Unimodular Random Measured Metric Spaces and Palm Theory on Them

The mass transport principle (MTP) was first used in [20] for studying percolation on trees. It was developed further in [7] for studying group-invariant percolation on graphs and in [8] for proving recurrence of local limits of planar graphs. [4] also established a property called involution invariance for local limits of finite graphs³³3The MTP for local limits of graphs was also observed in [8].. This property turned out to be a special case of the MTP, and in fact, equivalent to it. Then, [3] defined unimodular random graphs by simply using the MTP as the definition, provided various general examples, and established many properties for them. The term unimodular was chosen in [3] because, in order that a fixed transitive graph $G$ satisfies the MTP, it is necessary and sufficient that the automorphism group of $G$ is a unimodular group.

It is useful to recall the MTP for unimodular graphs. A mass transport or a transport function is a function $g$ that takes as input a graph $G$ and two vertices of $G$ and outputs a value in $\mathbb{R}^{\geq 0}$. We think of $g(G,u,v)$ as the mass going from $u$ to $v$. The assumptions on $g$ are measurability (to be defined suitably) and invariance, where the latter means here that $g$ is invariant under isomorphisms. A random rooted graph $[\boldsymbol{G},\boldsymbol{o}]$ (where the graph $\boldsymbol{G}$ and the root vertex $\boldsymbol{o}$ are random) is called unimodular if it satisfies the following MTP for any transport function $g$:

Similar versions of the MTP exist for point processes in $\mathbb{R}^{d}$. For a recent application, one can mention the use of the MTP in constructing perfect matching between point processes and fair tessellations (see e.g., [25] and [13]). These MTPs are special cases of much older theorems in stochastic geometry; e.g., Mecke’s formula, Mecke’s theorem and Neveu’s exchange formula, which had been widely used in stochastic geometry. In particular, Mecke’s theorem (see e.g., [33]) can be rephrased as follows: If $\Phi_{0}$ is the Palm version of a stationary point process (i.e., conditioned to contain the origin, or equivaletnly, seeing the point process form a typical point), then

This equation looks similar to (1.1) with the difference that the random objects have different natures and the invariance condition for the transport function $g=g(\Phi,u,v)$ is the invariance under Euclidean translations. In fact, (1.2) characterizes a larger class of point processes which are called point-stationary point processes in [33] (e.g., the zero set or the graph of the simple random walk). See Subsection 4.1 for further discussion.

Due to the similarity of the MTP in (point-) stationary point processes and unimodular random graphs, a lot of connections have been observed between the two theories. In particular, [3] noted that any graph that is constructed on a stationary point process (with some equivariance and measurability condition) gives rise to a unimodular random graph. Also, various results and proof techniques in one theory can be translated into the other one with minor modifications (see [6]). The previous work [6] unifies these two notions by defining unimodular random discrete spaces.

The theorems mentioned above for stationary point processes are in fact more general and apply to stationary random measures. In particular, Mecke’s theorem in the general form can be rephrased equivalently in a MTP form as follows: If $\Psi_{0}$ is the Palm version of a stationary random measure on $\mathbb{R}^{d}$, then

for every function $g=g(\Psi_{0},u,v)\geq 0$ that is translation-invariant and measurable. As in the previous case, (1.3) characterizes a larger class of random measures, which are called mass-stationary random measures in [33]. See Subsection 4.1 for further discussion.

As mentioned above, the MTP (1.1) is satisfied for local limits of graphs. Similar properties have been established for some instances of scaling limits of graphs, where scaling limit means that the graph-distance metric is scaled by some small factor and the limit means the limit of a sequence of random metric spaces (sometime, the metric spaces are rooted; i.e., have a distinguished point, and/or measured; i.e., have a distinguished measure). Most notably, various instances that have a compact rooted measured scaling limit satisfy the so called re-rooting invariance property; i.e., the distribution is invariant under changing the root randomly (according to the distinguished measure on the model). This is the case for the Brownian continuum random tree [2], stable trees [15] and the Brownian map [34]. Also, in a few examples where the scaling limit is not compact, an MTP similar to (1.3) has been observed (e.g., in [10]). However, a general rigorous MTP result for scaling limits seems to be missing in the literature. This is done in this work by providing a general form of the MTP and by showing that it is preserved under weak limits. This general result readily implies the MTP and re-rooting invariance in the known examples of scaling limits. We will also provide some re-rooting invariance properties in the non-compact case as well.

