# Scalar curvature for metric spaces:
Defining curvature for Quantum Gravity without coordinates

Agustín Silva111Contact<EMAIL_ADDRESS>${}^{*{\dagger}}\\!$High Energy
Physics Department, Institute for Mathematics, Astrophysics, and Particle
Physics, Radboud University, Nijmegen, The Netherlands.  Jesse van der
Duin222Contact<EMAIL_ADDRESS>${}^{*{\dagger}}\\!$High Energy Physics
Department, Institute for Mathematics, Astrophysics, and Particle Physics,
Radboud University, Nijmegen, The Netherlands.

###### Abstract

Geometrical properties of spacetime are difficult to study in non-perturbative
approaches to quantum gravity like Causal Dynamical Triangulations (CDT),
where one uses simplicial manifolds to define the gravitational path integral,
instead of Riemannian manifolds. In particular, in CDT one only relies on two
mathematical tools, a distance measure and a volume measure. In this paper, we
define a notion of scalar curvature, for metric spaces endowed with a volume
measure or a random walk, without assuming nor using notions of tensor
calculus. Furthermore, we directly define the Ricci scalar, without the need
of defining and computing the Riemann or the Ricci tensor a priori. For this,
we make use of quantities, like the surface of a geodesic sphere, or the
return probability of scalar diffusion processes, that can be computed in
these metric spaces, as in a Riemannian manifold, where they receive scalar
curvature contributions. Our definitions recover the classical results of
scalar curvature when the sets are Riemannian manifolds. We propose seven
methods to compute the scalar curvature in these spaces, and we compare their
features in natural implementations in discrete spaces. The defined
generalized scalar curvatures are easily implemented on discrete spaces, like
graphs. We present the results of our definitions on random triangulations of
a 2D sphere and plane. Additionally, we show the results of our generalized
scalar curvatures on the quantum geometries of 2D CDT, where we find that all
our definitions indicate a flat ground state of the gravitational path
integral.

##  1 Introduction

Imagine the situation where someone puts you in a space where you are only
given a ruler to measure distances in that space, a way of measuring the
volume of a region in that space. You are allowed to walk wherever you want
inside the space, but not outside. Then, whoever put you in that space,
commands you to determine if the space you are in has intrinsic curvature or
not. You do not have any extra structure, like light rays bent by a near
curvature source, nor any matter fields that you can use. This would be akin
to living in a simplicial manifold or even a graph. What would you do? In this
paper, we define notions of scalar curvature for these kinds of situations.
More formally, we define coarse-grained notions of scalar curvature in metric
spaces endowed with a volume measure or a random walk. In particular, our
measures are best suited for discrete spaces, like simplicial manifolds, or
graphs. But before we dive into these definitions, we address the question:
why one should be interested in defining curvature in these spaces?

The long search for a theory of quantum gravity has provided us with a large
variety of ideas and different approaches to join general relativity with
quantum mechanics [1]. Among those approaches is Causal Dynamical
Triangulations (CDT), a non-perturbative approach to quantize gravity. In a
nutshell, CDT is a lattice approach to study the underlying quantum theory of
gravity. Analogue to lattice QCD, it uses discrete quantum field theory to
study the gravitational path integral [2]. It discretizes spacetime using
simplicial building blocks that respect causality, distinguishing between
time- and space-like distances in its construction. In this approach the
gravitational path integral is given by the continuum limit of the sum over
all possible simplicial manifolds without the need for coordinates, making the
theory manifestly diffeomorphism invariant.

The notion of curvature is fundamental to the discussion of spacetime in the
framework of General Relativity. In order to study the geometry of the
spacetimes appearing in non-perturbative quantum gravity, one can expect
curvature to be an important quantity to understand [3]. However, studying
curvature in the discrete setting of simplicial manifolds is challenging,
because the metric of the simplicial manifolds is not smooth. To be able to
study curvature in discrete approaches to quantum gravity like CDT, different
methods have to be considered. In this work we introduce generalizations of
the Ricci scalar curvature based on the volume of spheres and the return
probability of a random walker, that are especially suited for numerical
computation. These generalized definitions of scalar curvature are motivated
by their use in CDT, but can also be applied to other discrete settings like a
graph.

Previously, several notions of generalized curvature in simplicial manifolds
have been proposed. A standard notion of scalar curvature in simplicial
manifolds was introduced with Regge calculus [4], which measures scalar
curvature using the deficit angle of the simplicial building blocks. When
applying this prescription to 4D CDT, the continuum limit of this notion of
curvature is not defined, because the curvature is given at the scale of the
UV cut-off, the lattice discretization length of spacetime. Therefore, for
this framework one has to study curvature using a different prescription.

Another generalized notion of curvature is the prescription of Ollivier [5].
Here the Ricci curvature tensor is defined by using the transport distance
between two geodesic spheres. In this paper, we take Ollivier’s idea of
defining curvature indirectly, not from taking derivatives of the metric
tensor, but instead, by computing well-defined quantities in metric measure
spaces, that depend on the Ricci scalar in a Riemannian manifold. In [6] an
adaption of Ollivier’s idea is made with the purpose of applying it to CDT,
and is successfully applied, obtaining promising results. In [6] a
generalization of the Ricci tensor is made, called by the authors the Quantum
Ricci Curvature (QRC), which is proportional to the Ricci curvature when
applied to a Riemannian manifold. The QRC offers a computationally feasible,
but expensive calculation in metric measure spaces; it requires the
computation of the distance between all pairs of points of two geodesic
spheres.

The idea behind the notions of curvature presented in [5] and [6] is to
compute a quantity that depends on the Ricci curvature in a Riemannian
manifold, and use this quantity to define curvature in a more general setting.
We would like to emphasize the well previously understood fact, that these are
particular cases of the general idea of: instead of defining curvature
directly, by taking derivatives of the metric tensor, it is possible to define
curvature indirectly, by computing quantities that depend on the curvature in
a Riemannian manifold, that only rely on a distance function and some
additional structure, like a volume measure or a diffusion process. This
indirect/implicit definition of curvature is the method we will use in the
following to establish some new ways to determine the scalar curvature of
metric spaces.

Our method to define scalar curvature is similar in nature to the QRC, so, it
is worth mentioning some of its features. More specifically, the QRC is
defined through the average sphere distance
$\bar{d}\left(S_{p}^{\delta},S_{p^{\prime}}^{\delta}\right)$, between a pair
of spheres with radius $\delta$, centred at points $p$ and $p^{\prime}$, which
are themselves separated by geodesic distance $\delta$. The average sphere
distance $\bar{d}\left(S_{p}^{\delta},S_{p^{\prime}}^{\delta}\right)$ is the
average distance between pairs of points, one in each sphere [6].

In general, the outcome of the calculation of the average sphere distance is
normalized by the scale $\delta$, and is parameterized as

$\frac{\bar{d}\left(S_{p}^{\delta},S_{p^{\prime}}^{\delta}\right)}{\delta}:=C\quantity\big(1-K\quantity(p,p^{\prime},\delta)).$
(1)

Here the quantity $K(p,p^{\prime},\delta)$ defines the _quantum Ricci
curvature_ , and it captures any deviation from constancy as a function of
$\delta$. The prefactor $C$ can depend on many things, like the two points $p$
and $p^{\prime}$, or the dimension of the underlying space. For example, it
has been shown that in different 2D regular lattices the prefactor $C$
changes. The QRC $K(p,p^{\prime},\delta)$ defines a notion of (coarse-grained)
curvature, including directional information. It can be shown that in a
Riemannian manifold it can be expanded in powers of curvature invariants,
containing at the lowest order the Ricci tensor in the direction joining the
two points $p,p^{\prime}$ [6]. The point-pair $p,p^{\prime}$ plays the role of
the vector indicating this direction. Any deviation from $C$ in (1) can be
interpreted as the existence of non-zero curvature of the space. Since it is
out of the scope of this work to fully introduce the QRC, we refer the reader
to the recent review [7].

Crucially, this way of capturing curvature has the disadvantage that the
prefactor $C$ is in general unknown. This is specifically and issue in cases
where analytical calculations of the QRC are out of reach, and one has to rely
on numerical methods. To determine the value of $C$ in the QRC, one takes the
smallest $\delta$ values accessible in the measurements, in CDT this is
typically around $\delta=5$, where one expects lattice artefacts to disappear
for $5\leq\delta$. But, if there are other agents influencing the numerical
result, like discretization artefacts or some other noise, one would never
know if the resulting number is close or far away from the actual value of
$C$, if there is any, since these artefacts usually dominate the small
$\delta$ region, and therefore it is hard to be conclusive in the results.
Furthermore, there has been no clean way of disentangling the prefactor $C$
from the quantum Ricci curvature $K$, at least in CDT applications, so one
only has access to the average sphere distance measurements.

The motivation for this discussion are correlation functions, which are
fundamental objects of quantum field theory. As such, it is interesting to
study these for curvature in the case of quantum gravity.

Using the average sphere distance (1) to compute the correlation functions has
the problem that the correlations of the prefactor $C$ are in general mixed
with the correlations of the QRC, and make more difficult the interpretation
of the correlations. We will later explain how our definitions of curvature
somehow bypass this problem by eliminating the presence of any point dependent
multiplicative prefactor, providing definitions of curvature where
correlations of the obtained quantities might be easier to interpret than
those of using (1).

The structure of this paper is the following. In section 2 we go through two
examples of construction a scalar curvature definition from scratch: one based
on sphere volumes and one based on return probabilities. In section 3 we
generalize this construction of a scalar curvature definition, and introduce
seven methods of doing so for a certain class of scale dependent quantities.
In section 4 we implement our definitions for triangulations of the sphere and
the plane and discuss the applicability of the different definitions. In
section 5 we apply our scalar curvature definitions to the quantum geometries
of 2D CDT. Finally, in section 6 we present our conclusions and discuss them.

##  2 Construction method of generalized scalar curvatures

We begin with a pedagogical introduction to our curvature definitions by
explicitly constructing them from scratch for two different types of spaces:
metric measure spaces, and metric spaces with a random walk.

To this end, we start by introducing two scale-dependent quantities, which can
be computed in a metric space, endowed with a measure or a random walk
respectively. The important property is that these quantities give the Ricci
scalar curvature at the lowest order of a curvature expansion, if computed in
a Riemannian manifold.

#### Measure space:

First we give an example for a metric space endowed with a volume measure.
Recall, a Metric Measure Space (MMS) [8], denoted by $(\mathcal{X},d,\mu)$, is
a mathematical space $\mathcal{X}$ that has a distance function $d$ and a
measure $\mu$. A sphere is defined as
$\mathcal{S}_{p}^{r}=\quantity{p^{\prime}\in\mathcal{X}\,:\,d(p,p^{\prime})=r}$,
with its volume given by
$\norm{\mathcal{S}_{p}^{r}}\doteq\mu(\mathcal{S}_{p}^{r})$. Consider the case
where $\mathcal{X}$ is a $D$-dimensional Riemannian manifold, $d$ is the
geodesic distance function, and $\mu$ is the invariant integration measure.
Then, $\norm{\mathcal{S}_{p}^{r}}$ has the expansion in $r$ [9]

$\norm{\mathcal{S}_{p}^{r}}=\frac{D\,\pi^{\frac{D}{2}}}{\Gamma\quantity(\frac{D+2}{2})}r^{D-1}\quantity(1-\frac{R_{p}}{6D}r^{2}+\order{r^{4}}),$
(2)

where $R_{p}$ is the Ricci scalar at the point $p$. This expansion can be used
to define the scalar curvature as the first scale-dependent correction. To
construct a general definition of curvature, we manipulate the sphere volume,
and define the following scale-dependent quantity333Note that the subscript
$1$ in $\mathcal{SC}_{1}$ is used to differentiate this scalar curvature
definition from the other definitions that are presented in the following
section.

$\mathcal{SC}_{1}\quantity(\norm{\mathcal{S}_{p}^{r}})\doteq
r\partialderivative{r}\quantity(\frac{r\partialderivative{\norm{\mathcal{S}_{p}^{r}}}{r}}{\norm{\mathcal{S}_{p}^{r}}}).$
(3)

In a Riemannian manifold

$\mathcal{SC}_{1}\quantity(\norm{\mathcal{S}_{p}^{r}})=-\frac{2R_{p}}{3D}r^{2}+\order{r^{4}},$
(4)

recovering the Ricci scalar in the limit $r\rightarrow 0$, up to a factor
$-\frac{2}{3D}r^{2}$.

Using these manipulations, we can generalize the notion of scalar curvature to
any metric measure space, and $\mathcal{SC}_{1}(\norm{\mathcal{S}_{p}^{r}})$
can be used as a measure of the coarse-grained scalar curvature at that point.
Of course, for large values of $r$ the interpretation of the coarse-grained
curvature $\mathcal{SC}_{1}(\norm{\mathcal{S}_{p}^{r}})$ can be more difficult
due to possible higher-order curvature corrections. However, in an appropriate
short-scale range the interpretation is straightforward. Note that, in cases
where $r$ is a discrete variable and/or $\norm{\mathcal{S}_{p}^{r}}$ is not
defined for every continuum value of $r$, an analytic continuation is
understood in the definition of $\mathcal{SC}_{1}(||\mathcal{S}_{p}^{r}||)$.
Discrete implementations of the derivatives are needed in explicit
calculations. We discuss such implementations for the discrete examples we
consider in the following sections.

