Cox Sprinklings and Causal Sets from Random Lorentzian Metrics

1. Introduction

Causal set theory proposes a fundamentally discrete description of spacetime [see, e.g., 1,2]. A causal set is a pair $(C,\prec )$ in which ≺ is a transitive and non-circular causal relation and every causal interval contains finitely many elements. The relation ≺ records the causal structure, while the number of elements is intended to encode spacetime volume in discrete units. This motivates the guiding principle that causal order and number together should retain geometric information. An abstract causal set, however, does not by itself establish that it approximates a continuum spacetime or identify the continuum geometry it may represent.

Poisson sprinkling provides the standard probabilistic bridge between a Lorentzian spacetime and a causal set [1]. For a fixed time-oriented Lorentzian metric, points are sampled with constant density relative to spacetime volume and ordered using the chronological relation. This construction gives a probabilistic discretisation of continuum spacetime and, in Minkowski spacetime, introduces no preferred inertial frame in distribution [3]. A modern measure-theoretic treatment of sprinkling on general spacetimes, including probabilities for finite causal set isomorphism classes, is given in [4]. After the sampled locations and labels are discarded, only the number of points and their unlabelled causal order remain. Although Poisson sprinkling is standard, the causal set continuum correspondence has also been considered for more general point processes on fixed Lorentzian spacetimes [5]. For deterministic Lorentzian geometries, the probabilities of obtaining finite causal set patterns from random sprinklings have been used to compare spacetimes and study their infinite density limit [6]. More recently, reconstruction results for deterministic and weighted spacetimes have been obtained from the probability distributions of sampled chronological relations under fixed metrics [7]. Recent numerical work has also studied causal set ensembles generated from varying conformal factors using label invariant graph observables [8].

This work allows the Lorentzian metric to be random while retaining ordinary Poisson sprinkling conditional on each realisation. The resulting location process is a Cox process directed by the random intensity measure obtained by scaling the realised spacetime volume measure by the sprinkling density [9], while the same realised metric determines the chronological relation. We call this location process a Cox sprinkling and its associated coordinate-free causal set a Cox-generated random causal set. Randomness in the volume measure can generate additional variation and dependence in point counts, while randomness in the light-cone structure can change the causal order. Since the same realised metric governs both the sampling distribution and the chronological relation, the number of sampled points and their causal order may jointly retain information about the latent geometry. Fixed geometry Poisson sprinkling is recovered as the special case in which the metric is deterministic.

In the remainder of the paper, we first define Cox sprinklings and Cox-generated random causal sets, develop the volume-order split and identify the log-Gaussian subclass. We then study number-order dependence and define the coordinate-free Cox sampling information. Finally, we establish high density, parametric and replicated recovery results.

2. Random Metric and Cox Sprinkling

2.1. From Poisson Sprinkling to Cox Sprinkling

Let M be a second countable smooth manifold of dimension $d\ge 2$, and let $W\subset M$ be a relatively compact Borel observation window. Let G be a measurable random element taking values in the space of $C^2$ Lorentzian metrics on M, equipped with the Borel $\sigma$-field induced by the compact-open $C^2$ topology. Each realisation is assumed to be time-orientable and equipped with a chosen time orientation.

For each realisation $G=g$, let $V_g$ denote the Lorentzian volume measure induced by g. For every Borel set $A\subset M$, define

$$V_g\left(A\right)={\int }_{A}d{V}_g\left(x\right).$$

In local coordinates,

$$d{V}_g\left(x\right)=\sqrt{|det(g_{\mu \nu }\left(x\right)\left)\right|}d{x}^1\cdots d{x}^d.$$

Accordingly, define

$$V_G\left(A\right)={\int }_{A}d{V}_G\left(x\right).$$

Under the measurability assumption below, $V_G{|}_W$ is a random measure on W, and $V_G\left(W\right)$ is the random spacetime volume of the observation window W. Let ${\prec }_G$ denote the random strict chronological relation induced by G. For each realisation $G=g$, let ${\prec }_g$ denote the corresponding realised relation. We assume that

1.

$0<V_G\left(W\right)<\infty$ almost surely, meaning that $\mathbb{P}(0<V_G\left(W\right)<\infty )=1$. Finiteness ensures that a conditional Poisson sprinkling contains finitely many points in W almost surely, while positivity excludes the degenerate zero-volume case.

2.

The map $g\mapsto V_g\left(A\right)$ is measurable for every Borel set $A\subset W$. Hence $V_G\left(A\right)$ is a random variable and $V_G{|}_W$ is a random measure.

3.

The map $(g,x,y)\mapsto 1\mathbf{1}\left\{x{\prec }_{g}y\right\}$ on the metric state space $\times M\times M$ is measurable. This ensures that the induced random order and its associated probabilities are measurable.

4.

For almost every realisation $G=g$, the spacetime $(M,g)$ is chronological, meaning that it contains no closed time-like curves.

Fix a sprinkling density $\rho >0$ and define the random measure ${\mathrm{\Lambda }}_G$ on W by

$${\mathrm{\Lambda }}_G\left(A\right)=\rho V_G\left(A\right),A\in \mathcal{B}\left(W\right).$$

Let $\mathrm{\Phi }$ be a point process on W, viewed as a random locally finite counting measure, and write $N=\mathrm{\Phi }\left(W\right)$ for the total number of points in W. Conditionally on G, assume that

$$\mathrm{\Phi }\mid G\sim PPP\left({\mathrm{\Lambda }}_G\right).$$

Conditionally on G,

$$N=\mathrm{\Phi }\left(W\right)\sim Poisson\left({\mathrm{\Lambda }}_G\left(W\right)\right)=Poisson\left(\rho V_G\left(W\right)\right).$$

Since $V_G\left(W\right)<\infty$ almost surely, $N<\infty$ almost surely. For almost every realisation $G=g$, the chronological relation ${\prec }_g$ is transitive, and chronology makes it non-circular. Local finiteness is automatic because $\mathrm{\Phi }$ is finite in W. Hence the induced ordered set is a finite causal set almost surely.

For every realised metric g and every $x\in M$, $V_g\left(\left\{x\right\}\right)=0$. Hence the conditional Poisson process has no repeated sampled locations almost surely. We therefore identify $\mathrm{\Phi }$ with its sampled point configuration and define the induced embedded causal set by

$${\tilde{\mathsf{C}}}_G=\left(\mathrm{\Phi },\{(x,y)\in \mathrm{\Phi }\times \mathrm{\Phi }:x{\prec }_{G}y\}\right)=\left(\mathrm{\Phi },{\prec }_G{|}_{\mathrm{\Phi }\times \mathrm{\Phi }}\right).$$

The notation ${\prec }_G{|}_{\mathrm{\Phi }\times \mathrm{\Phi }}$ denotes the restriction of the relation ${\prec }_G$, which is defined on $M\times M$, to pairs of sampled points, so only chronological relations between points of $\mathrm{\Phi }$ are retained. After discarding the sampled locations and point labels, define the observed causal set by

$${\mathsf{C}}_G=\left[(\mathrm{\Phi },{\prec }_G{|}_{\mathrm{\Phi }\times \mathrm{\Phi }})\right],$$

where the brackets denote the equivalence class under order isomorphisms. The coordinate-free observation is $(N,{\mathsf{C}}_G)$. Although N is determined by the number of elements of ${\mathsf{C}}_G$, we display it separately to distinguish the point count from the causal order pattern conditional on that count.