In this occasion, we should also mention the mass transport principle in the theory of countable Borel equivalence relations. This will be introduced in the next subsection and will be discussed further in Subsection 4.6.

In the previous subsection, we recalled that the theories of unimodular graphs, point processes, random measures and scaling limits exhibit the mass transport principle in various forms. The similarity of these formulas provide fruitful connections between these theories. So, it is natural to ask for a general theory of the MTP that unifies these notions. We saw that the discrete cases (unimodular graphs and point processes) can be unified by the notion of unimodular discrete spaces [6] (these cases are also tightly connected to the theory of countable Borel equivalence relations).

The first main task of this work is providing a unification of the MTP in the discrete and non-discrete cases. The unification is provided by introducing unimodular random rooted measured metric spaces, or in short, unimodular rmm spaces in Section 2. This can be thought of a common generalization of unimodular graphs, (point-) stationary point processes, (mass-) stationary random measures and the random continuum spaces arising in scaling limits. In this part, establishing the MTP for general scaling limits is novel and is one of the main motivations of this work.

Unimodular rmm spaces are defined as random triples $[\boldsymbol{X},\boldsymbol{o},\boldsymbol{\mu}]$, where $\boldsymbol{X}$ is a boundedly-compact metric space, $\boldsymbol{o}$ is a point of $\boldsymbol{X}$ called the root, and $\boldsymbol{\mu}$ is a boundedly-finite Borel measure on $\boldsymbol{X}$. The Gromov-Hausdorff-Prokhorov metric, given in the most general form in [31], provides the measure-theoretic requirements for this definition. Then, the definition of unimodularity in this setting is given by modifying the MTP (1.3) accordingly. Note that a distinguished measure is needed on $\boldsymbol{X}$ in order that the MTP makes sense. Many specific examples and general categories of unimodular rmm spaces are discussed in Section 4.

In Sections 3 and 5, we provide general results on unimodular spaces. In particular, we prove that unimodularity is preserved under weak convergence. This implies that all (measured) scaling limits are unimodular (which is one of the main motivations of this work), and hence, in the compact case satisfy the re-rooting invariance property. We also study re-rooting in the non-compact case and the properties of random walks on a unimodular rmm space. We define ergodicity and prove an ergodic decomposition theorem. Also, various definitions of amenability are provided and proved to be equivalent, based on similar definitions in other fields of mathematics; e.g., definition by local means, approximate means, hyperfinitenss and Folner condition. It is also proved that the number of topological ends of a unimodular rmm space belongs to $\{0,1,2,\infty\}$ almost surely.

We should mention that the discrete cases of the MTP, discussed above, are connected to the theory of countable Borel equivalence relations (CBER), which has totally different roots (ergodic theory of dynamical systems, group actions and orbit equivalence theory)⁴⁴4This theory is almost as old as Mecke’s theorem and is very useful in probability theory, but still deserves to be more recognized in the probability community. Some results in this theory have been re-invented later by probabilists; e.g., the existence of one-ended treeing in the amenable case (see e.g., [12], [24] and [40]) and factor point processes (see [32]).. There exists a MTP-like formula in this theory, which in fact defines when an equivalence relation is measure-preserving; i.e., the measure is invariant. It is a generalization of the measure-preserving property for dynamical systems and group actions. As discussed in [3], this notion is connected to unimodular graphs via graphings. While the two theories have substantial overlap, their view point and motivations are different. In fact, the theory of CBERs can be thought of as the mathematical ground under the theory of unimodular graphs; just like the distinction of measure theory and probability theory. The unification of the MTP in this work does not cover CBERs since we focus on random metric spaces. We will still use this theory in proving some of the results. This will be discussed further in Subsection 4.6.

In this work, we also consider random measures and point processes on a given unimodular rmm space. In this view, the base space can be thought of a generalization of the Euclidean or hyperbolic spaces. Heuristically, the intensity of a point process $\Phi$ on $[\boldsymbol{X},\boldsymbol{o},\boldsymbol{\mu}]$ is the expected number of points of $\Phi$ per unit measure (measured by $\boldsymbol{\mu}$). Also, the mean of some quantity on the points of $\Phi$ is the expectation of that quantity at a typical point of $\Phi$. These notions will be formalized by generalizing the Palm theory in Section 6, which is the second main task of this work. We will generalize various notions and theorems in stochastic geometry; e.g., the Campbell measure, Pam distribution, the Campbell formula, Mecke’s theorem, Neveu’s exchange formula and the existence of balancing transport kernels. We also show that the Palm theory generalizes Palm inversion at the same time (i.e., reconstructing the probability measure from the Palm distribution) and also generalizes various constructions of unimodular graphs by adding vertices and edges to another given unimodular graph (e.g., the dual of a unimodular planar graph).