#### Random walk:

Let’s use the previous technique to define curvature from a different
perspective. One can use a similar method to obtain a scalar curvature
definition using the return probability density of the diffusion process.
Diffusion processes can be constructed from Markov chains, where the
probability of the next event in the chain depends only on the current event
of the chain. More specifically, one can model a diffusion process on a metric
space by defining a Random Walk in it [10]. The probability density of the
random walker to move from point to point, is equivalent to the probability
density of a particle that is being diffused to move from point to point on
the space itself [11]. More formally, a Metric Space with a Random Walk
(RWMS), denoted by $\quantity(\mathcal{X},d,\quantity{X_{i}})$, is a
mathematical space $\mathcal{X}$ that has a distance function $d$ and a set of
random variables $\quantity{X_{i}}$ for each point $i$ in $\mathcal{X}$, where
each random variable $X_{i}$ has to be integrable, and have a finite first
moment [5]. The return probability density $P_{p}^{n}$ of the random walk, is
defined as the probability of the random walk to return to $p$ after walking
time $n$. Notice that for the random walk to represent a diffusion process,
the random walk time $n$ has to be a linear function of the diffusion time.

On a Riemannian manifold, a diffusion process with constant diffusion
coefficient is a solution of the heat equation. This means that if one finds a
solution for the return probability density of a random walker, that evolves
satisfying these conditions at every point in the space, one is finding a
solution for the heat kernel of the Laplacian on that space [10].

In these cases, the return probability density of the random walker
$P_{p}^{n}$, is a solution to the heat equation, and therefore is equal to the
heat kernel $K(p,p;\tau)$, evaluated from $p$ to $p$ at diffusion time $\tau$:

$P_{p}^{n}=K(p,p;\tau).$ (5)

We use this correspondence between a random walk and a diffusion process to
construct a definition of scalar curvature in a RWMS. Note that we will not
assume that every metric space with a random walk satisfies this identity, and
we only use it to show that our calculations recover the Ricci scalar in the
case of a Riemannian manifold, with the previous diffusion conditions
satisfied.

For this construction, note that in a D-dimensional Riemannian manifold, the
heat kernel $K(p,p;\tau)$ has the well-known expansion [12]

$K(p,p;\tau)=\frac{1}{(4\pi\tau)^{\frac{D}{2}}}\quantity(1+\frac{R_{p}}{6}\tau+\order{\tau^{2}}).$
(6)

To this expression, we apply several operations to extract the scalar
curvature, analogous to the construction used for the sphere volume. With this
in mind, we define the coarse-grained curvature

$\mathcal{SC}_{1}(P_{p}^{n})\doteq
n\partialderivative{n}(\frac{n\partialderivative{P_{p}^{n}}{n}}{P_{p}^{n}}).$
(7)

In the cases where $n$ is a discrete variable, one is to understand this
definition as an analytical continuation in $n$. Again, we will return to the
discretization of this definition in the following sections. Notice that on a
Riemannian manifold (7) takes the form

$\mathcal{SC}_{1}(P_{p}^{n})=n\partialderivative{n}(\frac{n\partialderivative{P_{p}^{n}}{n}}{P_{p}^{n}})=\tau\partialderivative{\tau}(\frac{\tau\partialderivative{K(p,p;\tau)}{\tau}}{K(p,p;\tau)})=\frac{R_{p}}{6}\tau+\order{\tau^{2}}.$
(8)

So we see that $\mathcal{SC}_{1}(P_{p}^{n})$ recovers the Ricci scalar in the
limit of $n\to 0$, up to a factor $\frac{\tau}{6}$.

Summarizing, the defined coarse-grained scalar curvatures
$\mathcal{SC}_{1}(P_{p}^{n})$ and $\mathcal{SC}_{1}(||\mathcal{S}_{p}^{r}||)$
both recover the Ricci scalar for a Riemannian manifold. In a general metric
space we can use these two quantities to define a scale-dependent curvature
scalar. For example, these scalar curvatures can be used in a graph where one
uses the link distance as distance function, and the number of nodes in a set
as the volume measure. In this space, sphere volumes can be measured with
discrete radius $r$ and can be used to compute the scalar curvature at every
point. Additionally, a random walk can be implemented to determine scalar
curvature in a different way. An advantage of these definitions, is that they
are scale dependent. $\mathcal{SC}_{1}(||\mathcal{S}_{p}^{r}||)$ depends on
the radius of a sphere and $\mathcal{SC}_{1}(P_{p}^{n})$ depends on the
diffusion step of the random walker. This scale dependence can be used to
renormalize the curvature by taking the continuum limit of the discrete
theory, as is done in lattice theories of quantum gravity like CDT. In CDT the
path integral over the gravitational degrees of freedom is regularized as a
sum over discrete spacetimes, and a continuum limit is taken by rescaling all
distances and taking the infinite-volume limit. Our definitions are perfectly
suited for these discrete spacetimes, as there is no notion of tensor calculus
defined on them.

Notice that if one wants to extract the precise dimensionful value of the
scalar curvature $R_{p}$ from (7), one has to know the exact linear relation
between $n$ and $\tau$. To the best of our knowledge, this relation is not
known in a general situation. Despite that, one can approximate this relation
by the well-known, and widely used, identity obtained for a Gaussian random
walk in a Euclidean space

$\operatorname{Var}\quantity(S_{n})=\sigma^{2}n=2\eta\tau,$ (9)

where $\sigma^{2}$ is the variance of the Gaussian variables describing each
step in the random walk, $\operatorname{Var}\quantity(S_{n})$ is the variance
of the sum of $n$ of those variables, and $\eta$ is the diffusion constant of
the process. Using this identification, one can obtain a numerical value for
$R_{p}$ from (7). Even though our implementations for a random walk will not
be in a Euclidean space, and will not use Gaussian variables, on the lack of a
better identification we will still use (7) to determine dimensionful
numerical values of the curvature, $R_{p}$. We will return to this in the next
sections.

##  3 Generalized scalar curvatures

Based on the constructions presented in the previous section, we present a
more general methodology to extract the Ricci scalar from scale-dependent
quantities. We define coarse-grained scalar curvatures for a more general
class of scale-dependent quantities $\mathcal{Q}_{p}^{s}$ centred at point
$p$, that depend on a length scale444For example, this length scale can be the
geodesic radius of a sphere $r$, or the diffusion time $\tau$ of a diffusion
process as we used in the previous section. $s$. Specifically, we consider the
class of quantities $\mathcal{Q}_{p}^{s}$, that in a Riemannian manifold, take
the functional form

$\mathcal{Q}_{p}^{s}=C_{p}s^{\alpha_{p}}(1+\mathcal{K}_{p}(s)),$ (10)

where $C_{p},\alpha_{p}\in\mathbb{R}$. $\mathcal{K}_{p}(s)$ is a scalar
function of $s$ that contains scalar curvature invariants, and it satisfies
$\mathcal{K}_{p}(s)\to 0$ when the curvature of the manifold go to zero, and
thus all curvature invariants go to zero, $R\to 0$. Examples of such
quantities are: the volume of a geodesic ball or sphere, the average distance
between point pairs in a geodesic ball or sphere, or the trace of the heat
kernel of the covariant Laplacian. In these cases the scale $s$ is given by
the sphere/ball radius $r$, or by the diffusion time $\tau$. For all of these
quantities $\mathcal{K}_{p}(s)=0$ in a flat manifold, but in a general
Riemannian manifold it is not. We will extract $\mathcal{K}_{p}(s)$ from
quantities of this class in a Riemannian manifold, and use this construction
to define a notion of coarse-grained scalar curvature in a metric space.

For the rest of this paper we will assume that
$\absolutevalue{\mathcal{K}_{p}(s)}<1$, to avoid singular behaviour of
$\mathcal{Q}$. This assumption allows us to only consider quantities
$\mathcal{Q}_{p}^{s}$, where $\alpha_{p}\geq 0$. If $\alpha_{p}<0$, like for
the heat kernel $K(p,p,\tau)$, we just take the inverse of $\mathcal{Q}$,
which will have the same functional form. This assumption is usually made in
Riemannian geometry, to compute each power in $s$ in the expansion of
$\mathcal{K}_{p}(s)$ at small scales $s$.

Then, we apply a series of operations to extract the curvature using integrals
and/or derivatives of the scale $s$. In a discrete setting, different
combinations of (discretizations of) integrals and derivatives have different
properties. So we propose a set of $7$ different methods to define scalar
curvature in table 1, the best of which can be picked for the specific use
case. We expect that among all of these, a balance between accuracy and
discretization artefacts can be found in each particular implementation. We
call each method a Generalized Scalar Curvature ($\mathcal{GSC}$)555We will
differentiate between the different prescriptions of generalized scalar
curvature with subscripts $\quantity{\mathcal{D},\mathcal{I},1,2,3,4,5}$. The
first to have a different type of subscript because they have an unknown
additive constant, unlike the rest, as we discuss later. We will use
$\mathcal{GSC}_{i}$ to refer to any of them..

Each of the methods requires certain regularity conditions for
$\mathcal{Q}_{p}^{s}$ as a function of $s$. Some prescriptions assume the
existence of one, and even two derivatives of $\mathcal{Q}_{p}^{s}$ with
respect to the scale $s$. Others assume the existence of some integrals that
require the quantity $\mathcal{Q}_{p}^{s}$ to be finite or even approach $0$
in a particular way, like for example, demanding that
$\frac{\mathcal{Q}_{p}^{s}}{s}$ remains finite in the limit of $s\to 0$. It is
up to each implementation that these assumptions should be checked. In the
cases studied in this paper, where we use $\norm{\mathcal{S}_{p}^{r}}$ and
$\quantity(P_{p}^{n})^{-1}$, we checked that these conditions hold in a
Riemannian manifold.