For a fixed time-oriented Lorentzian metric g such that $(M,g)$ is chronological, a Poisson sprinkling of density $\rho >0$ in W is a point process satisfying ${\mathrm{\Phi }}_g\sim PPP\left(\rho V_g{|}_W\right)$. Its embedded causal set is

$${\tilde{\mathsf{C}}}_g=\left({\mathrm{\Phi }}_g,{\prec }_g{|}_{{\mathrm{\Phi }}_g\times {\mathrm{\Phi }}_g}\right),$$

which retains the sampled locations. After discarding the locations and point labels, its coordinate-free projection is

$${\mathsf{C}}_g=\left[{\tilde{\mathsf{C}}}_g\right].$$

Since $V_g\left(W\right)<\infty$, ${\mathsf{C}}_g$ is finite almost surely and is called a Poisson-generated random causal set. Its coordinate-free observation is $(N_g,{\mathsf{C}}_g)$, where $N_g={\mathrm{\Phi }}_g\left(W\right)$.

By analogy with Poisson sprinkling, we define Cox sprinkling and the associated Cox-generated random causal set as follows.

Definition 1

(Cox sprinkling and Cox-generated random causal set). The location process Φ defined above is called a Cox sprinkling directed by ${\mathrm{\Lambda }}_G$, and ${\mathsf{C}}_G$ is called its associated Cox-generated random causal set. For each joint realisation $(G,\mathrm{\Phi })=(g,\varphi )$, its realised value is

$$\mathsf{C}(g,\varphi )=\left[(\varphi ,{\prec }_g{|}_{\varphi \times \varphi })\right].$$

Then ${\mathsf{C}}_G=\mathsf{C}(G,\mathrm{\Phi })$.

For the Poisson-generated random causal set ${\mathsf{C}}_g$, randomness arises only from the sprinkling ${\mathrm{\Phi }}_g$. For the Cox-generated random causal set ${\mathsf{C}}_G$, randomness arises from the conditional Poisson sprinkling and may also arise from the random metric G. Conditionally on G, the location process $\mathrm{\Phi }$ is an ordinary Poisson point process with intensity measure ${\mathrm{\Lambda }}_G$. Since the conditional distribution of $\mathrm{\Phi }$ given G depends on G only through ${\mathrm{\Lambda }}_G$,

$$\mathrm{\Phi }\mid {\mathrm{\Lambda }}_G\sim PPP\left({\mathrm{\Lambda }}_G\right).$$

Hence $\mathrm{\Phi }$ is a Cox process directed by ${\mathrm{\Lambda }}_G$. The random measure ${\mathrm{\Lambda }}_G$ specifies the conditional distribution of the sampled locations but does not, in general, determine their chronological relations. For a general random metric, the conditional probability distribution of the complete ordered construction is therefore specified by the joint random pair $({\mathrm{\Lambda }}_G,{\prec }_G)$. If $G=e^{2\mathrm{\Omega }}{g}_0$ almost surely for some random scalar field $\mathrm{\Omega }$ and a fixed time-oriented Lorentzian metric $g_0$, with every realisation inheriting the time orientation of $g_0$, then ${\prec }_G={\prec }_{g_0}$ almost surely and the metric randomness enters the ordered construction only through ${\mathrm{\Lambda }}_G$.

Under the preceding assumptions, the point process $\mathrm{\Phi }$ has the following properties.

1.

The point process $\mathrm{\Phi }$ is a Cox process with directing random measure ${\mathrm{\Lambda }}_G$.

2.

The point process $\mathrm{\Phi }$ is a Poisson point process when ${\mathrm{\Lambda }}_G=\nu$ almost surely for some deterministic measure $\nu$. Let ${\mu }_0$ be a fixed deterministic locally finite reference measure on W. If $d\nu =\lambda d{\mu }_0$, then $\mathrm{\Phi }$ is homogeneous with respect to ${\mu }_0$ if there exists a constant $c\ge 0$ such that ${\mu }_0\left(\{x\in W:\lambda (x)\ne c\}\right)=0$, and inhomogeneous otherwise.

A Lorentzian metric G determines both a volume measure $V_G$ and a chronological relation ${\prec }_G$. The volume measure governs the distribution of the sampled locations, while the chronological relation determines the order among the sampled points. Consequently, if $V_G{|}_W=\mu$ almost surely for some deterministic measure $\mu$ while ${\prec }_G$ remains random, then $\mathrm{\Phi }$ is a Poisson point process, although the associated random causal set may still depend on G through its order structure.

2.2. Volume-Order Split

To construct the Volume-order split, we fix a deterministic time-oriented $C^2$ Lorentzian reference metric $g_0$. For each realisation $G=g$, define the relative log-volume field $Z_g$ by

$$d{V}_g=e^{Z_g}d{V}_{g_0},$$

or equivalently,

$$Z_g=log{\displaystyle \frac{d{V}_g}{d{V}_{g_0}}}.$$

So $Z_g\left(x\right)$ describes the local change in spacetime volume under g relative to the reference metric $g_0$. We want to separate the two roles of the realised metric g. Its effect on the distribution of sampled locations is encoded by $d{V}_g=e^{Z_g}d{V}_{g_0}.$ We must also retain the light-cone structure of g, which determines the chronological relation. We therefore choose $\tilde{g}$ to be conformally related to g, so that ${\prec }_{\tilde{g}}={\prec }_g$, and normalise its volume measure by requiring $d{V}_{\tilde{g}}=d{V}_{g_0}.$

Suppose that $\tilde{g}=ag$ for a positive scalar field a. Then $det\tilde{g}=a^{d}detg$. In local coordinates, the volume element of a d-dimensional Lorentzian metric g is

$$d{V}_g=\sqrt{|detg|}d{x}^1\cdots d{x}^d.$$

Hence,

$$d{V}_{\tilde{g}}=a^{d/2}d{V}_g.$$

Since $d{V}_g=e^{Z_g}d{V}_{g_0}$, we have

$$d{V}_{\tilde{g}}=a^{d/2}d{V}_g=a^{d/2}{e}^{Z_g}d{V}_{g_0}.$$

Requiring $d{V}_{\tilde{g}}=d{V}_{g_0}$ therefore gives $a^{d/2}{e}^{Z_g}=1$ and $a=e^{-2Z_g/d}$. We then define

$$\tilde{g}=e^{-2Z_g/d}g$$

and equip $\tilde{g}$ with the time orientation inherited from g. Since $\tilde{g}$ is a positive conformal rescaling of g, ${\prec }_{\tilde{g}}={\prec }_g$. The original metric can be recovered from $(Z_g,\tilde{g})$ through $g=e^{2Z_g/d}\tilde{g}.$ Therefore, for each realised metric g, $Z_g$ records the volume variation relative to $g_0$, while $\tilde{g}$ retains the light-cone structure with its volume measure normalised to $V_{g_0}$. Applying this construction to the random metric G defines the random objects $Z_G$ and $\tilde{G}$. The field $Z_G$ determines the random intensity measure ${\mathrm{\Lambda }}_G$, while $\tilde{G}$ determines the chronological relation ${\prec }_G={\prec }_{\tilde{G}}$.