An important application of the Palm theory in this work is to construct a countable Borel equivalence relation equipped with an invariant measure. This is done by means of adding a Poisson point process and considering its Palm version and is used to prove some of the theorems (e.g., amenability and invariant disintegration) using the results of the theory of countable Borel equivalence relations. These applications are included in Section 7.

If $X$ is a metric space, the closed ball of radius $r$ centered at $x\in X$ is denoted by $\overline{B}_{r}(x):=\{y\in X:d(x,y)\leq r\}$. The open ball is denoted by $B_{r}(x)$. Also, all measures are assumed to be Borel measures. If $f:X\to\mathbb{R}^{\geq 0}$ is measurable, the measure $f\mu$ on $X$ is defined by $(f\mu)(A):=\int_{A}fd\mu$. If $\mu$ is a probability measure, biasing $\mu$ by $f$ means considering the probability measure $\frac{1}{c}(f\mu)$, where $c=\int_{X}fd\mu$. The Prokhorov metric between finite measures on $X$ is denoted by $d_{P}$.

A rooted measured metric space, abbreviated by a rmm space, is a tuple $(X,o,\mu)$ where $X$ is a metric space, $\mu$ is a non-negative Borel measure on $X$ and $o$ is a distinguished point of $X$ called the root. For simplicity of notation, the metric on $X$ is always denoted by $d$ if there is no ambiguity, otherwise, it will be denoted by $d^{X}$. In this paper, we always assume that the metric space is boundedly compact (i.e. every closed ball is compact) and $\mu$ is boundedly finite (i.e. every ball has finite measure under $\mu$). Similarly, a doubly-rooted measured metric space is a tuple $(X,o_{1},o_{2},\mu)$, where $\mu$ is as before and $o_{1}$ and $o_{2}$ are two ordered distinguished points of $X$. Note that we do not require that $X$ is a geodesic space. It can be even disconnected.

An isomorphism (or a GHP-isometry) between two rmm spaces $(X,o,\mu)$ and $(X^{\prime},o^{\prime},\mu^{\prime})$ is a bijective isometry $\rho:X\rightarrow X^{\prime}$ such that $\rho(o)=o^{\prime}$ and $\rho_{*}\mu=\mu^{\prime}$, where $\rho_{*}(\mu)$ represents the push forward of the measure $\mu$ under $\rho$. If such $\rho$ exists, then $(X,o,\mu)$ and $(X^{\prime},o^{\prime},\mu^{\prime})$ are called isomorphic. Isomorphisms between doubly-rooted spaces are defined similarly. Under this equivalence relation, the equivalence class containing $(X,o,\mu)$ (resp. $(X,o_{1},o_{2},\mu)$) is denoted by $[X,o,\mu]$(resp. $[X,o_{1},o_{2},\mu]$).

Let $\mathcal{M}_{*}$ be the set of equivalence classes of rmm spaces under isomorphisms. Similarly, define $\mathcal{M}_{**}$ for doubly-rooted measured metric spaces. These sets become Polish spaces under the Gromov-Hausdorff-Prokhorov (GHP) metric (see [31] for the general case of the GHP metric and its history). To keep focus on the main goals, we skip the definition of the metric and we just mention the GHP topology (see Section 3 of [29]): A sequence $[X_{n},o_{n},\mu_{n}]$ converges to $[X,o,\mu]$ when these spaces can be embedded isometrically in a common boundedly-compact metric space $Z$ such that, after the embedding, $o_{n}$ converges to $o$, $X_{n}$ converges to $X$ as closed subsets of $Z$ (with the Fell topology) and $\mu_{n}$ converges to $\mu$ in the vague topology (convergence against compactly-supported continuous functions).

Polishness of $\mathcal{M}_{*}$ allows one to use classical tools in probability theory for studying random elements of $\mathcal{M}_{*}$, which are called random rmm spaces here. In particular, every random rooted graph defines a random rmm space (equipped with the graph-distance metric and the counting measure on the vertices). The same is true for point processes. Also, $\mathbb{R}^{d}$, rooted at the origin and equipped with a random measures on $\mathbb{R}^{d}$, defines a random rmm space. The condition of boundedly-compactness matches the conditions of locally-finiteness in each example.