Implicit Scalar Curvature from quantities
$\mathcal{Q}_{p}^{s}=C_{p}s^{\alpha_{p}}(1+\mathcal{K}_{p}(s))$
---
Definition |  Expansion up to $O(\mathcal{K}_{p})$
[.5]() $\mathcal{GSC}_{\mathcal{D}}(\mathcal{Q}_{p}^{s})\doteq 1+\dfrac{s\cdot\partial_{s}\mathcal{Q}_{p}^{s}}{\mathcal{Q}_{p}^{s}}$ |  $1+\alpha_{p}+s\partial_{s}\mathcal{K}_{p}(s)$
[.5]0 $\mathcal{GSC}_{\mathcal{I}}(\mathcal{Q}_{p}^{s})\doteq\dfrac{s\cdot\mathcal{Q}_{p}^{s}}{\int_{0}^{s}\mathcal{Q}_{p}^{x}dx}$ |  $1+\alpha_{p}+\left((1+\alpha_{p})\mathcal{K}_{p}(s)-\frac{(1+\alpha_{p})^{2}}{s^{1+\alpha_{p}}}\int_{0}^{s}x^{\alpha_{p}}\mathcal{K}_{p}(x)dx\right)$
[.5]0 $\mathcal{GSC}_{1}(\mathcal{Q}_{p}^{s})\doteq 1+s\cdot\partial_{s}(\dfrac{s\cdot\partial_{s}\mathcal{Q}_{p}^{s}}{\mathcal{Q}_{p}^{s}})$ |  $1+(s\partial_{s}\mathcal{K}_{p}(s)+s^{2}\partial_{s}^{2}\mathcal{K}_{p}(s))$
[.5]0 $\mathcal{GSC}_{2}(\mathcal{Q}_{p}^{s})\doteq s\cdot(\dfrac{s\cdot\partial_{s}\mathcal{Q}_{p}^{s}}{\mathcal{Q}_{p}^{s}})(\int_{0}^{s}(\dfrac{x\cdot\partial_{x}\mathcal{Q}_{p}^{x}}{\mathcal{Q}_{p}^{x}})dx)^{-1}$ |  $1+\frac{1}{\alpha_{p}}(s\partial_{s}\mathcal{K}_{p}(s)-\frac{1}{s}\int_{0}^{s}x\textbf{}\partial_{x}\mathcal{K}_{p}(x)dx)$
[.5]0 $\mathcal{GSC}_{3}(\mathcal{Q}_{p}^{s})\doteq\dfrac{(\int_{0}^{s}\mathcal{Q}_{p}^{x}dx)\cdot s\cdot\partial_{s}(s\cdot\mathcal{Q}_{p}^{s})}{(s\cdot\mathcal{Q}_{p}^{s})^{2}}$ |  $1+\left(\frac{1+\alpha_{p}}{s^{1+\alpha_{p}}}\int_{0}^{s}x^{\alpha_{p}}\partial_{x}\mathcal{K}_{p}(x)dx+\frac{s\partial_{s}\mathcal{K}_{p}(s)}{1+\alpha_{p}}-\mathcal{K}_{p}(s)\right)$
[.5]0 $\mathcal{GSC}_{4}(\mathcal{Q}_{p}^{s})\doteq\dfrac{(\mathcal{Q}_{p}^{s}-\dfrac{\int_{0}^{s}\mathcal{Q}_{p}^{x}dx}{s})(\mathcal{Q}_{p}^{s}+\int_{0}^{s}(\dfrac{\mathcal{Q}_{p}^{x}}{x})dx)}{(\mathcal{Q}_{p}^{s})^{2}}$ |  $1+\frac{1}{\alpha_{p}(1+\alpha_{p})}(\mathcal{K}_{p}(s)+\int_{0}^{s}\frac{(s\alpha_{p}^{2}x^{\alpha_{p}-1}-(1+\alpha_{p})^{2}x^{\alpha_{p}})\mathcal{K}_{p}(x)}{s^{1+\alpha_{p}}}dx)$
[.5]0 $\mathcal{GSC}_{5}(\mathcal{Q}_{p}^{s})\doteq 1+s\cdot\partial_{s}(\dfrac{\partial_{s}\mathcal{Q}_{p}^{s}}{\mathcal{Q}_{p}^{s}}\int_{0}^{s}\dfrac{x\cdot\partial_{x}\mathcal{Q}_{p}^{x}}{\mathcal{Q}_{p}^{x}}dx)$ |  $1+\alpha_{p}\left(s^{2}\mathcal{K}_{p}^{\prime\prime}(s)+\frac{1}{s}\int_{0}^{s}x\mathcal{K}_{p}^{\prime}(x)dx-2s\mathcal{K}_{p}^{\prime}(s)\right)$
Table 1: In the first column, the generalized scalar curvatures
$\mathcal{GSC}_{i}$ are defined in terms of a scale-dependent quantity
$\mathcal{Q}_{p}^{s}=C_{p}s^{\alpha_{p}}(1+\mathcal{K}_{p}(s))$. Each of these
defines a coarse-grained scalar curvature
$\mathcal{SC}_{i}(\mathcal{Q}_{p}^{s})$ as
$\mathcal{GSC}_{i}(\mathcal{Q}_{p}^{s})\doteq
1+\alpha_{p}+\mathcal{SC}_{i}(\mathcal{Q}_{p}^{s})$ for
$i=\mathcal{D},\mathcal{I}$ and as
$\mathcal{GSC}_{i}(\mathcal{Q}_{p}^{s})\doteq
1+\mathcal{SC}_{i}(\mathcal{Q}_{p}^{s})$ for $i=1,2,3,4,5$ In the second
column are the expansions of $\mathcal{GSC}_{i}$ to the lowest order in the
curvature scalar $\mathcal{K}_{p}(s)$.

In the second column of table 1, we present the expansion of each generalized
scalar curvature at the lowest order in $\mathcal{K}_{p}(s)$ for a Riemannian
manifold. The definitions are such that each generalized coarse-grained scalar
curvature $\mathcal{GSC}_{i}$ has the property that any deviation from
constancy can be interpreted as the presence of curvature. So, for spaces with
zero curvature the generalized scalar curvatures are all constant. Notice that
$\mathcal{GSC}_{\mathcal{I}}$ and $\mathcal{GSC}_{\mathcal{D}}$ contain and
additive term $\alpha_{p}$, which is in general point dependent. All other
$\mathcal{GSC}_{i}$ only have an additive term of $1$ as the zeroth-order.

These definitions also apply to the quantities discussed in the previous
section, $\norm{\mathcal{S}_{p}^{r}}$ and $P_{p}^{n}$. In case we work in a
metric measure space using the sphere volume $\norm{\mathcal{S}_{p}^{r}}$, we
have

$\mathcal{Q}_{p}^{s}\to\norm{\mathcal{S}_{p}^{r}},\quad\text{where }s\to r.$
(11)

In case we work in a metric space endowed with a random walk, we can use
$P_{p}^{n}$ to determine the scalar curvature, with a small modification. The
return probability $P_{p}^{n}$ has a negative leading power in a Riemannian
manifold, so we invert $P_{p}^{n}$ for the integrals in the generalized scalar
curvature definitions of table 1 to be well-defined. So we define the
generalized scalar curvatures $\mathcal{GSC}_{i}$ with respect to the inverse
return probability $(P_{p}^{n})^{-1}$. In a Riemannian manifold, this has an
expansion

$(P_{p}^{n})^{-1}=(4\pi\tau)^{\frac{D}{2}}\quantity(1-\frac{R_{p}}{6}\tau+\order{\tau^{2}}),$
(12)

which is a quantity of the form (10). So, for the generalized scalar
curvatures based on the inverse return probability, we use

$\mathcal{Q}_{p}^{s}\to\quantity(P_{p}^{n})^{-1},\quad\text{where }s\to\tau.$
(13)

If one uses (9), then $s=\frac{n\sigma^{2}}{2\eta}$. The expansion for the
generalized scalar curvature based on these quantities are presented in table
2.

Using the generalized scalar curvatures based on these quantities, we can
define scalar curvature, using $\mathcal{GSC}_{i}$ on a general metric space.
And we can interpret deviations from constancy as the presence of curvature.
For $\mathcal{GSC}_{\mathcal{D}}$ and $\mathcal{GSC}_{\mathcal{I}}$, the
interpretation is slightly more difficult. Unless the quantity being studied
is known to be a constant with no scale dependence in a flat spacetime, like
the normalized averaged distance introduced in the first section,
$\mathcal{GSC}_{\mathcal{D}}$ and $\mathcal{GSC}_{\mathcal{I}}$ are not
particularly useful to determine the scalar curvature, since the leading term
will be an unknown scale independent prefactor. For the other
$\mathcal{GSC}_{i}$, any deviation from $1$ can be 666After statistical noise
and lattice artefacts are taken into account. interpreted as the presence of
curvature. The advantage of using $||\mathcal{S}_{p}^{r}||$ and
$(P_{p}^{n})^{-1}$ to define coarse-grained curvatures, is that the continuum
expressions in a Riemannian manifold are known. So positive or negative
deviations from $1$ can be directly related to the _sign_ of the curvature of
the manifold. Using table 2, we can see that in for these
$\mathcal{Q}_{p}^{s}$, any $\mathcal{GSC}_{i}$, with $i=1,2,3,4,5$ below $1$
indicates positive curvature, and above $1$ indicates negative curvature.

Surface and Spectral Scalar Curvatures
---
Method |  $||\mathcal{S}_{p}^{r}||$ |  $(P_{p}^{n})^{-1}$
[.5]() $\mathcal{GSC}_{\mathcal{D}}(\mathcal{Q}_{p}^{s})$ |  $D-\frac{1}{3D}R_{p}r^{2}+O(r^{4})$ |  $\frac{D+2}{2}-\frac{1}{6}R_{p}\tau+O(\tau^{2})$
[.5]0 $\mathcal{GSC}_{\mathcal{I}}(\mathcal{Q}_{p}^{s})$ |  $D-\frac{1}{3(2+D)}R_{p}r^{2}+O(r^{4})$ |  $\frac{D+2}{2}-\frac{2+D}{6(4+D)}R_{p}\tau+O(\tau^{2})$
[.5]0 $\mathcal{GSC}_{1}(\mathcal{Q}_{p}^{s})$ |  $1-\frac{2}{3D}R_{p}r^{2}+O(r^{4})$ |  $1-\frac{1}{6}R_{p}\tau+O(\tau^{2})$
[.5]0 $\mathcal{GSC}_{2}(\mathcal{Q}_{p}^{s})$ |  $1-\frac{2}{9D(D-1)}R_{p}r^{2}+O(r^{4})$ |  $1-\frac{1}{6D}R_{p}\tau+O(\tau^{2})$
[.5]0 $\mathcal{GSC}_{3}(\mathcal{Q}_{p}^{s})$ |  $1-\frac{2}{3D^{2}(D+2)}R_{p}r^{2}+O(r^{4})$ |  $1-\frac{2}{3(8+6D+D^{2})}R_{p}\tau+O(\tau^{2})$
[.5]0 $\mathcal{GSC}_{4}(\mathcal{Q}_{p}^{s})$ |  $1-\frac{2}{3D^{5}+6D^{4}-3D^{3}-6D^{2}}R_{p}r^{2}+O(r^{4})$ |  $1-\frac{8}{3D(2+D)^{2}(4+D)}R_{p}\tau+O(\tau^{2})$
[.5]0 $\mathcal{GSC}_{5}(\mathcal{Q}_{p}^{s})$ |  $1-\frac{8(D-1)}{9D}R_{p}r^{2}+O(r^{4})$ |  $1-\frac{D}{8}R_{p}\tau+O(\tau^{2})$
Table 2: Short-scale expansion of each $\mathcal{GSC}_{i}$ in a Riemannian
manifold, using the sphere volume $\norm{\mathcal{S}_{p}^{r}}$ and the inverse
return probability $(P_{p}^{n})^{-1}$. The timescale $\tau$ is linear in the
scale $n$. If one uses 9, then $\tau=\frac{n\sigma^{2}}{2\eta}$.

Furthermore, it is useful to compare $\mathcal{GSC}_{i}$ for all length scales
$s$, not just in the limit, to the results in a constantly curved Riemannian
manifold. For a $D$-dimensional constantly curved Riemannian manifold, this
can be done for the quantity $\norm{\mathcal{S}_{p}^{r}}$. For the return
probability this is unfortunately impossible, because so far only asymptotic
expansions are known for the heat kernel. The full continuum expression for
each $\mathcal{GSC}_{i}(||\mathcal{S}_{p}^{r}||)$ on a $D$-dimensional sphere
and a hyperbolic $D$-space can be found in appendix A. All expressions depend
on at most 2 parameters, the dimension $D$ and the radius of the space $R$.
Surprisingly, $\mathcal{GSC}_{2}(||\mathcal{S}_{p}^{r}||)$ only depends on
$R$, and not on $D$. This makes it the ideal reference to compare the results
of the generalized scalar curvatures for spaces where one does not want to
assume a priory a value for $D$. In some cases, one might want to keep this
parameter free, determining some sort of "effective dimension" of the studied
space, derived from our curvature definitions. Something similar to this was
done using the QRC in 2D Euclidean dynamical triangulations in [13].

By comparing the results of $\mathcal{GSC}_{2}(||\mathcal{S}_{p}^{r}||)$ in
those space with the results of appendix A, one can determine the resemblance
to a sphere or a hyperbolic space. Furthermore, if there is a strong
resemblance between the measured $\mathcal{GSC}_{2}(||\mathcal{S}_{p}^{r}||)$
and the plot 9 for a constantly curved Riemannian manifold, an effective
radius can be assigned without needing to assume a value for $D$, by fitting
the measured $\mathcal{GSC}_{2}(||\mathcal{S}_{p}^{r}||)$ to the formulas (32)
or (40) given in the appendix.

##  4 Implementations and examples of use

Now that we have defined the generalized curvatures $\mathcal{GSC}_{i}$, we
will proceed to implement them in discrete metric spaces, employing numerical
methods. More specifically, we will study triangulations of known
2-dimensional surfaces, a 2-sphere and a periodic square section of the plane.
These spaces are, by construction, similar to the underlying smooth space in
terms of their metric properties, and therefore we will use them to study the
properties and limitations of our definitions in a controlled setting.

To construct triangulations of the plane and the sphere, we will make use of
the so-called Delaunay triangulations. Since we will only work with
triangulations of 2D surfaces embedded into euclidean space, we will restrict
our description of Delaunay triangulations to them. Delaunay triangulations of
a surface embedded in Euclidean space, are triangulations of a sample set of
points contained in the surface, that have the property that no point in the
sample is contained inside the circumcircle of any triangle of the
triangulation [14]. In this work, we construct the triangulation by using
Poisson Disk sampled points [15]. The distance function used to create the
sample of points is the induced metric on the surfaces (plane and sphere) from
the embedding in euclidean space. For simplicity, we denote the space being
triangulated (plane and sphere) by $M$, and we will denote a Delaunay
triangulation in it as $\mathcal{D}_{\mathcal{T}}(M)$. The resulting Delaunay
triangulation $\mathcal{D}_{\mathcal{T}}(M)$ is a random geometry, in the
sense that its vertices are randomly sampled.