The Volume-order split depends on the choice of reference metric. For a deterministic $h\in C^2\left(M\right)$, replacing $(g_0,Z_G,\tilde{G})$ by $(e^{2h/d}{g}_0,Z_G-h,e^{2h/d}\tilde{G})$ changes the representation but leaves G, its random intensity measure ${\mathrm{\Lambda }}_G$ and its chronological relation ${\prec }_G$ unchanged. Throughout this paper, $g_0$ is fixed, so $Z_G$ and $\tilde{G}$ are always interpreted relative to this choice.

2.3. The Location Process as a log-Gaussian Cox Process

Let $Z_G$ denote the random log-volume field obtained from the preceding split and suppose that $Z_G$ is a Gaussian random field. The random intensity measure is given by

$${\mathrm{\Lambda }}_G\left(dx\right)=\rho e^{Z_G\left(x\right)}d{V}_{g_0}\left(x\right).$$

Its intensity density relative to $V_{g_0}$ is

$${\lambda }_G\left(x\right)={\displaystyle \frac{d{\mathrm{\Lambda }}_G}{d{V}_{g_0}}}\left(x\right)=\rho e^{Z_G\left(x\right)}.$$

Taking its logarithm gives

$$log{\lambda }_G\left(x\right)=log\left(\rho e^{Z_G\left(x\right)}\right)=log\rho +Z_G\left(x\right).$$

Since $Z_G$ is Gaussian, $log{\lambda }_G$ is also Gaussian. Hence the location process $\mathrm{\Phi }$ is a log-Gaussian Cox process relative to $V_{g_0}$ [10]. The Gaussian assumption constrains only $Z_G$ and imposes no restriction on the normalised metric $\tilde{G}$. Hence the location process may be log-Gaussian Cox while the chronological relation ${\prec }_G={\prec }_{\tilde{G}}$ remains random.

3. Information Recovery

3.1. Conditioning on the Number of Sampled Points in Cox Sprinklings

The coordinate-free observation is $(N,{\mathsf{C}}_G)$, where $N=\mathrm{\Phi }\left(W\right)$ and ${\mathsf{C}}_G$ is the unlabelled causal set obtained after discarding the sampled locations and point labels. Conditionally on G, $\mathrm{\Phi }$ is a Poisson point process on W with intensity measure ${\mathrm{\Lambda }}_G$. We assume that $0<{\mathrm{\Lambda }}_G\left(W\right)<\infty$ almost surely. In particular,

$$N\mid G\sim Poisson\left({\mathrm{\Lambda }}_G\left(W\right)\right).$$

Given G and $N=n$, the sampled locations can be represented by independent random points $X_1,\dots ,X_n$ satisfying

$$\mathbb{P}(X_i\in A\mid G,N=n)={\displaystyle \frac{{\mathrm{\Lambda }}_G\left(A\right)}{{\mathrm{\Lambda }}_G\left(W\right)}}$$

for every $A\in \mathcal{B}\left(W\right)$. Let $N_A=\mathrm{\Phi }\left(A\right)$ denote the number of sampled points in A. We also have

$$N_A\mid G,N=n\sim Binomial\left(n,{\displaystyle \frac{{\mathrm{\Lambda }}_G\left(A\right)}{{\mathrm{\Lambda }}_G\left(W\right)}}\right).$$

Therefore, ${\mathrm{\Lambda }}_G\left(W\right)$ determines the conditional Poisson distribution, and in particular the conditional mean, of the number of sampled points, whereas the ratios ${\mathrm{\Lambda }}_G\left(A\right)/{\mathrm{\Lambda }}_G\left(W\right)$ determine their conditional distribution within W given G and $N=n$.

Given G and $N=n$, the sampled locations induce a causal order by declaring $i\prec j$ if and only if $X_i{\prec }_G{X}_j$. Let ${\mathcal{C}}_n$ denote the set of isomorphism classes of causal sets with n elements. For $C\in {\mathcal{C}}_n$, define

$$p_C\left(G\right)=\mathbb{P}({\mathsf{C}}_G=C\mid G,N=n).$$

Therefore, for $C\in {\mathcal{C}}_n$,

$$\mathbb{P}(N=n,{\mathsf{C}}_G=C\mid G)=\mathbb{P}(N=n\mid G)\mathbb{P}({\mathsf{C}}_G=C\mid G,N=n)=e^{-{\mathrm{\Lambda }}_G\left(W\right)}{\displaystyle \frac{{\mathrm{\Lambda }}_G{\left(W\right)}^n}{n!}}{p}_C\left(G\right).$$

Taking expectations over G gives

$$\mathbb{P}(N=n,{\mathsf{C}}_G=C)=\mathbb{E}\left[e^{-{\mathrm{\Lambda }}_G\left(W\right)}{\displaystyle \frac{{\mathrm{\Lambda }}_G{\left(W\right)}^n}{n!}}{p}_C\left(G\right)\right],$$

while

$$\mathbb{P}(N=n)=\mathbb{E}\left[e^{-{\mathrm{\Lambda }}_G\left(W\right)}{\displaystyle \frac{{\mathrm{\Lambda }}_G{\left(W\right)}^n}{n!}}\right].$$

Consequently,

$$\mathbb{P}({\mathsf{C}}_G=C\mid N=n)={\displaystyle \frac{\mathbb{E}\left[e^{-{\mathrm{\Lambda }}_G\left(W\right)}{\mathrm{\Lambda }}_G{\left(W\right)}^n{p}_C\left(G\right)\right]}{\mathbb{E}\left[e^{-{\mathrm{\Lambda }}_G\left(W\right)}{\mathrm{\Lambda }}_G{\left(W\right)}^n\right]}}.$$

Equivalently, if $\mathbb{P}(N=n)>0$, the tower property gives

$$\mathbb{P}({\mathsf{C}}_G=C\mid N=n)=\mathbb{E}[p_C\left(G\right)\mid N=n].$$

$p_C\left(G\right)$ is a latent random probability, and observing $N=n$ updates its distribution through the random metric G. In a fixed metric g, $p_C\left(g\right)$ is deterministic, so no corresponding update of a latent geometry occurs. The same update can be written relative to the prior mean as

$$\mathbb{P}({\mathsf{C}}_G=C\mid N=n)=\mathbb{E}\left[p_C\left(G\right)\right]+{\displaystyle \frac{Cov\left(e^{-{\mathrm{\Lambda }}_G\left(W\right)}{\mathrm{\Lambda }}_G{\left(W\right)}^n,p_C\left(G\right)\right)}{\mathbb{E}\left[e^{-{\mathrm{\Lambda }}_G\left(W\right)}{\mathrm{\Lambda }}_G{\left(W\right)}^n\right]}}.$$

Hence conditioning on $N=n$ changes the probability of the causal set pattern whenever $Cov\left(e^{-{\mathrm{\Lambda }}_G\left(W\right)}{\mathrm{\Lambda }}_G{\left(W\right)}^n,p_C\left(G\right)\right)\ne 0$. If ${\mathrm{\Lambda }}_G\left(W\right)$ is independent of $p_C\left(G\right)$, the covariance vanishes and

$$\mathbb{P}({\mathsf{C}}_G=C\mid N=n)=\mathbb{E}\left[p_C\left(G\right)\right].$$

For a simple two-geometry example, suppose that $G=g_A$ and $G=g_B$ have equal prior probabilities, with

$${\mathrm{\Lambda }}_{g_A}\left(W\right)=100,{\mathrm{\Lambda }}_{g_B}\left(W\right)=1.$$

For some $C\in {\mathcal{C}}_{50}$, let

$$p_C\left(g_A\right)=0.9,p_C\left(g_B\right)=0.1.$$

After observing $N=50$, the likelihood ratio is

$${\displaystyle \frac{\mathbb{P}(N=50\mid G=g_A)}{\mathbb{P}(N=50\mid G=g_B)}}=e^{-99}{100}^{50}\approx {10}^{57}.$$

Thus the posterior probability of $g_A$ is nearly one, and consequently

$$\mathbb{P}({\mathsf{C}}_G=C\mid N=50)\approx 0.9,$$

whereas the prior mean is $\mathbb{E}\left[p_C\left(G\right)\right]=0.5$. Although $N=50$ is a lower-tail event under $g_A$, it is approximately ${10}^{57}$ times more likely under $g_A$ than under $g_B$.