In this subsection, we define unimodular rmm spaces and a few basic examples.

In words, the expected outgoing mass from the root should be equal to the expected incoming mass to the root. The case $\boldsymbol{\mu}:=0$ trivially satisfies the MTP and is not very interesting. However, there are some interesting classes of non-proper examples (e.g., when extending a unimodular graph, Example 6.23, or $\mathbb{R}^{d}$ or any other space equipped with a point process or a random measure, Example 4.1 and Definition 6.1). Another basic example is when $[\boldsymbol{X},\boldsymbol{o}]$ is arbitrary and $\boldsymbol{\mu}:=\delta_{\boldsymbol{o}}$ is the Dirac measure at $\boldsymbol{o}$. More generally, finite measures provide basic examples explained below.

Although the infinite case is more interesting for our purpose, the finite case appears in many important examples of scaling limits when the limiting space is compact (see Subsection 4.3). We will see that the re-rooting invariance property in these examples is a quick corollary of the fact that weak convergence preserves unimodularity (Lemma 3.1).

A heuristic interpretation of unimodularity is that the root is a typical point of the measure $\boldsymbol{\mu}$. When $\boldsymbol{\mu}$ is a finite measure, this heuristic is rigorous according to Example 2.4. In the general case, the heuristic is that the expectation of any (equivariant) quantity evaluated at the root is an interpretation of the average of that quantity over the points of the metric space. More precisely, for measurable functions $f(\boldsymbol{o}):=f(\boldsymbol{X},\boldsymbol{o},\boldsymbol{\mu})$, the expectation $\mathbb{E}\left[f(\boldsymbol{o})\right]$ is an interpretation of the average of $f(\boldsymbol{X},\cdot,\boldsymbol{\mu})$ over the points of $\boldsymbol{X}$, where the average is taken according to the measure $\boldsymbol{\mu}$ (in fact, this should be modified in the non-ergodic case; see Subsection 5.4). See Corollary 5.8 for a rigorous statement.

This is similar to the case of unimodular graphs, but the proof is more involved since a convergent sequence of rmm spaces does not necessarily stabilize in a given window. This will be handled by a generalization of Strassen’s theorem.

Lemma 3.1 and Example 2.4 give naturally the following generalization of soficity:

This generalizes Question 10.1 of [3], which involves unimodular graphs. Note that every such weak limit is unimodular. Also, one can assume that $X_{n}$ is a finite metric space and $\mu_{n}$ is a multiple of the counting measure without loss of generality.

Unimodularity means heuristically that the root is a typical point. In particular, every property of the points that has zero chance to be seen at the root, is observed at almost no other point. This is formalized in the following easy but important lemma.

This is a generalization of Lemma 2.3 of [3]. It can also be generalized by allowing extra randomness as well, but some care is needed that will be discussed in Subsection 5.1.

By letting $\boldsymbol{S}:=\mathrm{supp}(\boldsymbol{\mu})$, one immediately obtains:

This generalizes Corollary 2.10 of [5] for unimodular graphs. As mentioned before Proposition 4 of [35], this is related to Poincaré’s recurrence theorem. See also Lemma 5.15 for a further generalization.

In the following proposition, re-rooting a unimodular rmm space is considered even in the non-compact case. Assume for each rmm space $(X,o,\mu)$ a probability measure $k_{o}=k_{(X,o,\mu)}$ is given on $X$. This will be considered as the law for changing the root from $o$ to a new root. By letting $(X,\mu)$ fixed and letting $o$ vary, this can be regarded as a Markovian kernel $k^{(X,\mu)}$ on $X$ (assuming the following measurability condition). It is called an equivariant Markovian kernel if it is invariant under the isomorphisms of rmm spaces and for each measurable set $A\subseteq\mathcal{M}_{**}$, the function $[X,o,\mu]\mapsto k_{o}(\{y\in X:[X,o,y,\mu]\in A\})$ is a measurable function on $\mathcal{M}_{*}$.

Given a deterministic pair $(X,\mu)$, an equivariant Markovian kernel $k$ transports $\mu$ to another measure on $X$ defined by $\mu^{\prime}(\cdot):=\int_{X}k_{y}(\cdot)d\mu(y)$. If $\mu^{\prime}=\mu$, then $\mu$ is called a stationary measure for the kernel on $X$.