Even though both the plane and the sphere have a canonical distance function
from the embedding, making them metric spaces, we only use that distance
function to construct the triangulation. To test the robustness of our
curvature definitions we will use a different distance function. For both the
sphere and the plane define the metric space
$(P(\mathcal{D}_{\mathcal{T}}(M)),d_{l})$, with the set of points
$P(\mathcal{D}_{\mathcal{T}}(M))$ constituting the vertices of the
triangulation, and as the distance function we use the link distance $d_{l}$,
which is the minimal number of links between two points in the triangulation.
The reason for using this distance function instead of the distance function
used to construct the triangulation itself, is because we would like to test
the capability of our curvature definitions to recover geometrical properties
of the underlying triangulated space without making direct use of the metric
tensor of the underlying continuum surface. The motivation for doing this is
twofold: First, we want to test the robustness of our curvature measures with
respect to discretization artefacts, so replacing the continuum distance
function from the embedding with a lattice distance function can be regarded
as a discrete approximation of such distance, and thus as a test ground for
discretization artefacts. Second, since our main goal is to apply these
curvature definitions into the CDT quantum geometries, where one does not even
know if there is an underlying continuum manifold being approximated, and one
only relies on the discrete setting of the geometries in the path integral,
using the link distance in $P(\mathcal{D}_{\mathcal{T}}(M))$ also works as a
preamble and test ground for implementing the generalized curvatures in those
quantum geometries.

We can endow the spaces denoted by $(P(\mathcal{D}_{\mathcal{T}}(M)),d_{l})$
with additional structure. To make these spaces metric measure spaces, we
define a Hausdorff measure $\mu_{h}$, which is given by $\mu_{h}(A)=\\#A$ for
any subset $A\subseteq P(\mathcal{D}_{\mathcal{T}}(M))$, where $\\#A$ is the
cardinality or number of elements in the set. We also define a random walk
$\\{X_{i}\\}$ for each point $i$ in $P(\mathcal{D}_{\mathcal{T}}(M))$. With
these two structures, we can put our generalized scalar curvatures
$\mathcal{GSC}_{i}$ to practice, using both the sphere volume and the return
probability.

As mentioned before, in this paper we constructed Delaunay triangulations for
a 2D plane and a 2D sphere, and we will implement our definitions there. The
link distance in these triangulations is expected to be a discretization of
the geodesic distance on these surfaces, up to proportionality factors. So, by
approximating these 2D spaces with triangulations, we expect our definitions
to recover the Ricci scalar curvature of these spaces in the limit of infinite
triangles. We show examples of Delaunay triangulations of the plane and the
sphere in Fig. 1.

When performing measurements of sphere volume and return probability on these
triangulations we make use of the vertex graph and dual graph of the
triangulations. The vertex graph is the graph where the nodes are given by the
vertices of the triangulation and the links between the nodes are given by the
edges of the triangulation. This means that in this graph, the graph distance
represents the link distance of the triangulation. The dual graph is the graph
where the nodes are given by the triangles of the triangulation and the links
between the nodes are the edges that are shared by the respective triangles.
Note that this graph is indeed the dual graph of the vertex graph.

(a) Poisson Disk Sampled Plane

(b) Poisson Disk Sampled Sphere

Figure 1: Delaunay triangulations of a plane and a sphere. The shown Delaunay
triangulations are constructed using Python and Mathematica, making use of
Poisson disk sampling of points on the surface of smooth 2D surfaces.

We are interested in studying spaces that are large enough to see curvature
effects before reaching finite-size effects. To this end, we studied Delaunay
triangulations with $150\,\mathrm{k}$ vertices. In the case of the plane this
is not as relevant as for the sphere, since we are making a periodic plane
using the triangulated square region. In the case of the sphere, we can make a
rough estimation of the effective radius of the Delaunay triangulation of the
sphere, by comparing the area of the triangulation, given by the number of
triangles, to the area of a continuum sphere $4\pi R^{2}$. We set the length
of links to $1$, so we estimate the effective radius to be about
$R_{\text{eff}}\simeq\sqrt{\frac{\frac{\sqrt{3}}{4}(2*150000)}{(4\pi)}}\simeq
102$. However, after scaling all the links to the same length, the resulting
triangulation is further away form the original surface, and is not clear if
this is an overestimate, or an underestimate for the area of the closest
continuum sphere. The original Delaunay triangulation is close to an
equilateral triangulation, which is the case by construction due to the
Poisson disk sampling. Still, the triangulation after all edges are taken to
be of length $1$ will resemble something more like a "fuzzy" plane or sphere,
with bumps everywhere. This "fuzziness" is essential to test the robustness of
our curvature definitions, as will be explained later. An example of how this
rescaling to equilateral triangles affects the triangulation can be seen in
Fig. 2.

Figure 2: Example of how rescaling all the links to length $1$ affects the
geometry of a triangulation. "Cusps" will appear all over the surfaces after
rescaling, introducing some "fuzziness" in the triangulated surfaces shown in
Fig. 1.

We do this rescaling because we want to measure a discrete version of the
generalized curvatures. Since we need to define spheres, it is much more
computationally convenient when all link lengths are equal to $1$ since to
construct a sphere of radius $r$, one simply needs to find all the points at
lattice distance equal to $r$, and the construction of a sphere is
straightforward. This would not be the case if the links would have different
length between them. Moreover, the building blocks used in 2D CDT are all
equilateral, so working with equilateral random surfaces is a natural first
step before the using CDT quantum geometries.

There are several ways to discretize the expressions for each
$\mathcal{GSC}_{i}$. We decided to discretize all derivatives using second-
order accurate finite differences, and to discretize all integrals using a
cumulative trapezoidal rule. Specifically, for a function $f(s)$, which we
discretize with $f[s]$, where $s=0,1,2,\dots$, we use the discrete derivative

$\derivative{f}{s}\longrightarrow\begin{dcases}f[1]-f[0]&\text{for }s=0\\\
\frac{f[s+1]-f[s-1]}{2}&\text{for }s\geq 1\end{dcases}$ (14)

and the discrete integral

$\int_{0}^{s}\differential{s^{\prime}}f(s^{\prime})\longrightarrow\begin{dcases}0&\text{for
}s=0\\\ \frac{1}{2}\sum_{i=1}^{s}\quantity\Big(f[i]+f[i-1])&\text{for }s\geq
1\end{dcases}.$ (15)

Additionally, we choose to set $\mathcal{Q}_{p}^{s=0}=0$ in the calculation of
the $\mathcal{GSC}_{i}$ to more closely resemble the continuum expressions of
the quantities we measure. With this choice, some quotients that appear in the
generalized curvatures definitions are ill-defined at $s=0$. In the continuum
this is not an issue, because the limit $s\rightarrow 0$ is well-defined; at
least for sphere volume and inverse return probability we consider. However,
in the discrete setting we cannot take such a limit, and we have to define the
value of these quotients that take the form $\frac{0}{0}$, for example
$\evaluated{\frac{s\cdot\partial_{s}\mathcal{Q}_{p}^{s}}{\mathcal{Q}_{p}^{s}}}_{s=0}$.
Only quotients that appear inside of integrals in our definitions are crucial
to define, as they affect the results of the final $\mathcal{GSC}_{i}$ for all
$s$. Other ill-defined quotients only affect the results for the first few
$s$, in which we are not interested because they are strongly affected by
lattice artefacts anyway. Due to the large arbitrariness in defining these
quotients, we decided that the most consistent choice with our formalism was
setting the value of these ill-defined quotients at $s=0$ to be equal to the
value they take in a Riemannian Manifold. The only significant impact this
choice makes is on the initial overshoot of the $\mathcal{GSC}$, where
nevertheless lattice artefacts dominate. In the case of
$\mathcal{GSC}_{i}(\norm{S^{r}_{p}})$ we set

$\evaluated{\frac{r\cdot\partial_{r}\norm{S^{r}_{p}}}{\norm{S^{r}_{p}}}}_{r=0}=(D-1),\quad\text{and}\quad\evaluated{\frac{\norm{S^{r}_{p}}}{r}}_{r=0}=\begin{dcases}2\pi&D=2\\\
0&D>2\end{dcases},$ (16)

where $D$ is the topological dimension. In the case of
$\mathcal{GSC}_{i}((P_{p}^{n})^{-1})$ we set

$\evaluated{\frac{n\cdot\partial_{n}(P_{p}^{n})^{-1}}{(P_{p}^{n})^{-1}}}_{n=0}=\frac{D}{2}\quad\text{and}\quad\evaluated{\frac{(P_{p}^{n})^{-1}}{n}}_{n=0}=\begin{dcases}4\pi&D=2\\\
0&D>2\end{dcases},$ (17)

Notice that an input for ill-defined quotients is only required for the
initial point of $\mathcal{GSC}_{2}$, $\mathcal{GSC}_{4}$ and
$\mathcal{GSC}_{5}$. In the other cases, the initial point can simply be
ignored.

As mentioned before, the numerous prescriptions to define a coarse-grained
scalar curvature introduced in Table 1, exist to accommodate the different
impacts combinations of derivatives and integrals can have on the numerical
results. Numerical integrals tend to be more robust with respect to noise than
numerical derivatives, but they have the disadvantage of enlarging short scale
lattice artefacts. A balance between introducing large lattice artefacts using
integrals, and introducing large noise using derivatives has to be taken into
account when implementing the generalized curvatures in a discrete setting. So
we present several generalized scalar curvatures $\mathcal{GSC}_{i}$ to be
able to compare their features regarding noise and lattice artefacts in
discrete implementations. Note that different $\mathcal{GSC}_{i}$ define the
same scalar curvature in a smooth Riemannian manifold. However, we cannot take
for granted that these multiple definitions are equivalent in a quantum
geometry, and we will verify their similarity in the quantum geometries of 2D
CDT in the next section.

###  4.1 Discrete diffusion process

The return probability $(P_{p}^{n})^{-1}$, requires some preliminary
explanation of its implementation. First, as mentioned before, if we want to
determine precise dimensionful values for the scalar curvature, and not only
its signature, we need an identification between the random walk time and the
corresponding diffusion time. The identification (9) is based on equally
distributed, independent, and integrable Gaussian random variables describing
each random step. In the case of a discrete geometry, a standard
implementation of a random walk is to start at a point in the space, and move
to a random neighbour of that point. The neighbour is chosen uniformly from
all direct neighbours. In the case of an irregular discrete space, a vertex
might not have the same number of neighbours all other vertices. Therefore,
the probability distribution of the random variables determining the random
walk is different for each point. To get closer to a situation that best
resembles the conditions that allow the identification (9) we will implement
the random walk in such a way that each step is equally distributed. This is
because, if the variables are not identically distributed, the variance
computed in (9) is not necessarily equal to a factor $\sigma^{2}$ times the
number of steps taken. In the case of not equally distributed random variables
describing each step, relation (9) could be regarded as an effective relation,
where the proportionality factor $\sigma^{2}$ would be some effective variance
of each step. Since in the case of not equally distributed steps one would
have to determine this relation by other means, we will restrict ourselves to
a situation where we can compute $\sigma^{2}$ exactly. This is the case of the
dual of our triangulations. In the dual lattice of the triangulation, each
vertex will have the same amount of nearest neighbours, and thus, the random
variables describing each step are equal to a uniform distribution among the
three nearest neighbours of each vertex. For a visualization of this effect on
a Delaunay triangulation of a sphere and its dual graph, see Fig. 3. If the
triangulation is equilateral, ($\sigma^{2}$) can be computed exactly, as we
will do later in this chapter.

Even though the random variables in this discrete setting are not Gaussian, we
appeal to the Central Limit Theorem and assume that at some large value of $n$
the identification (9) will hold approximately, despite the fact that our
variables are uniformly distributed instead of being Gaussians. Therefore, we
will use identification (9) for our diffusion processes in the dual lattice,
despite the fact that we are not exactly in the situation where this
identification is exact. We remark that this implementation in the dual is not
necessary if is interested in detecting the presence of curvature by using our
prescriptions. The only reason we do this, is because we need a precise
identification between diffusion steps and diffusion time, to obtain
predictions for the dimensionful numerical value of curvature in our discrete
spaces. If one were only interested in determining the presence and signature
of curvature in a space, it is sufficient to apply our prescriptions to a
diffusion process in the lattice.

Figure 3: On the left, a Delaunay triangulation of a sphere constructed from a
sample of $500$ points on a continuum sphere. On the right, the associated
dual graph of the triangulation. Notice that each node in the dual graph has
coordination number $3$, even though the vertex graph of the triangulation
does not have a fixed coordination number.

Since in the Delaunay triangulation the edge length was set to $1$ everywhere,
the edge length in the dual graph is going to be $\frac{1}{\sqrt{3}}$ so in
the dual we will also have the same "fuzziness" as in the Delaunay
Triangulation. From now on, when we speak of the random walk in the Delaunay
triangulations, we will be speaking of a random walk on the dual graph of that
Delaunay triangulation, after rescaling all the edges to length $1$.