In a fixed-geometry Poisson sprinkling, the metric is fixed, so observing $N=n$ does not change the geometry used to generate the causal order. In a Cox sprinkling, however, the same random metric G determines the total intensity ${\mathrm{\Lambda }}_G\left(W\right)$, which governs the distribution of the number of sampled points N, the normalised sampling measure ${\mathrm{\Lambda }}_G(·)/{\mathrm{\Lambda }}_G\left(W\right)$, which governs the sampled locations given $N=n$, and the chronological relation ${\prec }_G$, which determines the order among those locations. Consequently, conditioning on $N=n$ can change the distribution of the latent metric through ${\mathrm{\Lambda }}_G\left(W\right)$ and may thereby change the probabilities of the n-element causal set patterns. This effect is absent, for example, when ${\mathrm{\Lambda }}_G\left(W\right)$ is independent of the pair $\left({\mathrm{\Lambda }}_G(·)/{\mathrm{\Lambda }}_G\left(W\right),{\prec }_G\right)$.

3.2. The Coordinate-Free Cox Sampling Information

Assume that $0<V_G\left(W\right)<\infty$ almost surely. For each $\rho >0$, define

$${\mathrm{\Lambda }}_{G,\rho }=\rho V_G{|}_W,{\mathrm{\Phi }}_{\rho }\mid G\sim PPP\left({\mathrm{\Lambda }}_{G,\rho }\right).$$

Keeping G random, ${\mathrm{\Phi }}_{\rho }$ is a Cox process directed by ${\mathrm{\Lambda }}_{G,\rho }$. Write $(N_{\rho },{\mathsf{C}}_{G,\rho })$ for the corresponding coordinate-free observation. Given G and $N_{\rho }=n$, the sampled locations are independent with probability measure $V_G(·)/V_G\left(W\right)$ and are ordered by ${\prec }_G$. Hence $p_C\left(G\right)$ is determined by $V_G(·)/V_G\left(W\right)$ and ${\prec }_G$, and does not depend on $\rho$.

Define the random coordinate-free Cox sampling information by

$${\mathcal{S}}_G=\left(V_G\left(W\right),{\left(p_C\left(G\right)\right)}_{n\ge 1,C\in {\mathcal{C}}_n}\right).$$

It is determined by the latent random metric G. The first component records the random total spacetime volume $V_G\left(W\right)$, while its remaining components record all finite causal order sampling probabilities after the sampled locations and labels have been discarded. For each realisation $G=g$, the random object ${\mathcal{S}}_G$ takes the realised value

$${\mathcal{S}}_g=\left(V_g\left(W\right),{\left(p_C\left(g\right)\right)}_{n\ge 1,C\in {\mathcal{C}}_n}\right),$$

where $p_C\left(g\right)$ is the probability that n independent points sampled according to $V_g(·)/V_g\left(W\right)$ induce the causal set pattern C under ${\prec }_g$.

For every $\rho >0$, $n\ge 1$ and $C\in {\mathcal{C}}_n$, the conditional Poisson distribution of $N_{\rho }$ and the definition of $p_C\left(G\right)$ give

$$\mathbb{P}(N_{\rho }=n,{\mathsf{C}}_{G,\rho }=C\mid G)=e^{-\rho V_G\left(W\right)}{\displaystyle \frac{{\left(\rho V_G\left(W\right)\right)}^n}{n!}}{p}_C\left(G\right).$$

Hence the conditional probability distribution of $(N_{\rho },{\mathsf{C}}_{G,\rho })$ given G is determined by the random coordinate-free Cox sampling information ${\mathcal{S}}_G$. For a realisation $G=g$, this formula becomes

$$\mathbb{P}(N_{\rho }=n,{\mathsf{C}}_{G,\rho }=C\mid G=g)=e^{-\rho V_g\left(W\right)}{\displaystyle \frac{{\left(\rho V_g\left(W\right)\right)}^n}{n!}}{p}_C\left(g\right).$$

At any fixed known $\rho >0$, two admissible realised metrics g and $g^{\prime }$ induce the same conditional probability distribution if and only if

$${\mathcal{S}}_g={\mathcal{S}}_{g^{\prime }}.$$

Equality of the Poisson count distributions implies $V_g\left(W\right)=V_{g^{\prime }}\left(W\right)$, while equality of the causal set distributions given $N_{\rho }=n$ implies $p_C\left(g\right)=p_C\left(g^{\prime }\right)$ for every $n\ge 1$ and $C\in {\mathcal{C}}_n$. Conversely, equality of ${\mathcal{S}}_g$ and ${\mathcal{S}}_{g^{\prime }}$ implies that $(N_{\rho },{\mathsf{C}}_{G,\rho })$ has the same conditional probability distribution under $G=g$ and $G=g^{\prime }$. For each realisation $G=g$, ${\mathcal{S}}_g$ completely characterises the conditional probability distribution of the number of sampled points and their unlabelled causal order. It need not determine $V_g{|}_W$, ${\prec }_g$ or g separately. This equivalence concerns the complete conditional probability distribution of $(N_{\rho },{\mathsf{C}}_{G,\rho })$, and a single finite observation does not determine all components of ${\mathcal{S}}_g$.

3.3. High Density Recovery of the Realised Cox Sampling Information

For the high density analysis, one realisation $G=g$ is drawn and retained as $\rho$ increases, so that only the sprinkling density changes. The aim is to determine whether the corresponding realised information ${\mathcal{S}}_g$ can be recovered from $N_{\rho }$ and ${\mathsf{C}}_{G,\rho }$. For each $k\ge 1$, let ${\mathcal{C}}_k$ denote the finite set of all causal set patterns containing exactly k elements, where two patterns are regarded as the same if they differ only in the names assigned to their elements. Let $c_{\mathrm{obs}}$ be an observed causal set with n elements, and let $C\in {\mathcal{C}}_k$ be a target causal set pattern. For $n\ge k$, define

$${\widehat{p}}_C\left(c_{\mathrm{obs}}\right)={\displaystyle \frac{\mathrm{number}\mathrm{of}\mathrm{subsets}\mathrm{of}{c}_{\mathrm{obs}}\mathrm{containing}\mathrm{exactly}k\mathrm{elements}\mathrm{that}\mathrm{have}\mathrm{pattern}C}{\mathrm{number}\mathrm{of}\mathrm{subsets}\mathrm{of}{c}_{\mathrm{obs}}\mathrm{containing}\mathrm{exactly}k\mathrm{elements}}}.$$