Let $[\boldsymbol{X},\boldsymbol{o},\boldsymbol{\mu}]$ be a unimodular rmm space and $k$ be an equivariant Markovian kernel. Conditional to $[\boldsymbol{X},\boldsymbol{o},\boldsymbol{\mu}]$, choose $\boldsymbol{o}^{\prime}\in\boldsymbol{X}$ randomly with distribution $k_{\boldsymbol{o}}(\cdot)$, which is regarded as a new root.

In particular, in the compact case, this theorem implies the re-rooting invariance mentioned in Example 2.4. This also generalizes the invariance of Palm distributions under bijective point-shifts and a similar statement for unimodular graphs (Proposition 3.6 of [5]). See also Subsection 5.2 for a generalization to the case where an initial biasing is considered.

Before proving the theorem, we need some lemmas. We will first use the MTP when $k(\boldsymbol{o})$ is absolutely continuous w.r.t. $\boldsymbol{\mu}$, and then, we will deduce the general case from the first case.

In fact, the proof shows that there always exists such a measurable Radon-Nikodym derivative $f$.

In fact, this is just the composition of the Markovian kernel corresponding to $g$ (given $(X,\mu)$) with its time-reversal. This ensures that the new kernel preserves $\mu$.

A point process in $\mathbb{R}^{d}$ is a random discrete subset of $\mathbb{R}^{d}$. More generally, a random measure on $\mathbb{R}^{d}$ is a random boundedly-finite measure on $\mathbb{R}^{d}$. A point process or random measure $\Phi$ is stationary if $\Phi+t$ has the same distribution as $\Phi$ for every $t\in\mathbb{R}^{d}$. If so, the Palm version of $\Phi$ can be defined in various ways and means heuristically seeing $\Phi$ from a typical point or conditioning on $0\in\Phi$. Formally, if $B\subseteq\mathbb{R}^{d}$ is an arbitrary bounded Borel set, bias by $\Phi(B)$ and move the origin to a random point with distribution ${\left.\kern-1.2pt\Phi\vphantom{\big{|}}\right|_{B}}$; i.e.,

where $\lambda_{\Phi}$ is a constant called the intensity of $\Phi$, which is assumed to be positive and finite here. For other methods to defined the Palm version, one can mention the use of the Campbell measure and tessellations, which will be generalized in Section 6.

The Palm version satisfies Mecke’s theorem [37], which can be rephrased equivalently⁵⁵5Mecke’s theorem is stated in different notations in the literature, but it is easy to transform the equation in this form. in the MTPs (1.2) and (1.3) (see also [30] and [33]). Mecke proved in addition that with some finite moment condition, this equation characterizes the Palm versions of stationary point process. Also, one can reconstruct the stationary version from the Palm version via a formula which is known as Palm inversion. Without the finite moment condition, the MTPs (1.2) and (1.3) define larger classes of point processes and random measures, which are called point-stationary point processes and mass-stationary random measures in [33].⁶⁶6The definition of point-stationarity in [33] is more complicated, but it is proved in [33] that it is equivalent to Mecke’s condition. For example, one can mention the zero set of the simple random walk, its graph, and the local time at zero of Brownian motion (see [6] for further examples). The MTPs (1.2) and (1.3) directly imply the following.

As mentioned in the introduction, a unimodular (random) graph [3] is a random rooted graph that satisfies the MTP (1.1). Also, the MTP provides many connections between unimodular graphs and (point-) stationary point process. As a common generalization, [6] defines unimodular discrete spaces, which are random boundedly-finite discrete metric spaces that satisfy a similar MTP. Since the Benjamini-Schramm topology for rooted graphs is consistent with the GHP topology (see e.g., [29]), The MTP implies the following.

In considering graphs as rmm spaces, some information might be lost (e.g., parallel edges and loops). However, one can keep these information by considering rmm spaces equipped with some additional structure (see Subsection 5.1). In this sense, unimodular rmm spaces generalize unimodular graphs.

More generally, [3] defines unimodular networks, which are unimodular graphs equipped with some marks on the vertices and edges. Also, [6] defines unimodular marked discrete spaces. Similarly to the above example, these can be regarded as unimodular rmm spaces equipped with suitable additional structures.