Let $P(p^{\prime},p;n)$ be the probability of a random walker moving from
point $p$ to $p^{\prime}$ on the dual graph of the triangulation, in $n$
steps. We start in our diffusion process with
$P(p^{\prime},p;0)=\delta_{p,p^{\prime}}$, where $\delta$ is the Kronecker
delta symbol; this signifies that the walk starts at point $p$ and is
normalized such that $\sum_{p^{\prime}}P(p^{\prime},p;n)=1$ for all $n$; The
diffusion process is then implemented using the evolution equation [16]

$P(p^{\prime},p;n+1)=(1-\eta)P(p^{\prime},p;n)+\frac{\eta}{3}\sum_{q\sim
p^{\prime}}P(q,p;n),$ (18)

where $q\sim p^{\prime}$ denotes the set of all direct neighbours of
$p^{\prime}$ in the graph. Also, $\eta\in\left(0,1\right]$ takes the role of a
discrete diffusion constant, which can be set smaller than $1$ to mitigate the
large differences between the return probability for odd and even $n$ for
small $n$; we use $\eta=0.9$ in our measurements. The return probability is
determined by measuring $P_{p}^{n}=P(p,p;n)$ after each diffusion step. If we
want to resemble calculations done with the heat kernel, exactly as we did for
$||S^{r}_{p}||$, we have to choose scale $n=0$ values for any undetermined
ratio that appears in the $\mathcal{GSC}$s.

###  4.2 Discrete scalar curvature

The results of the implementations for each of the scalar curvatures
$\mathcal{GSC}_{i}$, both using $||S^{r}_{p}||$ and $(P_{p}^{n})^{-1}$, on the
Delaunay triangulations of a 2-sphere and a periodic plane can be seen in Fig.
4 and 5 respectively. The calculation of the $\mathcal{GSC}_{i}$ is done at
each point, for each value of $||S^{r}_{p}||$ and $(P_{p}^{n})^{-1}$. The
results for the $\mathcal{GSC}_{i}(||S_{p}^{r}||)$ are more susceptible to the
discretization of derivatives and integrals than the
$\mathcal{GSC}_{i}((P_{p}^{n})^{-1})$, regarding the initial overshoot and the
noisiness of the scalar curvature. For the $\mathcal{GSC}_{i}(||S_{p}^{r}||)$,
the dashed line curves in Fig. 4 is a $1$-parameter fit of the continuum
curves (see appendix A), where radius is the only fitting parameter ($D=2$).
The quality of the fittings depends on the $\mathcal{GSC}_{i}$ in question.
The generalized scalar curvatures with more integrals, like
$\mathcal{GSC}_{2,3,4}$ seem to fit a constantly curved manifold more closely
at large scales. As smaller scales they deviate significantly, due to lattice
artefacts. The results for $\mathcal{GSC}_{1,5}$ that contain second
derivatives, show precisely the opposite: noise dominates these scalar
curvatures at large scales.

Given that the underlying geometry is a sphere, we decided to compute
numerical values for the radius instead of the scalar curvature, being
equivalent in this situation. The fitted radius of the
$\mathcal{GSC}_{i}(||S_{p}^{r}||)$, using the fitting functions provided in
appendix A, give $R_{\text{eff},\mathcal{D}}=94.96\pm 0.01$,
$R_{\text{eff},\mathcal{I}}=95.367\pm 0.001$, $R_{\text{eff},1}=92.4\pm 5$,
$R_{\text{eff},2}=93.97\pm 0.01$, $R_{\text{eff},3}=93.28\pm 0.01$,
$R_{\text{eff},4}=93.973\pm 0.001$, and $R_{\text{eff},5}=92.8\pm 0.6$. The
error bars presented in the results are simply the errors of the fitting. It
is important to remark that since the fitting functions used do not have the
initial overshoot present in the numerical results, and are forced to converge
to the Riemannian manifold values at short scales, the fitting functions, and
therefore the estimated parameters from them, are much more accurate at larger
scales than shorter ones. Unfortunately, there is no correct answer, since the
triangulation is only topologically a sphere but does not reliably approximate
a constantly curved sphere due to the scaling of the edges to be all equal to
$1$. The fitting values obtained differ from each others roughly by a $3\%$,
and at the most by a $9\%$ from the very rough estimation of
$R_{\text{eff}}\simeq 102$ made earlier, where we assumed a sphere perfectly
described by equilateral triangles.

Figure 4: Sphere-volume scalar curvatures
$\mathcal{GSC}_{i}(\norm{S_{p}^{r}})$ on a Delaunay triangulation of a plane
and sphere of around $150\,\text{k}$ vertices, including the best fit
continuum curves for the sphere. The displayed error bars all represent
$95\%$-significance or $2\sigma$ intervals, based on the fluctuations between
all $20\,\text{k}$ points on the triangulations.

The $\mathcal{GSC}$ are designed for defining curvature, based on small scale
properties of scale-dependent quantities like the sphere surface or the return
probability, not necessarily by fitting $\mathcal{GSC}_{i}$ with an analytical
expression. So we also analyse the small scale behaviour of the measured
scalar curvatures to obtain an estimation of the radius of the underlying
sphere. To define curvature using the knowledge of the continuum expansion of
our $\mathcal{GSC}$, we have to be sure that we are not in a range of lattice
artefacts. In our case, we believe that for
$\mathcal{GSC}_{\mathcal{I}}(||S_{p}^{r}||)$ and
$\mathcal{GSC}_{\mathcal{D}}(||S_{p}^{r}||)$ the size of lattice artefacts
seems to be larger than the rest of the $\mathcal{GSC}$. This emphasizes our
argument that having this unknown scale-independent prefactor in the
observables is an undesired feature. We concluded that lattice artefacts are
no longer dominating the numerical results at $r>40$. For
$\mathcal{GSC}_{1}(||S_{p}^{r}||)$, $\mathcal{GSC}_{2}(||S_{p}^{r}||)$,
$\mathcal{GSC}_{3}(||S_{p}^{r}||)$, $\mathcal{GSC}_{4}(||S_{p}^{r}||)$ and
$\mathcal{GSC}_{5}(||S_{p}^{r}||)$, lattice artefacts seem to be negligible
for $r>15$. Using this ranges, one can use the knowledge of the expansions
(see table 2) to fit a quadratic function to each
$\mathcal{GSC}_{i}(||S_{p}^{r}||)$ in the appropriate range, and extract an
estimate of the scalar curvature. In this case, we extract the radius of the
underlying sphere. Based on the continuum formulas for the
$\mathcal{GSC}_{i}(||S_{p}^{r}||)$ in appendix A, we find that for
$\frac{r}{R}\leq\frac{1}{2}$, a quadratic approximation of the expansion
deviates less than $1\%$ from its actual value for every
$\mathcal{GSC}_{i}(||S_{p}^{r}||)$. So, it can be used as a decent
approximation to compute the curvature in this range. In this range we
estimate the scalar curvature as

$\frac{\mathcal{SC}_{i}(||S_{p}^{r}||)}{\beta_{i}(D)\,r^{2}}\simeq
R_{p}+\mathcal{O}(r^{2}),$ (19)

where

$\displaystyle\beta_{i}(D)=\\{\frac{-1}{3D},\frac{-1}{3(2+D)},\frac{-2}{3D},\frac{-2}{9D(D-1)},\frac{-2}{3D^{2}(D+2)},\frac{-2}{3D^{5}+6D^{4}-3D^{3}-6D^{2}},\frac{-8(D-1)}{9D}\\}$
$\displaystyle\quad\text{for }i=\\{\mathcal{D},\mathcal{I},1,2,3,4,5\\}$ (20)

We decided to apply this approximation to the value of $r=50$ in all our
$\mathcal{GSC}_{i}(||S_{p}^{r}||)$ using $D=2$. We chose this radius since we
believe that it is the smallest value of $r$ at which all $\mathcal{GSC}_{i}$
can be assumed to have negligible lattice artefacts. The large value is due to
$\mathcal{GSC}_{3}(||S_{p}^{r}||)$ and $\mathcal{GSC}_{4}$, that posses a
large range of lattice artefacts. With the obtained value of the curvature
$R_{p}$, one can estimate the more intuitive dimensionful value, which is the
associated effective radius defined as
$R_{p}\doteq\frac{D(D-1)}{R_{eff}^{2}}$, that provides an estimation for the
size of the approximate sphere being triangulated. The results for the
estimations of $R_{\text{eff}}$ are $R_{\text{eff},1}=85\pm 16$,
$R_{\text{eff},2}=81\pm 1$, $R_{\text{eff},3}=85\pm 1$,
$R_{\text{eff},4}=75\pm 1$, and $R_{\text{eff},5}=84\pm 11$, where
$R_{\text{eff},i}$ denotes the estimate of $R_{\text{eff}}$ based on the
results of $\mathcal{GSC}_{i}$. The error bars presented for the estimated
effective radius are the propagated errors coming form the statistical error
in the $\mathcal{GSC}_{i}$s. Except for the estimated radius using
$\mathcal{GSC}_{4}$, the other estimations agree with each others within error
bars. We believe that the significant deviation in the estimation of
$R_{\text{eff},4}$ comes from the fact that the two integrals involved
introduce, small, but non-vanishing discretization artefacts even at large
scales. This renders $\mathcal{GSC}_{4}$ not suitable for estimations of the
curvature using approximation (19). Other sources of error, like a systematic
error from the truncation to a quadratic function, or a method that takes into
account how different our equilateral triangulation is from a continuum sphere
might change these results, making them overlap with each others and with the
global fit used previously. Later, we will see that the results using the
return probability are more accurate, and overlap with each others and with
the global fit of the continuum sphere. Note that in this case
$i=\mathcal{D},\mathcal{I}$ cannot be used at this short distance $r=50$ as
the lattice artefact are large for these $\mathcal{GSC}$, such that each
$\mathcal{GSC}$ is still to strongly deviating from the continuum result at
$r=50$.

This would correspond to a significantly larger or even negative curvature, so
we will not associate a sphere radius in these cases. This means
$\mathcal{GSC_{D}}$ and $\mathcal{GSC_{I}}$ are not well suited for extracting
the scalar curvature using (19). We believe the main reason for this is the
leading unknown constant different from $1$ that dominates the numerical value
of $\mathcal{GSC_{D}}$ and $\mathcal{GSC_{I}}$. This emphasizes the use of the
other five methods to determine the curvature of the underlying space, since
they do not contain this unknown leading constant by construction.

Unfortunately, the quadratic approximations for every
$\mathcal{GSC}_{i}(||S_{p}^{r}||)$ are not good approximations if we compare
the results with the global fit obtained from the continuum formulas. At the
used scales, they provide the effective radius with an error of roughly
$10\%$, compared with the continuum fit. Furthermore, since the results coming
from some of them don’t overlap with each others, it is difficult to determine
which one is the most suitable for such approximation. This means that
estimations of the curvature using a quadratic approximation of
$\mathcal{GSC}_{i}(||S_{p}^{r}||)$ are nor numerically reliable. They can
provide an estimate of the actual curvature radius, but one should be aware
that, at least in the scales analysed in this work, there is roughly a $10\%$
error when estimating the effective radius. We therefore conclude that the
$\mathcal{GSC}_{i}(||S_{p}^{r}||)$ are more useful for determining large scale
curvature, by comparing the continuum formulas with the numerical results
obtained, instead of being precise for determining curvature at each point.
This picture changes when studying $\mathcal{GSC}_{i}((P_{p}^{n})^{-1})$.

Overall, our conclusion is that the $\mathcal{GSC}_{2}(||S_{p}^{r}||)$ is the
most robust definition of curvature in this case. The reason is that
$\mathcal{GSC}_{2}(||S_{p}^{r}||)$ seems to have less noise at large scales
than the $\mathcal{GSC}_{i}(||S_{p}^{r}||)$ with two derivatives, and at small
scales, the curved and the flat manifold cases start deviating from each other
faster than the $\mathcal{GSC}_{i}(||S_{p}^{r}||)$ with two integrals.
Furthermore, as shown in appendix A, $\mathcal{GSC}_{2}(||S_{p}^{r}||)$ has
the property of being independent of the dimension of the space in a
constantly curved $D$-dimensional manifold. This makes it ideal to apply in
fractal spaces where the dimension is unknown, and very useful to compare with
constantly curved spacetimes. Whether this holds for more general manifolds is
unknown to us.

In Fig. 5, we discuss the results of $\mathcal{GSC}_{i}((P_{p}^{n})^{-1})$,
after averaging over a sample of $200$ points, to eliminate the "fuzziness".
There is an overall increase in the robustness of the generalized scalar
curvatures compared to using the sphere surface. Furthermore, we studied
diffusion steps up to values $2$ orders of magnitude larger than the lattice
distance used with the sphere surface. Nonetheless, we see that the measured
range shows only a linear correction to flatness; an exception is
$\mathcal{GSC}_{1}$, where we see higher-order corrections at the largest
measured scales. Therefore, it seems that the diffusion process explores
distance scales much smaller than the sphere surface. Because of that, it
became computationally unfeasible to study equivalent distance scales with
both methods. This is not a problem, since both methods can be used
complementary to study different scale ranges in more detail in the same
metric space. This indicates that $\mathcal{GSC}_{i}((P_{p}^{n})^{-1})$ is
ideal to define the scalar curvature at short scales, since the higher-order
contributions are negligible until large diffusion times.