Here A ranges over all subsets of $c_{\mathrm{obs}}$ containing exactly k elements, and $c_{\mathrm{obs}}{|}_A$ denotes the causal order restricted to A. We write $c_{\mathrm{obs}}{|}_A\cong C$ when this restricted causal set has the same causal order pattern as C. Since there are $\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)$ such subsets,

$${\widehat{p}}_C\left(c_{\mathrm{obs}}\right)={\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)}^{-1}\sum _{\begin{array}{c}A\subseteq c_{\mathrm{obs}}\\ \left|A\right|=k\end{array}}\mathbf{1}1\{c_{\mathrm{obs}}{|}_A\cong C\},$$

where the indicator equals one when the condition holds and zero otherwise. When $n<k$, set ${\widehat{p}}_C\left(c_{\mathrm{obs}}\right)=0$. For a two-element chain, this statistic gives the proportion of causally related pairs. For a three-element chain, it gives the proportion of subsets that can be ordered as $x\prec y\prec z$. For an anti-chain, it gives the proportion of subsets whose elements are pairwise causally unrelated.

To formulate recovery of the complete coordinate-free Cox sampling information, enumerate all finite causal set patterns as $C_1,C_2,\dots$. Denote the space $[0,\infty )\times {[0,1]}^{\mathbb{N}}$ by $\mathcal{S}$. A sampling information object can then be represented as an element

$$(a,\mathbf{u})=(a,u_1,u_2,\dots )\in \mathcal{S},$$

where $a\ge 0$ represents the total spacetime volume and $u_j\in [0,1]$ represents the probability associated with the causal set pattern $C_j$. To measure the distance between two sampling information objects $(a,\mathbf{u})$ and $(b,\mathbf{v})$, equip $\mathcal{S}$ with the metric

$$d_{\mathcal{S}}\left((a,\mathbf{u}),(b,\mathbf{v})\right)={\displaystyle \frac{1}{2}}min\{1,|a-b\left|\right\}+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{j=1}^{\infty }}{2}^{-j}|u_j-v_j|.$$

Since $|u_j-v_j|\le 1$ and ${\sum }_{j=1}^{\infty }{2}^{-j}=1$, the infinite series converges and is at most one. Hence $d_{\mathcal{S}}$ takes values in $[0,1]$, and it equals zero exactly when $a=b$ and $u_j=v_j$ for every $j\ge 1$.

Under the preceding enumeration, the previously defined random coordinate-free Cox sampling information can be written as

$${\mathcal{S}}_G=\left(V_G\left(W\right),{\left(p_{C_j}\left(G\right)\right)}_{j\ge 1}\right),$$

and we define its empirical version by

$${\widehat{\mathcal{S}}}_{G,\rho }=\left({\displaystyle \frac{N_{\rho }}{\rho }},{\left({\widehat{p}}_{C_j}\left({\mathsf{C}}_{G,\rho }\right)\right)}_{j\ge 1}\right).$$

Theorem 1

(Recovery of the coordinate-free Cox sampling information). Suppose that $0<V_G\left(W\right)<\infty$ almost surely and that, for every $\rho >0$, ${\mathrm{\Phi }}_{\rho }\mid G\sim PPP\left(\rho V_G{|}_W\right).$ Assume that the same random metric G, and hence the same volume measure $V_G{|}_W$ and chronological relation ${\prec }_G$, is used as ρ increases. As $\rho \to \infty$,

$$d_{\mathcal{S}}\left({\widehat{\mathcal{S}}}_{G,\rho },{\mathcal{S}}_G\right)\stackrel{\mathbb{P}}{\to }0.$$

In particular,

$${\displaystyle \frac{N_{\rho }}{\rho }}\stackrel{\mathbb{P}}{\to }{V}_G\left(W\right),$$

and, for every fixed finite causal set pattern C, with $C\in {\mathcal{C}}_k$ for some $k\ge 1$,

$${\widehat{p}}_C\left({\mathsf{C}}_{G,\rho }\right)\stackrel{\mathbb{P}}{\to }{p}_C\left(G\right).$$

So the complete random coordinate-free Cox sampling information ${\mathcal{S}}_G$ is recovered in probability. The result remains valid when $V_G\left(W\right)$, the normalised sampling measure $V_G(·)/V_G\left(W\right)$ and the chronological relation ${\prec }_G$ are statistically dependent through the same random metric G.

Proof. 

Conditionally on G, $N_{\rho }\sim Poisson\left(\rho V_G\left(W\right)\right)$, so

$$\mathbb{E}\left({\displaystyle \frac{N_{\rho }}{\rho }}\mid G\right)=V_G\left(W\right),Var\left({\displaystyle \frac{N_{\rho }}{\rho }}\mid G\right)={\displaystyle \frac{V_G\left(W\right)}{\rho }}.$$

For $\epsilon >0$, define

$$A_{\rho ,\epsilon }=\left\{\left|{\displaystyle \frac{N_{\rho }}{\rho }}-V_G\left(W\right)\right|>\epsilon \right\}.$$

Since $V_G\left(W\right)<\infty$ almost surely, for almost every realisation $G=g$, Chebyshev’s inequality gives

$$r_{\rho }\left(g\right)=\mathbb{P}(A_{\rho ,\epsilon }\mid G=g)\le {\displaystyle \frac{V_g\left(W\right)}{\rho {\epsilon }^2}}⟶0.$$

By the law of total probability,

$$\mathbb{P}\left(A_{\rho ,\epsilon }\right)=\mathbb{E}\left[\mathbb{P}(A_{\rho ,\epsilon }\mid G)\right]=\mathbb{E}\left[r_{\rho }\left(G\right)\right].$$

Since $0\le r_{\rho }\left(G\right)\le 1$ and $r_{\rho }\left(G\right)\to 0$ almost surely, the dominated convergence theorem gives

$$\mathbb{P}\left(A_{\rho ,\epsilon }\right)=\mathbb{P}\left(\left|{\displaystyle \frac{N_{\rho }}{\rho }}-V_G\left(W\right)\right|>\epsilon \right)=\mathbb{E}\left[r_{\rho }\left(G\right)\right]⟶0.$$

Therefore,

$${\displaystyle \frac{N_{\rho }}{\rho }}\stackrel{\mathbb{P}}{\to }{V}_G\left(W\right).$$

Since $V_G\left(W\right)>0$ almost surely, the preceding convergence also implies that $N_{\rho }\to \infty$ in probability as $\rho \to \infty$.

