices of $K$ containing $x$. More generally, we define the $p$-degree $\deg_p(x)$ of $x$ as the number of $p$-simplices of $K$ containing $x$. If the $p$-degree is bounded then also the $(p+1)$-degree is. We will use the notation $K(p)$ for the set of $p$-simplices of $K$.

The *degree* $\deg(\mu)$ of an random rooted simplicial complex $\mu$ on ${\mathcal{SC}_*}$ is the $\mu$-essential supremum of the maximum degree: $$\begin{aligned}
&\deg(\mu)=\inf\Big\lbrace d\in\mathbb{R}\;|\; \mu\big\{[K,x]\in{\mathcal{SC}_*}\;|\;\mathrm{deg}_{max}([K,x]) >d\big\}=0\Big\rbrace.\end{aligned}$$ We say a measure 