Given that there is no closed formula for the return probability on a sphere,
only asymptotic expansions, we do not have closed analytical expression of
$\mathcal{GSC}_{i}((P_{p}^{n})^{-1})$ to fit to. Nevertheless, we can still
estimate the curvature and the radius of the underlying sphere using the
formula

$\frac{2\eta}{\sigma^{2}}\frac{\mathcal{SC}_{i}((P_{p}^{n})^{-1})}{\gamma_{i}(D)\,n}\simeq
R_{p}+\mathcal{O}(n^{2}),$ (21)

where

$\displaystyle\gamma_{i}(D)=\\{\frac{-1}{6},\frac{-(2+D)}{6(4+D)},\frac{-1}{6},\frac{-1}{6D},\frac{-2}{3(8+6D+D^{2})},\frac{-8}{3D(2+D)^{2}(4+D)},\frac{-D}{8}\\}$
$\displaystyle\quad\text{for }i=\\{\mathcal{D},\mathcal{I},1,2,3,4,5\\}$ (22)

We apply this approximation for all $\mathcal{SC}_{i}((P_{p}^{n})^{-1})$ using
$D=2$, and at $n=5000$, because we believe it is roughly the smallest value of
$n$ at which lattice artefacts can be neglected in all profiles.

To estimate of the effective radius of the underlying sphere, we have to
determine the value of the standard deviation $\sigma^{2}$ in (9).

In the dual of a triangulation, this value can be computed exactly, to
be777This can be computed as the standard deviation of a uniform distribution
that can only take 3 possible values, moving at points at distance
$\frac{1}{\sqrt{3}}$ from the centre. $\sigma^{2}=\frac{1}{3}$.

This leads us to the following estimations of the radius of the sphere from
formula 21, $R_{\text{eff},\mathcal{D}}=102\pm 4$,
$R_{\text{eff},\mathcal{I}}=109\pm 7$, $R_{\text{eff},1}=96\pm 4$,
$R_{\text{eff},2}=91\pm 4$, $R_{\text{eff},3}=91\pm 4$,
$R_{\text{eff},4}=100\pm 8$, and $R_{\text{eff},5}=94\pm 4$, where
$R_{\text{eff},i}$ denotes the estimate of $R_{\text{eff}}$ based on the
results of $\mathcal{GSC}_{i}$.

(a)  (b)

(c)  (d)

(e)  (f)

(g)  (h)

Figure 5: Scalar curvature on a Delaunay triangulation of a plane and sphere
based on inverse return probability. The continuum plane curve indicates what
flat curvature looks like in the underlying continuum plane. The error bands
give a $95\%$-confidence interval, which is based on the fluctuations between
$50$ points on the triangulations.

Except for the predictions using
$\mathcal{GSC}_{\mathcal{D}}((P_{p}^{n})^{-1})$ and
$\mathcal{GSC}_{\mathcal{I}}((P_{p}^{n})^{-1})$, these values are close to the
estimated values obtained from the continuum fit formulas using sphere
surface, and we interpret these results as a confirmation of the robustness of
the method and of our definitions. Furthermore, these results overlap with
each others within error bars and are also close to the rough expectation of
$R_{\text{eff}}\simeq 102$ mentioned before. Besides, also using this spectral
method to determine the curvature,
$\mathcal{GSC}_{\mathcal{D}}((P_{p}^{n})^{-1})$ and
$\mathcal{GSC}_{\mathcal{I}}((P_{p}^{n})^{-1})$ deviate larger from the rest,
as was to be expected since the leading constant of these quantities is in
principle undetermined, emphasizing the fact that they are not suitable for
extracting the scalar curvature, at least using the return probability.

The range where the initial lattice artefacts dominate is in accordance with
the range where lattice artefacts disappeared using the sphere surface. To see
this, one can make use if identification 9 and determine the approximated
relation between $r$ and $n$ as
$\operatorname{Var}\quantity(S_{n})=\sigma^{2}n\simeq r^{2}$. Since in case of
using the sphere surface the lattice artefacts disappeared in a range between
$r=35$ and $r=40$, this means that lattice artefacts for the return
probability should disappear in a range between $n=3675$ and $4800$, precisely
as wee observe in our measurements.

All the generalized scalar curvatures exhibit noise at large scales, and we
observe no significant difference between the small scale lattice artefacts
among them. So, we conclude that $\mathcal{GSC}_{5}((P_{p}^{n})^{-1})$ is the
best to measure scalar curvature, because of its linearity at a large range of
scales and for having the largest absolute deviation from constancy, with
respect to the other $\mathcal{GSC}_{i}((P_{p}^{n})^{-1})$ with no unknown
leading constant.

Ideally, used together, $\mathcal{GSC}_{i}((P_{p}^{n})^{-1})$ will prove short
scale distance curvature, and $\mathcal{GSC}_{i}(||S_{p}^{r}||)$ will prove
larger distance scales, providing a large range of scales at which scalar
curvature can be defined and computed in a metric space. If one is interested
in the large scale geometry of the space, then we suggest to use
$\mathcal{GSC}_{2}(||S_{p}^{r}||)$, and compare the results with the
constantly curved manifold profiles. If one is interested in a more detailed
information about the curvature at short scales, like to measure correlations
of scalar curvature at two different points, then we suggest to use
$\mathcal{GSC}_{5}((P_{p}^{n})^{-1})$.

One last remark is in order. We expect that the numerical differences in the
estimation of the scalar curvature between the $7$ different $\mathcal{GSC}$s
disappears when taking the limit of vanishing lattice spacing. Since we did
not perform a study of the scaling of our results at different triangulation
volumes, this remains a conjecture.

In what follows, we will use these definitions to determine the curvature of
the quantum geometries that appear in the path integral of 2D Causal Dynamical
Triangulations. Since these geometries are also triangulations, we use the
same prescriptions as used for the Delaunay triangulations in this section.

##  5 Quantum scalar curvature

As discussed in the introduction, the main motivation for our generalized
definitions of curvature is its use in a quantum geometry setting. Here, there
is no notion of tensor calculus, but we can define distances and a diffusion
process. In order to create valid observables for the quantum geometries it
must be diffeomorphism invariant, which means that point dependence has to be
eliminated. This is because the points in the quantum geometries are
fluctuating, and no identification between points in different configurations
can be made. As such, we will define an average quantum scalar curvature
observable, for which we will remove the point dependence of our curvature
definitions by taking a manifold average. To that end, we define the _Quantum
Scalar Curvature_ $\mathcal{QSC}_{i}(\mathcal{Q}^{s})$ based on an $s$ scale-
dependent local observable $\mathcal{Q}_{p}^{s}$ to be:

$\mathcal{QSC}_{i}(\mathcal{Q}^{s})=\frac{\int\differential[D]{p}\sqrt{\absolutevalue{g}}\,\mathcal{GSC}_{i}(\mathcal{Q}_{p}^{s})}{\int\differential[D]{p}\sqrt{\absolutevalue{g}}},$
(23)

where the manifold average is taken over manifold $M$ with metric $g$, where
$\absolutevalue{g}$ denotes the metric determinant. Then, we want to compute
the expectation value of this quantum scalar curvature
$\mathcal{QSC}_{i}(\mathcal{Q}^{s})$ observable,

$\expectationvalue{\mathcal{QSC}_{i}(\mathcal{Q}^{s})}=\frac{1}{Z}\int\mathcal{D}\quantity[g]\,e^{-S_{\text{eu}}[g]}\,\mathcal{QSC}_{i}(\mathcal{Q}^{s}),$
(24)

which is the Euclidean expectation value given as a Euclidean path integral
over all possible geometries. The measure $\mathcal{D}[g]$ here, denotes that
the integral is taken over distinct geometries, which is given by the measure
over all metrics $\mathcal{D}g$, dividing out the diffeomorphisms of the
geometry [17]. The Euclidean action $S_{\text{eu}}(g)$ is dictated by the
particular model one is working with. Furthermore, we denote the partition
function as $Z=\int\mathcal{D}\quantity[g]\,e^{-S_{\text{eu}}[g]}$. For this
paper we choose the setting of two-dimensional Causal Dynamical Triangulations
(CDT) [18].

In general, CDT is a non-perturbative method to quantize gravity. It uses
lattice techniques to regularize the sum over all possible metrics on a
manifold, by replacing and defining it as a sum over all possible simplicial
discretizations of it, with some constraints on the possible gluings between
the constituting simplices. These constraints are the ones responsible for
allowing a well-defined Wick rotation in CDT, unlike other approaches where
sometimes it is not even clear if this rotation exists. The causal dynamical
triangulations in the name of the approach are the simplicial manifolds that
represent curved, Lorentzian spacetimes appearing in the gravitational path
integral. The causality is imposed by a distinction between time and space
like edges of the simplices, and a particular way of gluing them such that at
each point there is a foliation of the space in the direction of the time-like
edges. By gluing these simplices, one obtains a simplicial manifold. Using
these discrete elements one can actually make sense of the path integral over
all geometries. The study of the resulting (discrete) quantum geometries is
usually done by means of Monte Carlo simulations. With the use of these
simulations, one can generate an ensemble of path integral configurations,
where each of them will be a simplicial manifold, and therefore a discrete
geometry. The expectation value of an observable over this ensemble, is later
on approximated by an average over a sample of configurations in this
ensemble. The ensemble of all these path integral configurations constitutes
the so-called quantum geometry under study.

Since it is out of the scope of this work to properly introduce all the
details of CDT, we refer the reader to some extensive reviews on the matter
[19, 2]. What the reader should retain from this brief and incomplete
introduction to CDT, is that from the CDT simulations, one can obtain an
ensemble of discrete geometries constituting a quantum geometry, where the
main interest is to study its geometrical properties, like curvature.

Figure 6: Visualization of a typical triangulation of 2D CDT with
$N_{2}=10\,{k}$ triangles and $\tau=123$ time-slices, based on a spring
embedding. Spatial slices of constant time are marked by dark lines.

In two-dimensional dynamical triangulations, the discretized version of the
Einstein-Hilbert action becomes very simple. In this case the Euclidean action
for a given triangulation $T$ is given by: $S_{\text{DT}}(T)=\lambda N(T)$,
where $\lambda$ is a dimensionless cosmological constant and $N(T)$ is the
number of triangles in the triangulation. The partition function over the
ensemble $\mathcal{T}$ is given by

$Z=\sum_{T\in\mathcal{T}}\frac{1}{C_{T}}e^{-\lambda N(T)},$ (25)

where $C_{T}$ is a symmetry factor suppressing highly symmetric triangulation.
In this case, the expectation value of the quantum scalar curvature is defined
as

$\expectationvalue{\mathcal{QSC}_{i}(\mathcal{Q}^{s})}=\frac{1}{Z}\sum_{T\in\mathcal{T}}\frac{1}{C_{T}}e^{-\lambda
N(T)}\,\frac{1}{\absolutevalue{T}}\sum_{p\in
T}\mathcal{GSC}_{i}(\mathcal{Q}^{s}_{p}),$ (26)

where $\absolutevalue{T}$ is the total number of points $p$ in the vertex (or
dual) graph of the triangulation. Instead of considering the ensemble of all
triangulations, we consider the ensemble of triangulations with a fixed total
volume $N(T)=N_{2}$. This fixed-volume ensemble is much more convenient to
work with computationally, making it the standard choice for numerical CDT
[19]. Furthermore, we only consider triangulations with a toroidal topology
without boundaries.

As mentioned before, we estimate the quantum scalar curvature by sampling
triangulations from the ensemble using Markov chain Monte Carlo methods. It is
possible to compute the triangulation average of
$\mathcal{GSC}_{i}(\mathcal{Q}^{s}_{p})$ in each triangulation, but
computationally rather expensive. Instead, we estimate the triangulation
average with a sample average, based on a uniform sample of points. To
decrease the effect caused by the compactness of the geometry we perform the
actual measurements on an unwrapped version of the simulation toroidal
triangulations. Specifically we cut the triangulations open along two closed
non-contractible loops and tile the unfolded triangulation to create an
infinite periodic triangulation of planar topology; for more details see [20].

Identical to the implementation in the Delaunay triangulations, the sphere
volume measurements are implemented by counting the number of points in vertex
graph at a given link distance $r$ from the origin. The computation of the
return probability of a random walker is also implemented in the way as for
the Delaunay triangulations. We use the dual graph of the CDT triangulations
and perform the random walk by updating the probability of the random walk
with (18). The return probability at a point $p$ after $n$ steps in a
configuration $T$, is computed by recording the value of $P_{T}(p,p;n)$ after
each diffusion step in (18).

Using the described methods we measured the quantum scalar curvatures for
sphere volumes $||S_{p}^{r}||$ in CDT. We performed measurements on $5\,{k}$
triangulations and approximated the manifold average by a sample average over
$500$ points. The resulting quantum scalar curvatures are displayed in Fig. 7.