Fix a finite causal set pattern $C\in {\mathcal{C}}_k$ and suppose that $N_{\rho }=n\ge k$. Given G and $N_{\rho }=n$, the sampled locations $X_1,\dots ,X_n$ are independent with probability measure $V_G(·)/V_G\left(W\right)$. Hence, for every subset A containing exactly k sampled points, the probability that the causal order restricted to A has pattern C is $p_C\left(G\right)$. Since ${\widehat{p}}_C\left({\mathsf{C}}_{G,\rho }\right)$ is the average of the corresponding indicators over all such subsets, linearity of expectation gives

$$\mathbb{E}\left({\widehat{p}}_C\left({\mathsf{C}}_{G,\rho }\right)\mid G,N_{\rho }=n\right)=p_C\left(G\right).$$

The n sampled points have $\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)$ subsets containing exactly k points. Fix one sampled point $X_i$. A subset containing $X_i$ is formed by keeping $X_i$ and choosing its remaining $k-1$ points from the other $n-1$ sampled points. Hence $X_i$ belongs to $\left({\displaystyle \genfrac{}{}{0pt}{}{n-1}{k-1}}\right)$ of the k-element subsets, and the proportion of subsets containing $X_i$ is

$${\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)}{\left(\genfrac{}{}{0pt}{}{n}{k}\right)}}={\displaystyle \frac{k}{n}}.$$

Let $X_i^{\prime }$ be an independent point with the same conditional probability measure $V_G(·)/V_G\left(W\right)$ as $X_i$. Replace the ith sampled location $X_i$ by $X_i^{\prime }$ while leaving all other sampled locations unchanged, and let ${\widehat{p}}_C^{\left(i\right)}$ denote the resulting value of the pattern frequency. Only the subset indicators involving the ith sampled point can change under this replacement. Each indicator takes values in $\{0,1\}$ and therefore changes by at most one. Since ${\widehat{p}}_C$ is the average of these indicators,

$$\left|{\widehat{p}}_C\left({\mathsf{C}}_{G,\rho }\right)-{\widehat{p}}_C^{\left(i\right)}\right|\le {\displaystyle \frac{\left(\genfrac{}{}{0pt}{}{n-1}{k-1}\right)}{\left(\genfrac{}{}{0pt}{}{n}{k}\right)}}={\displaystyle \frac{k}{n}}.$$

Applying the Efron–Stein inequality [11,12] conditionally on G and $N_{\rho }=n$ to the statistic ${\widehat{p}}_C\left({\mathsf{C}}_{G,\rho }\right)$ gives

$$\begin{array}{cc}\hfill Var\left({\widehat{p}}_C\left({\mathsf{C}}_{G,\rho }\right)\mid G,N_{\rho }=n\right)& \le {\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^n}\mathbb{E}\left[{\left({\widehat{p}}_C\left({\mathsf{C}}_{G,\rho }\right)-{\widehat{p}}_C^{\left(i\right)}\right)}^2\mid G,N_{\rho }=n\right]\hfill \\ & \le {\displaystyle \frac{1}{2}}{\displaystyle \sum _{i=1}^n}{\left({\displaystyle \frac{k}{n}}\right)}^2={\displaystyle \frac{k^2}{2n}}\le {\displaystyle \frac{k^2}{n}}.\hfill \end{array}$$

Let $m\ge k$ be an arbitrary fixed sample-size threshold. For every $n\ge m$, the conditional mean and variance satisfy

$$\mathbb{E}\left({\widehat{p}}_C\left({\mathsf{C}}_{G,\rho }\right)\mid G,N_{\rho }=n\right)=p_C\left(G\right),Var\left({\widehat{p}}_C\left({\mathsf{C}}_{G,\rho }\right)\mid G,N_{\rho }=n\right)\le {\displaystyle \frac{k^2}{n}}.$$

By Chebyshev’s inequality, for every $\epsilon >0$,

$$\mathbb{P}\left(\left|{\widehat{p}}_C\left({\mathsf{C}}_{G,\rho }\right)-p_C\left(G\right)\right|>\epsilon \mid G,N_{\rho }=n\right)\le {\displaystyle \frac{k^2}{n{\epsilon }^2}}\le {\displaystyle \frac{k^2}{m{\epsilon }^2}}.$$

When $N_{\rho }<m$, the conditional error probability is at most one, whereas when $N_{\rho }\ge m$, the preceding Chebyshev’s bound is at most $k^2/\left(m{\epsilon }^2\right)$. The conditional law of total probability therefore gives

$$\mathbb{P}\left(\left|{\widehat{p}}_C\left({\mathsf{C}}_{G,\rho }\right)-p_C\left(G\right)\right|>\epsilon \mid G\right)\le \mathbb{P}(N_{\rho }<m\mid G)+{\displaystyle \frac{k^2}{m{\epsilon }^2}}.$$

Taking expectations over G gives

$$\mathbb{P}\left(\left|{\widehat{p}}_C\left({\mathsf{C}}_{G,\rho }\right)-p_C\left(G\right)\right|>\epsilon \right)\le \mathbb{P}(N_{\rho }<m)+{\displaystyle \frac{k^2}{m{\epsilon }^2}}.$$

Since $N_{\rho }\to \infty$ in probability, $\mathbb{P}(N_{\rho }<m)\to 0$ for every fixed m. Letting first $\rho \to \infty$ and then $m\to \infty$ gives

$${\widehat{p}}_C\left({\mathsf{C}}_{G,\rho }\right)\stackrel{\mathbb{P}}{\to }{p}_C\left(G\right).$$

Finally, by the definition of $d_{\mathcal{S}}$,

$$d_{\mathcal{S}}\left({\widehat{\mathcal{S}}}_{G,\rho },{\mathcal{S}}_G\right)={\displaystyle \frac{1}{2}}min\left\{1,\left|{\displaystyle \frac{N_{\rho }}{\rho }}-V_G\left(W\right)\right|\right\}+{\displaystyle \frac{1}{2}}{\displaystyle \sum _{j=1}^{\infty }}{2}^{-j}\left|{\widehat{p}}_{C_j}\left({\mathsf{C}}_{G,\rho }\right)-p_{C_j}\left(G\right)\right|.$$

The preceding convergence $N_{\rho }/\rho \stackrel{\mathbb{P}}{\to }{V}_G\left(W\right)$ implies

$${\displaystyle \frac{1}{2}}min\left\{1,\left|{\displaystyle \frac{N_{\rho }}{\rho }}-V_G\left(W\right)\right|\right\}\stackrel{\mathbb{P}}{\to }0.$$

Similarly, for every fixed j, the preceding convergence ${\widehat{p}}_{C_j}\left({\mathsf{C}}_{G,\rho }\right)\stackrel{\mathbb{P}}{\to }{p}_{C_j}\left(G\right)$ implies

$${\displaystyle \frac{1}{2}}{2}^{-j}\left|{\widehat{p}}_{C_j}\left({\mathsf{C}}_{G,\rho }\right)-p_{C_j}\left(G\right)\right|\stackrel{\mathbb{P}}{\to }0.$$

Fix $\epsilon >0$ and choose J sufficiently large that

$${\displaystyle \frac{1}{2}}\sum _{j>J}{2}^{-j}<{\displaystyle \frac{\epsilon }{2}}.$$

Since the total-volume term and the first J pattern-probability terms converge to zero in probability, their finite sum also converges to zero in probability. Moreover, since ${\widehat{p}}_{C_j}\left({\mathsf{C}}_{G,\rho }\right),p_{C_j}\left(G\right)\in [0,1]$, the contribution of all coordinates with $j>J$ is bounded by ${\displaystyle \frac{1}{2}}{\sum }_{j>J}{2}^{-j}<\epsilon /2$. Consequently,

$$\mathbb{P}\left(d_{\mathcal{S}}\left({\widehat{\mathcal{S}}}_{G,\rho },{\mathcal{S}}_G\right)>\epsilon \right)⟶0,$$

and therefore

$$d_{\mathcal{S}}\left({\widehat{\mathcal{S}}}_{G,\rho },{\mathcal{S}}_G\right)\stackrel{\mathbb{P}}{\to }0.$$

□

The theorem states that, after one random metric G is realised and held fixed, increasing $\rho$ allows its coordinate-free Cox sampling information to be recovered. The scaled number of sampled points $N_{\rho }/\rho$ estimates $V_G\left(W\right)$, while the observed causal set pattern frequencies estimate $p_C\left(G\right)$ for every finite pattern C. Each $p_C\left(G\right)$ depends jointly on the normalised volume measure $V_G(·)/V_G\left(W\right)$, which determines how the points are sampled within W, and the chronological relation ${\prec }_G$, which determines their causal order. Therefore, when ${\prec }_G$ is random, $p_C\left(G\right)$ cannot be interpreted as information about the Cox intensity measure alone. Moreover, different realised metrics may have the same coordinate-free Cox sampling information, so recovery of ${\mathcal{S}}_G$ does not by itself imply recovery of G.