Let $\boldsymbol{G}_{n}$ be a sequence of finite graphs (or metric spaces), which might be deterministic or random, and let $\boldsymbol{o}_{n}$ be a vertex in $\boldsymbol{G}_{n}$ chosen uniformly at random. A (measured) scaling limit of $\boldsymbol{G}_{n}$ is the weak limit of a sequence of the form $[\epsilon_{n}\boldsymbol{G}_{n},\boldsymbol{o}_{n},\delta_{n}\boldsymbol{\mu}_{n}]$ as random elements in $\mathcal{M}_{*}$. Here, $\boldsymbol{\mu}_{n}$ is the uniform measure on the vertices of $\boldsymbol{G}_{n}$ and $\epsilon_{n}\boldsymbol{G}_{n}$ means that the graph-distance metric is scaled by $\epsilon_{n}$. Likewise, a subsequential (measured) scaling limit is defined as the limit along a subsequence. The coefficients $\epsilon_{n}$ and $\delta_{n}$ may also depend on the non-rooted graph $\boldsymbol{G}_{n}$ (but not on $\boldsymbol{o}$). Since $\boldsymbol{o}_{n}$ is chosen uniformly, $[\boldsymbol{G}_{n},\boldsymbol{o}_{n}]$ is a unimodular graph (in fact, it is enough to have the re-rooting invariance property, see Example 2.4).

Another setting for scaling limits is zooming-out a given unimodular graph or rmm space $[\boldsymbol{X},\boldsymbol{o},\boldsymbol{\mu}]$; i.e., (subsequential) limits of $[\epsilon_{n}\boldsymbol{X},\boldsymbol{o},\delta_{n}\boldsymbol{\mu}]$ when the factors $\epsilon_{n}$ and $\delta_{n}$ do not depend on $\boldsymbol{o}$ and converge to zero appropriately. There are also examples of zooming-in a given rmm space (even compact) at the root, which is the case when $\epsilon_{n},\delta_{n}\to\infty$ appropriately. Various examples will be mentioned below.

In each of these settings, Lemma 3.1 implies the following.

Non-compact scaling limits have also some re-rooting invariance property, which is stated in Theorem 3.8.

In this subsection, we investigate when a deterministic rmm space $[X,o,\mu]$ is a unimodular rmm space. By Lemma 3.4, it is necessary that $(X,\mu)$ is transitive; i.e., $[X,o,\mu]$ is isomorphic to $[X,y,\mu]$ for every $y\in X$. However, transitivity is not enough. The following generalizes the analogous result about transitive graphs (see [7]).

The first part is proved in a more general form in Theorem  4.18. The second part can also be proved similarly and the proof is skipped for brevity.

It is also easy to prove this corollary by verifying the MTP directly.

Let $(X,\mu)$ be a deterministic measured metric space. In this subsection, we investigate when one can choose a random root $\boldsymbol{o}\in X$ such that $[X,\boldsymbol{o},\mu]$ is unimodular. This generalizes the case of quasi-transitive graphs;⁷⁷7This is the only motivation for the title of this subsection. see [7] and Section 3 of [3].

Let $\Gamma$ be the automorphism group of $(X,\mu)$. If $\Gamma$ contains only the identity function, then it is straightforward to see that $\mu$ should be a finite measure and the distribution of $\boldsymbol{o}$ should be proportional to $\mu$ (see Example 2.4). However, if $\Gamma$ is nontrivial, the situation is more involved.

The orbit of $x\in X$ is the set $\Gamma x=\{\gamma x:\gamma\in\Gamma\}$. Boundedly-compactness of $X$ implies that $\Gamma$ is a locally-compact topological group. Let $\left|\cdot\right|$ be a left-invariant Haar measure on $\Gamma$. Let $B$ be an arbitrary open set that intersects every orbit in a nonempty bounded set. For instance, one can let $B:=\bigcup_{n\geq 1}B_{n}(o)\setminus\Gamma\overline{B}_{n-2}(o)$ for an arbitrary $o\in X$. For $x\in X$, define

The assumptions imply $0<h(x)<\infty$. Also, for $\alpha\in\Gamma$, one has $\Gamma_{\alpha x,B}=\alpha\Gamma_{x,B}$, and hence, $h(\alpha x)=h(x)$.

This generalizes Theorem 3.1 of [3]. Indeed, in the case where $X$ is a graph or network and $\mu$ is the counting measure, one can choose $B$ by choosing exactly one point from every orbit. In this case, for $x\in B$, $h(x)$ is the inverse of the measure of the stabilizer of $x$. One can also generalize the if side of the claim to the case where $\Gamma$ is a closed subgroup of the automorphism group that acts transitively.