From Fig. 7 we see some interesting features of the resulting quantum scalar
curvatures. First, all of them exhibit a plateau. This means that all
prescriptions are compatible with a zero curvature space. This is remarkable
and is not to be expected a priori, since the configurations themselves are by
no means something one would consider flat. In fact, a typical configuration
of toroidal topology looks like Fig. 6.

One remark is in order. The measured values for
$\mathcal{GSC}_{\mathcal{D}}(||S_{p}^{r}||)$ and
$\mathcal{GSC}_{\mathcal{I}}(||S_{p}^{r}||)$ have a plateau at a value
different from $2$, and the plateau is rather at around $\simeq 2.2$.
Furthermore, the expectation value $\expectationvalue{||S_{p}^{r}||}$ differs
greatly from the $2$-dimensional Delaunay plane and sphere.

There are two reasons for this. First, as mentioned before, the multiplying
factor to $||S_{p}^{r}||$ is unknown, and can be different in different
spaces, changing the absolute value of $||S_{p}^{r}||$ in different discrete
spaces. Furthermore, from $\mathcal{GSC}_{\mathcal{D}}(||S_{p}^{r}||)$ and
$\mathcal{GSC}_{\mathcal{I}}(||S_{p}^{r}||)$ one can see that the expectation
value of the parameter determining the leading power is not exactly equal to
$2$ but rather $\simeq 2.2$. This parameter can be interpreted as the scaling
dimension of the geodesic spheres, and by fitting a power law directly to the
expectation value of $||S_{p}^{r}||$ one extracts the so called Hausdorff
dimension.

(a)  (b)

(c)  (d)

(e)  (f)

(g)  (h)

Figure 7: The quantum scalar curvatures of the sphere volume
$\expectationvalue{\mathcal{QSC}_{i}(\norm{S_{p}^{r}})}$ on 2D CDT with
$N_{2}=300\,\text{k}$ triangles and $\tau=219$ time-slices. As a comparison
the generalized scalar curvature results $\mathcal{GSC}_{i}(\norm{S_{p}^{r}})$
of the Delaunay triangulations as already presented in Fig. 4 are also
included with the same number of triangles, $300\,\text{k}$, as well as the
continuum expectation for a plane, reflecting no curvature. The error bars
represent $95\%$-confidence intervals; for CDT this is based on the
fluctuations in $\mathcal{QSC}_{i}$ over $5000$ CDT configurations.

The theoretical value for the Hausdorff dimension in 2D CDT is $2$, but it has
been shown that is remarkably difficult in simulations to recover this value
numerically. This seems to be the case in our observables, where our leading
scaling parameter is also not exactly equal to $2$ as expected from the
topological dimension of the triangulation. Since this is only a problem for
determining the curvature from $\mathcal{GSC}_{\mathcal{D}}$ and
$\mathcal{GSC}_{\mathcal{I}}$, but not from the rest of the
$\mathcal{GSC}_{i}$s, and since it is outside the scope of this work to dive
into the details of the dimension of these fractal spaces, we refer the reader
previous investigations in the calculation of the Hausdorff dimension, and
other notions of dimensions [19, 16, 21, 22]. The fact that after our
manipulations the generalized curvatures between CDT and the Delaunay plane
overlap, despite having possible different leading powers and different
leading multiplying constants is a confirmation that our manipulations and
definitions indeed eliminate these differences, keeping only the curvature
scale dependent contributions.

In conclusion, all our measurements of $\mathcal{GSC}_{i}(||S_{p}^{r}||)$ seem
to indicate that, at least up to the measured scales, the ground state of the
2D CDT path integral exhibits the properties of a space with zero average
curvature, when extracting curvature using the geodesic sphere surface.

Similarly, we performed measurements of the quantum scalar curvature based on
the inverse of the return probability $(P^{n}_{p})^{-1}$, measured using the
previously described methods. We performed measurements on $1370$
triangulations of the fixed-volume ensemble, and we approximated the manifold
average by a sample average over $100$ points. The resulting
$\mathcal{QSC}_{i}((P^{n}_{p})^{-1})$ are presented in Fig. 8. Similar to the
sphere volume measurements we see that after a region of discretization
artefacts the $\mathcal{QSC}_{i}((P^{n}_{p})^{-1})$ are compatible with
flatness within error bars, again indicating $0$ average curvature. Compared
to the QSC based on sphere volumes the statistical errors are a smaller in
absolute terms, even though a lot less samples are used. This seems to
indicate that using the return probability is more robust than using the
sphere surface. Nevertheless, we also see in this case, that there are large
overshoots for every $\mathcal{GSC}_{i}((P^{n}_{p})^{-1})$ at $n<1000$. Even
though in absolute value these deviations are rather small, one should also
take into account that extracting the curvature using this method one will
always obtain very small deviations from $1$. In particular, we saw that for a
Delaunay triangulation of a sphere with the same number of triangles, the
deviations from $1$ where roughly up to $0.1$, which means that lattice
artefacts of this order are large. Despite this, at $n>5000$ these effects
seem to be washed out, and results compatible with flatness were found.
Interestingly, unlike in the case of the sphere surface,
$\mathcal{GSC}_{\mathcal{D}}((P^{n}_{p})^{-1})$ and
$\mathcal{GSC}_{\mathcal{I}}((P^{n}_{p})^{-1})$ seem to have a plateau at a
value much closer to $2$ than the ones using the sphere surface, indicating
that using a random walker, the leading power of the return probability is
compatible with a spectral dimension of 2D CDT to be equal to $2$ (recall the
leading term in each $\mathcal{GSC}$ in Tab. 2). Nevertheless, if one were to
fit a constant value to the results of
$\mathcal{GSC}_{\mathcal{D}}((P^{n}_{p})^{-1})$ and
$\mathcal{GSC}_{\mathcal{I}}((P^{n}_{p})^{-1})$, it would be slightly above
$2$. This slight deviation in the leading power of $(P^{n}_{p})^{-1}$,
together with a difference in the leading multiplying factor of
$(P^{n}_{p})^{-1}$ between different discrete spaces, can also be seen in the
plots of $(P^{n}_{p})^{-1}$, where one can see deviations from the Delaunay
plane. Again, the fact that after our manipulations the generalized curvatures
between CDT and Delaunay plane overlap despite having possible different
leading powers and different leading multiplying constants, is another
confirmation that our manipulations and definitions indeed eliminate these
differences, keeping only the curvature scale dependent contributions.

Summarizing, we conclude that all our definitions of scalar curvature indicate
that the path integral ground state of 2D CDT has average curvature equal to
$0$. This could go in contradictions with the studies performed in [23], where
it was found that the QRC measured in 2D CDT seemed to be non-constant. Their
result could either indicate the presence of curvature, or of some slowly
decaying lattice artefact, that vanishes for scales larger scales than the
ones studied in [23]. Despite the remarkable precision in their measurements,
the QRC still has the problem of the unknown leading prefactor multiplying the
results, as mentioned in the introduction. The problem in interpreting the
results of the QRC, is that even if a non-constant region of the scale-
dependent QRC is observed, one does not know how far away from flatness it is.
Therefore, the actual numerical value of the QRC is meaningless unless one
knows the multiplying prefactor, and because of that, one cannot know if the
results are actually converging to it or diverging from it.

(a)  (b)

(c)  (d)

(e)  (f)

(g)  (h)

Figure 8: The quantum scalar curvatures of the inverse return probability
$\expectationvalue{\mathcal{QSC}_{i}((P_{p})^{-1}))}$ on 2D CDT with
$N_{2}=300\,\text{k}$ triangles and $\tau=219$ time-slices. As a comparison
the generalized scalar curvature results $\mathcal{GSC}_{i}(\norm{S_{p}^{r}})$
of the Delaunay triangulations as already presented in Fig. 4 are also
included with the same number of triangles, $300\,\text{k}$, as well as the
continuum result for generalized scalar curvature in a plane. The error bands
represent $95\%$-confidence intervals; for CDT this is based on the
fluctuations in $\mathcal{QSC}_{i}$ over $1370$ CDT configurations, and for
the Delaunay triangulations this is based on the fluctuation between $50$
different points on the triangulation.

We believe that we have solved that problem in our curvature definitions, by
eliminating this unknown leading constants and powers. This is because by
construction, since our observables are ratios of derivatives or integrals of
the same quantity, any multiplying scale independent factor disappears,
leaving only the scale-dependent part of it. Furthermore, even if the leading
scaling power is unknown, $\mathcal{GSC}_{i}$ with $i=1,2,3,4,5$ will also
eliminate this leading scaling, preserving only the curvature contributions of
the quantity used. In these generalizations of the scalar curvature, the
reference value is always $1$, independently of the used quantity. This makes
the interpretation of the results easier than in the presence of unknown
leading powers or multiplying constants, since any deviation from $1$ is
interpreted as the presence of curvature, and its sign depends on the quantity
being used.

##  6 Conclusions

In this paper we define and implement generalized notions of Ricci scalar
curvature, for metric spaces endowed with a measure and/or with a random walk,
with the aim of providing computationally feasible methods of determining
curvature that can be implemented in the non-perturbative approach to quantum
gravity called Causal Dynamical Triangulations (CDT). The definitions are
successfully implemented in 2D CDT. This is done without the need of defining
or computing intermediate, more complex, quantities like the Riemann or Ricci
tensor. We do not assume the existence of tensor calculus at any stage in our
definitions of curvature in these spaces. Moreover, our constructions do not
assume that the considered spaces are smooth in any sense. Indeed, they are
designed to be used in discrete and fractal spaces, like the ones appearing in
CDT. Furthermore, our definitions are made intentionally without the use of
coordinates, since this is situation that is encountered in the geometries of
CDT.

For the construction of these generalized notions of scalar curvature
$\mathcal{GSC}$, we use quantities that can be computed both in a Riemannian
manifold, and in a metric space with a measure and/or a random walk. In
particular, we use the volume of a sphere $||S_{p}^{r}||$ with a radius $r$,
centred at a point $p$, and we use the return probability $P_{p}^{n}$ of a
random walker at a point $p$ after $n$ steps. In a Riemannian manifold, both
of these quantities define the Ricci scalar curvature by the leading-order
scale-dependent correction from the result in Euclidean space. In a nutshell,
the procedure used to define a $\mathcal{GSC}$ is the following: 1) Find a
scale dependent quantity that can be computed both in a Riemannian manifold
and in a metric space with a measure and/or a random walk. This quantity
should be such that in a Riemannian manifold, its first curvature correction
with respect to the Euclidean calculation defines the Ricci scalar. Examples
are $||S_{p}^{r}||$ and $P_{p}^{n}$. 2) Apply a certain number of integrals
and derivatives to this scale dependent quantity, to extract the leading order
curvature correction in a Riemannian manifold. 3) Replicate these calculations
on the metric space in question, and use the result as a definition of scalar
curvature, associated with this quantity.

Our definitions, for a general scale-dependent quantity, are shown in table 1.
Their expansions in a Riemannian manifold, using $||S_{p}^{r}||$ and
$P_{p}^{n}$, are shown in table 2. We introduce $7$ different methods, that
involve different combinations of integrals and derivatives. The reason for
doing this is that different methods introduce different undesired numerical
effects like noise and lattice artefacts in discrete spaces. The methods are
compared in order to find the one that has the most desirable features, which
can be different in a different setting. We tested these definitions on random
geometries generated by constructing Delaunay triangulations of a sphere and a
periodic plane. Our results seemed to indicate that applying these
prescriptions to determine the generalized scalar curvature of the
corresponding Delaunay triangulations, match remarkably well with the Ricci
scalar of the underlying Riemannian manifolds. We conclude that this is a
confirmation that our generalized notions of curvature can indeed be applied
reliably to discrete metric spaces, and that their result can be interpreted
as a notion of scalar curvature.

To determine the curvature of the ground state of the path integral in 2D
Causal Dynamical Triangulations, we also measured our generalized notions of
scalar curvature in this setting. We estimated the expectation value of our
$\mathcal{GSC}$s associated with $||S_{p}^{r}||$ and $P_{p}^{n}$, and
determined that the expectation value of all of our curvature definitions,
both using $||S_{p}^{r}||$ and $P_{p}^{n}$, are compatible with flatness. This
seems to suggest that the ground state of the 2D CDT path integral is flat.
The results are shown in Fig. 7 and 8.