For the LGCP special case, the preceding recovery theorem applies directly to the LGCP subclass introduced above. As $\rho \to \infty$,

$${\displaystyle \frac{N_{\rho }}{\rho }}\stackrel{\mathbb{P}}{\to }{V}_G\left(W\right),{\widehat{p}}_C\left({\mathsf{C}}_{G,\rho }\right)\stackrel{\mathbb{P}}{\to }{p}_C\left(G\right)$$

for every fixed finite causal set pattern C. Consequently, the coordinate-free observations $(N_{\rho },{\mathsf{C}}_{G,\rho })$ recover ${\mathcal{S}}_G$, which in this special case is the realised coordinate-free LGCP sampling information. The Gaussian assumption specifies the LGCP subclass but does not change the recovery argument.

3.4. Parametric Recovery from Coordinate-Free Cox Sampling

Suppose that the random metric has the parametric form $G=g_{\mathrm{\Theta }}$, where $\mathrm{\Theta }$ takes values in a compact metric parameter space $\mathcal{T}$ and $\theta \mapsto g_{\theta }$ is a measurable family of deterministic metrics satisfying the preceding assumptions. For each $\theta \in \mathcal{T}$, define its coordinate-free Cox sampling information by

$$\mathcal{S}\left(\theta \right)={\mathcal{S}}_{g_{\theta }}.$$

Assume that $\theta \mapsto \mathcal{S}\left(\theta \right)$ is continuous and injective. Draw $\mathrm{\Theta }$ once and retain the same realised value as $\rho$ increases. Conditionally on $\mathrm{\Theta }$, let

$${\mathrm{\Phi }}_{\rho }\mid \mathrm{\Theta }\sim PPP\left(\rho V_{g_{\mathrm{\Theta }}}{|}_W\right),$$

and order the sampled points using ${\prec }_{g_{\mathrm{\Theta }}}$. Keeping $\mathrm{\Theta }$ random, ${\mathrm{\Phi }}_{\rho }$ is a Cox process.

Let ${\widehat{\mathrm{\Theta }}}_{\rho }$ be a measurable minimiser satisfying

$${\widehat{\mathrm{\Theta }}}_{\rho }\in \underset{\theta \in \mathcal{T}}{arg\; min}{d}_{\mathcal{S}}\left({\widehat{\mathcal{S}}}_{G,\rho },\mathcal{S}\left(\theta \right)\right).$$

By definition, ${\widehat{\mathrm{\Theta }}}_{\rho }$ minimises $d_{\mathcal{S}}\left({\widehat{\mathcal{S}}}_{G,\rho },\mathcal{S}\left(\theta \right)\right)$ over all $\theta \in \mathcal{T}$. Therefore,

$$d_{\mathcal{S}}\left({\widehat{\mathcal{S}}}_{G,\rho },\mathcal{S}\left({\widehat{\mathrm{\Theta }}}_{\rho }\right)\right)\le d_{\mathcal{S}}\left({\widehat{\mathcal{S}}}_{G,\rho },\mathcal{S}\left(\theta \right)\right)$$

for every $\theta \in \mathcal{T}$. Taking $\theta =\mathrm{\Theta }$ gives

$$d_{\mathcal{S}}\left({\widehat{\mathcal{S}}}_{G,\rho },\mathcal{S}\left({\widehat{\mathrm{\Theta }}}_{\rho }\right)\right)\le d_{\mathcal{S}}\left({\widehat{\mathcal{S}}}_{G,\rho },\mathcal{S}\left(\mathrm{\Theta }\right)\right).$$

By the triangle inequality,

$$d_{\mathcal{S}}\left(\mathcal{S}\left({\widehat{\mathrm{\Theta }}}_{\rho }\right),\mathcal{S}\left(\mathrm{\Theta }\right)\right)\le d_{\mathcal{S}}\left(\mathcal{S}\left({\widehat{\mathrm{\Theta }}}_{\rho }\right),{\widehat{\mathcal{S}}}_{G,\rho }\right)+d_{\mathcal{S}}\left({\widehat{\mathcal{S}}}_{G,\rho },\mathcal{S}\left(\mathrm{\Theta }\right)\right).$$

By the symmetry of $d_{\mathcal{S}}$ and the minimum-distance property of ${\widehat{\mathrm{\Theta }}}_{\rho }$,

$$d_{\mathcal{S}}\left(\mathcal{S}\left({\widehat{\mathrm{\Theta }}}_{\rho }\right),{\widehat{\mathcal{S}}}_{G,\rho }\right)=d_{\mathcal{S}}\left({\widehat{\mathcal{S}}}_{G,\rho },\mathcal{S}\left({\widehat{\mathrm{\Theta }}}_{\rho }\right)\right)\le d_{\mathcal{S}}\left({\widehat{\mathcal{S}}}_{G,\rho },\mathcal{S}\left(\mathrm{\Theta }\right)\right).$$

Substituting this bound into the triangle inequality gives

$$d_{\mathcal{S}}\left(\mathcal{S}\left({\widehat{\mathrm{\Theta }}}_{\rho }\right),\mathcal{S}\left(\mathrm{\Theta }\right)\right)\le 2d_{\mathcal{S}}\left({\widehat{\mathcal{S}}}_{G,\rho },\mathcal{S}\left(\mathrm{\Theta }\right)\right).$$

By Theorem 1, the right-hand side converges to zero in probability. Since a continuous injective map from a compact metric space into a metric space has a continuous inverse on its image, it follows that

$${\widehat{\mathrm{\Theta }}}_{\rho }\stackrel{\mathbb{P}}{\to }\mathrm{\Theta }.$$

Therefore an identifiable parameter of the realised random metric can be recovered from a single high density coordinate-free Cox observation. If $\theta \mapsto \mathcal{S}\left(\theta \right)$ is not injective, the observation can recover at most the equivalence class defined by

$$\theta \sim {\theta }^{\prime }⟺\mathcal{S}\left(\theta \right)=\mathcal{S}\left({\theta }^{\prime }\right).$$

3.5. Replicated Recovery of the Distribution of Cox Sampling Information

The high density recovery result above recovers ${\mathcal{S}}_G$ by holding one realised metric fixed while increasing $\rho$. Since ${\mathcal{S}}_G$ takes values in the previously defined sampling-information space $\mathcal{S}$, write $\mathcal{L}\left({\mathcal{S}}_G\right)$ for its probability distribution on $\mathcal{S}$. Here each observation is generated from an independent metric, and the objective is to recover $\mathcal{L}\left({\mathcal{S}}_G\right)$.

For probability measures P and Q on $\mathcal{S}$, define the bounded-Lipschitz distance by

$$d_{\mathrm{BL}}(P,Q)=\underset{f}{sup}\left|\int fdP-\int fdQ\right|,$$

where f ranges over all functions $f:\mathcal{S}\to [-1,1]$ satisfying $|f\left(s\right)-f\left(t\right)|\le d_{\mathcal{S}}(s,t)$. The bounded-Lipschitz distance measures the difference between P and Q across all such functions. Let $R\ge 1$ denote the number of independent Cox-generated causal set observations. For each R, all observations use the same sprinkling density ${\rho }_R>0$, where ${\rho }_R\to \infty$ as $R\to \infty$. Draw independent metrics $G_{1,R},\dots ,G_{R,R}$ with the same probability distribution as G. Conditionally on these metrics, generate the point processes ${\mathrm{\Phi }}_{1,R},\dots ,{\mathrm{\Phi }}_{R,R}$ independently according to

$${\mathrm{\Phi }}_{i,R}\mid G_{i,R}\sim PPP\left({\rho }_R{V}_{G_{i,R}}{|}_W\right),i=1,\dots ,R.$$

For the ith observation, let ${\mathcal{S}}_{i,R}={\mathcal{S}}_{G_{i,R}}$ denote its unobserved true Cox sampling information and let ${\widehat{\mathcal{S}}}_{i,R}={\widehat{\mathcal{S}}}_{G_{i,R},{\rho }_R}$ denote the corresponding estimate obtained from the observed causal set. Define the empirical distribution of the estimated sampling information by

$${\widehat{P}}_R={\displaystyle \frac{1}{R}}{\displaystyle \sum _{i=1}^R}{\delta }_{{\widehat{\mathcal{S}}}_{i,R}},$$

where ${\delta }_s$ denotes the probability measure that assigns probability one to s.

Theorem 2

(Replicated recovery). Under the assumptions of Theorem 1 and the replicated sampling scheme above, suppose that ${\rho }_R\to \infty$ as $R\to \infty$. Then

$$d_{\mathrm{BL}}\left({\widehat{P}}_R,\mathcal{L}\left({\mathcal{S}}_G\right)\right)\stackrel{\mathbb{P}}{\to }0.$$

In particular, for every $f:\mathcal{S}\to [-1,1]$ satisfying $|f\left(s\right)-f\left(t\right)|\le d_{\mathcal{S}}(s,t)$,

$${\displaystyle \frac{1}{R}}{\displaystyle \sum _{i=1}^R}f\left({\widehat{\mathcal{S}}}_{i,R}\right)\stackrel{\mathbb{P}}{\to }\mathbb{E}\left[f\left({\mathcal{S}}_G\right)\right].$$

Proof. 

Introduce the unobserved empirical distribution

$$P_R^0={\displaystyle \frac{1}{R}}{\displaystyle \sum _{i=1}^R}{\delta }_{{\mathcal{S}}_{i,R}},$$

which is used only in the proof. For every function f used in the definition of $d_{\mathrm{BL}}$, the Lipschitz condition gives

$$\left|{\displaystyle \frac{1}{R}}{\displaystyle \sum _{i=1}^R}f\left({\widehat{\mathcal{S}}}_{i,R}\right)-{\displaystyle \frac{1}{R}}{\displaystyle \sum _{i=1}^R}f\left({\mathcal{S}}_{i,R}\right)\right|\le {\displaystyle \frac{1}{R}}{\displaystyle \sum _{i=1}^R}{d}_{\mathcal{S}}({\widehat{\mathcal{S}}}_{i,R},{\mathcal{S}}_{i,R}).$$

By the definition of $d_{\mathrm{BL}}$, this implies

$$d_{\mathrm{BL}}({\widehat{P}}_R,P_R^0)\le {\displaystyle \frac{1}{R}}{\displaystyle \sum _{i=1}^R}{d}_{\mathcal{S}}({\widehat{\mathcal{S}}}_{i,R},{\mathcal{S}}_{i,R}).$$

Each term on the right has the same probability distribution as $d_{\mathcal{S}}({\widehat{\mathcal{S}}}_{G,{\rho }_R},{\mathcal{S}}_G)$. Theorem 1 gives convergence of this quantity to zero in probability. Since it takes values in $[0,1]$, its expectation also converges to zero, so

$$\mathbb{E}\left[d_{\mathcal{S}}({\widehat{\mathcal{S}}}_{i,R},{\mathcal{S}}_{i,R})\right]⟶0.$$

The preceding bound and Markov’s inequality give

$$\mathbb{P}\left(d_{\mathrm{BL}}({\widehat{P}}_R,P_R^0)>\epsilon \right)\le {\displaystyle \frac{1}{\epsilon R}}{\displaystyle \sum _{i=1}^R}\mathbb{E}\left[d_{\mathcal{S}}({\widehat{\mathcal{S}}}_{i,R},{\mathcal{S}}_{i,R})\right]={\displaystyle \frac{\mathbb{E}\left[d_{\mathcal{S}}({\widehat{\mathcal{S}}}_{1,R},{\mathcal{S}}_{1,R})\right]}{\epsilon }}⟶0.$$

For every R, $({\mathcal{S}}_{1,R},\dots ,{\mathcal{S}}_{R,R})$ has the same probability distribution as an independent sample of size R from $\mathcal{L}\left({\mathcal{S}}_G\right)$. The law of large numbers for empirical probability measures therefore gives

$$d_{\mathrm{BL}}\left(P_R^0,\mathcal{L}\left({\mathcal{S}}_G\right)\right)\stackrel{\mathbb{P}}{\to }0.$$

Finally,

$$d_{\mathrm{BL}}\left({\widehat{P}}_R,\mathcal{L}\left({\mathcal{S}}_G\right)\right)\le d_{\mathrm{BL}}({\widehat{P}}_R,P_R^0)+d_{\mathrm{BL}}\left(P_R^0,\mathcal{L}\left({\mathcal{S}}_G\right)\right),$$

and both terms on the right converge to zero in probability. The stated convergence for each function f follows directly from the definition of $d_{\mathrm{BL}}$. □

Independent high density Cox-generated causal sets recover $\mathcal{L}\left({\mathcal{S}}_G\right)$. If G takes values almost surely in a measurable model class on which the map $g\mapsto {\mathcal{S}}_g$ is injective and has a measurable inverse on its image, then $\mathcal{L}\left({\mathcal{S}}_G\right)$ determines $\mathcal{L}\left(G\right)$. Without such an identifying restriction, metrics with the same Cox sampling information remain indistinguishable. No particular relationship between R and ${\rho }_R$ is required and both need only tend to infinity. Whether $\mathcal{L}\left({\mathcal{S}}_G\right)$ can be identified when the sprinkling density is fixed and only the number of independent observations increases is not addressed here.

4. Outlook

Several questions remain open, one is to determine when the coordinate-free Cox sampling information uniquely identifies a metric. Further work is also needed on finite-sample error bounds, convergence rates and practical inference from finite Cox-generated causal sets. Identifiability from replicated observations at a fixed sprinkling density is another open problem. Further study of concrete non-conformal models and physically motivated distributions for the random metric may clarify which geometric fluctuations remain visible in coordinate-free number and order statistics. A general Cox extension of the causal set Hauptvermutung also remains open.