Despite the fact that in a Riemannian manifold, all our generalized scalar
curvatures are equivalent in the sense that they all define the Ricci scalar
curvature at small scales, we conclude that two of our definitions of
curvature are better than the others in the discrete cases we considered. Our
conclusions are based in the implementations on the Delaunay triangulations.
We considered as better the curvature definitions where there is faster
convergence to the continuum curves used to fit the Delaunay results, together
with a reduction of noise at short and large scales in the numerical results.
When using the sphere surface $||S_{p}^{r}||$, we conclude the best definition
of curvature is $\mathcal{GSC}_{2}(||S_{p}^{r}||)$, for its robustness
properties, its fast convergence towards the continuum fit, and because in a
constantly curved Riemannian manifold it is independent of the dimension of
the space. This feature makes it interesting to apply it to fractal spaces.
$\mathcal{GSC}_{2}(||S_{p}^{r}||)$ appears to be more suitable to estimate the
curvature at large scales, by comparing the numerical results with the
continuum formulas given in appendix A, and not very reliable for estimating
curvature at short scales. When using the return probability of random walker
$P_{p}^{n}$, we conclude that $\mathcal{GSC}_{5}((P_{p}^{n})^{-1})$ is the
best prescription to obtain the curvature of the space. This is because of its
balance between robustness and its sensitivity to curvature. This sensitivity
can be seen in the results in the Delaunay triangulation of a sphere, since
for the same sphere, the largest absolute numerical deviation form $1$ among
all the $\mathcal{GSC}$s is associated with $\mathcal{GSC}_{5}$. Furthermore,
$\mathcal{GSC}_{5}((P_{p}^{n})^{-1})$ seems to be more suitable to estimate
curvature at short scales than any $\mathcal{GSC}_{i}(||S_{p}^{r}||)$.

Aside from our particular implementations and test spaces, the defined
$\mathcal{GSC}$s are ideal for the calculation of $n$-point functions. The
reason is that they explicitly define scalar curvature, without additive or
multiplicative unknown constants. Furthermore, our definitions allow us to
determine more than just the presence or the absence of curvature on a space.
Since there are no unknown multiplicative constants in our observables, we can
extract the scalar curvature of the given space, under some approximations
explained in the previous sections. This was done for the Delaunay
triangulation of the sphere.

Given that our definitions are scale-dependent, in the sense of defining a
notion of curvature at a given distance scale on the space, they can be
renormalized. This is extremely useful when the continuum limit of a space is
desired. This is the situation of CDT, where after performing calculations on
a fixed-size discrete triangulations, one studies the scaling of the results
when changing the size of the universe itself. This was not done in our
calculations since we were virtually using an infinite size 2D universe, by
using a periodically identified section of the plane. Therefore, our results
are already approximately equal to the ones in an infinite size 2D CDT
universe, and there was no need of rescaling. Nevertheless, it would be
interesting to study the dependence of our results with the size of the system
in question. As a matter of fact, even in the case of a Delaunay triangulation
of a 2-sphere, one could study the convergence, or the lack of, of the
numerical results for the curvature estimated using the different methods
proposed here. We expect that at least in a classical space like the Delaunay
triangulations, the numerical differences are just a discretization artefact,
and once the limit of vanishing lattice spacing is taken, all prescriptions
converge to the same curvature value.

In a future work, it would be interesting to apply these definitions to
determine the $2$-point function of the scalar curvature in CDT, obtaining a
non-perturbative result for the correlation function of the scalar curvature
in quantum gravity.

## Acknowledgements

The authors would like to thank Renate Loll for many useful discussions and
for her detailed feedback during several iterations of this manuscript.

##  Appendix A $\mathcal{GSC}_{i}(||\mathcal{S}_{p}^{r}||)$ in constant
curvature manifolds

In the case where the underlying space is a D-sphere of radius $R$, one can
compute exactly the functional form of
$\mathcal{GSC}_{i}(||\mathcal{S}_{p}^{r}||)$ as a function of $r$, $R$ and
$D$. This case is interesting since it could be used for direct comparison of
$\mathcal{GSC}_{i}(||\mathcal{S}_{p}^{r}||)$ of a general metric space with
that of a D-sphere.

On a D-sphere, one can compute the surface $||\mathcal{S}_{p}^{r}||$ of a
geodesic sphere of radius $r$ on top of it

$||\mathcal{S}_{p}^{r}||=2\pi A_{D}R^{D-1}\sin^{D-1}\left(\frac{r}{R}\right),$
(27)

where $A_{D}$ is a dimensional dependent constant of the form

$A_{D}=\prod_{i=2}^{D-1}(\int_{0}^{\pi}\sin^{D-i}(x)dx).$ (28)

Using this, one can compute exactly all the generalized curvature definitions

$\mathcal{GSC}_{\mathcal{D}}(||\mathcal{S}_{p}^{r}||)=1+(D-1)\,(\frac{r}{R})\,\cot\left(\frac{r}{R}\right),$
(29)
$\mathcal{GSC}_{\mathcal{I}}(||\mathcal{S}_{p}^{r}||)=\frac{D\,\frac{r}{R}\,\cot\left(\frac{r}{R}\right)}{\sqrt{\cos^{2}\left(\frac{r}{R}\right)}\,{}_{2}F_{1}\left(\frac{1}{2},\frac{D}{2},\frac{2+D}{2},\sin^{2}\left(\frac{r}{R}\right)\right)},$
(30)
$\mathcal{GSC}_{1}(||\mathcal{S}_{p}^{r}||)=1+(D-1)\,(\frac{r}{R})\,(\cot\left(\frac{r}{R}\right)-(\frac{r}{R})\,\csc^{2}\left(\frac{r}{R}\right)),$
(31)
$\mathcal{GSC}_{2}(||\mathcal{S}_{p}^{r}||)=\frac{12i\,(\frac{r}{R})^{2}\cot\left(\frac{r}{R}\right)}{6(\frac{r}{R})^{2}-\pi^{2}+12i\,(\frac{r}{R})\,\log\left(1-e^{2i(\frac{r}{R})}\right)+6\text{PolyLog}_{2}\left(e^{2i(\frac{r}{R})}\right)},$
(32)
$\mathcal{GSC}_{3}(||\mathcal{S}_{p}^{r}||)=\frac{\cos(\frac{r}{R})\,{}_{2}F_{1}\left(\frac{1}{2},\frac{D}{2},\frac{2+D}{2},\sin^{2}\left(\frac{r}{R}\right)\right)\left((D-1)+(\frac{r}{R})^{-1}\,\tan\left(\frac{r}{R}\right)\right)}{D},$
(33)
$\mathcal{GSC}_{4}(||\mathcal{S}_{p}^{r}||)=\frac{1}{D\sin^{D+1}\left(\frac{r}{R}\right)}\Big{[}\left((\int_{0}^{(\frac{r}{R})}\frac{\sin^{D-1}\left(y\right)}{y}\,dy)\sin^{2}\left(\frac{r}{R}\right)+\sin^{D+1}\left(\frac{r}{R}\right)\right)\\\
\times\left(D-(\frac{r}{R})^{-1}\,\sqrt{\cos^{2}\left(\frac{r}{R}\right)}\,{}_{2}F_{1}\left(\frac{1}{2},\frac{D}{2},\frac{2+D}{2},\sin^{2}\left(\frac{r}{R}\right)\right)\tan\left(\frac{r}{R}\right)\right)\Big{]},$
(34)
$\mathcal{GSC}_{5}(||\mathcal{S}_{p}^{r}||)=1+(D-1)^{2}\Bigg{[}(\frac{r}{R})^{2}\cot^{2}\left(\frac{r}{R}\right)\\\
+\frac{i(\frac{r}{R})\csc^{2}\left(\frac{r}{R}\right)\left(6(\frac{r}{R})^{2}-\pi^{2}+12i(\frac{r}{R})\log\left(1-e^{2i(\frac{r}{R})}\right)+6\text{PolyLog}_{2}\left(e^{2i(\frac{r}{R})}\right)\right)}{12}\Bigg{]},$
(35)

where $\int_{0}^{\frac{r}{R}}\frac{\sin^{D-1}\left(y\right)}{y}\,dy$ was kept
in integral form, but can be computed exactly for any $D\geq 2$.

Some comments are in order. The dimensional dependence of each of the
$\mathcal{GSC}_{i}(||\mathcal{S}_{p}^{r}||)$ appears to be different in each
of them. In particular, the dimension $D$ of the sphere appears inside
Hypergeometric functions in
$\mathcal{GSC}_{\mathcal{I}}(||\mathcal{S}_{p}^{r}||)$,
$\mathcal{GSC}_{3}(||\mathcal{S}_{p}^{r}||)$ and
$\mathcal{GSC}_{4}(||\mathcal{S}_{p}^{r}||)$. In
$\mathcal{GSC}_{\mathcal{D}}(||\mathcal{S}_{p}^{r}||)$,
$\mathcal{GSC}_{1}(||\mathcal{S}_{p}^{r}||)$ and
$\mathcal{GSC}_{5}(||\mathcal{S}_{p}^{r}||)$ it appears inside a
multiplicative factor in front of a curvature dependent function.
Interestingly, $\mathcal{GSC}_{2}(||\mathcal{S}_{p}^{r}||)$ is independent of
the dimension of the underlying sphere, which makes it suitable to compare
spaces with possible fractal dimensionality. In particular, it has been said
that $2D$ euclidean dynamical triangulations have a ground state that shares
some properties with a 5 dimensional sphere. It would be interesting if this
is also the case for this generalized curvatures. In particular, if it
resembles the properties of a sphere of any dimension, by comparing the
$\mathcal{GSC}_{2}(||\mathcal{S}_{p}^{r}||)$ for the system of interest with
the $D$-sphere.

Consider a hyperbolic $D$-space with scalar curvature equal to
$\frac{D(1-D)}{R^{2}}$, such that we define the radius $R$ to be such that
this holds. In this case, one can also compute the surface of a sphere
$||\mathcal{S}_{p}^{r}||$, where one obtains

$||\mathcal{S}_{p}^{r}||=2\pi
A_{D}R^{D-1}\sinh^{D-1}\left(\frac{r}{R}\right),$ (36)

where it differs from the sphere case since there is a hyperbolic sine
function instead of a sine function.

Figure 9: $\mathcal{GSC}_{2}(||\mathcal{S}_{p}^{r}||)$ for a D-Sphere of
scalar curvature $\frac{D(D-1)}{R^{2}}$, a hyperbolic D-space of scalar
curvature $\frac{D(1-D)}{R^{2}}$, and euclidean D-dimensional space.

Using this, one can compute exactly all the generalized curvature definitions

$\mathcal{GSC}_{\mathcal{D}}(||\mathcal{S}_{p}^{r}||)=1+(D-1)\,(\frac{r}{R})\,\coth\left(\frac{r}{R}\right),$
(37)
$\mathcal{GSC}_{\mathcal{I}}(||\mathcal{S}_{p}^{r}||)=\frac{D\,\frac{r}{R}\,\cot\left(\frac{r}{R}\right)}{\sinh\left(\frac{r}{R}\right)\,{}_{2}F_{1}\left(\frac{1}{2},\frac{D}{2},\frac{2+D}{2},-\sinh^{2}\left(\frac{r}{R}\right)\right)},$
(38)
$\mathcal{GSC}_{1}(||\mathcal{S}_{p}^{r}||)=1+(D-1)\,(\frac{r}{R})\,(\coth\left(\frac{r}{R}\right)-(\frac{r}{R})\,\csch^{2}\left(\frac{r}{R}\right)),$
(39)
$\mathcal{GSC}_{2}(||\mathcal{S}_{p}^{r}||)=\frac{12\,(\frac{r}{R})^{2}\coth\left(\frac{r}{R}\right)}{-6(\frac{r}{R})^{2}-\pi^{2}+12\,(\frac{r}{R})\,\log\left(1-e^{2(\frac{r}{R})}\right)+6\text{PolyLog}_{2}\left(e^{2(\frac{r}{R})}\right)},$
(40)
$\mathcal{GSC}_{3}(||\mathcal{S}_{p}^{r}||)=\frac{\sqrt{\cosh^{2}\left(\frac{r}{R}\right)}\,{}_{2}F_{1}\left(\frac{1}{2},\frac{D}{2},\frac{2+D}{2},-\sinh^{2}\left(\frac{r}{R}\right)\right)\left((D-1)+(\frac{r}{R})^{-1}\,\tanh\left(\frac{r}{R}\right)\right)}{D},$
(41)
$\mathcal{GSC}_{4}(||\mathcal{S}_{p}^{r}||)=\frac{1}{D\sinh^{D+1}\left(\frac{r}{R}\right)}\Big{[}\left((\int_{0}^{(\frac{r}{R})}\frac{\sinh^{D-1}\left(y\right)}{y}\,dy)\sinh^{2}\left(\frac{r}{R}\right)+\sinh^{D+1}\left(\frac{r}{R}\right)\right)\\\
\times\left(D-(\frac{r}{R})^{-1}\,\sqrt{\cosh^{2}\left(\frac{r}{R}\right)}\,{}_{2}F_{1}\left(\frac{1}{2},\frac{D}{2},\frac{2+D}{2},-\sinh^{2}\left(\frac{r}{R}\right)\right)\tanh\left(\frac{r}{R}\right)\right)\Big{]},$
(42)
$\mathcal{GSC}_{5}(||\mathcal{S}_{p}^{r}||)=1+\frac{(D-1)^{2}}{12}(\frac{r}{R})\csch^{2}\left(\frac{r}{R}\right)\Big{[}6(\frac{r}{R})^{2}+\pi^{2}+6(\frac{r}{R})(1-2i\pi)+6(\frac{r}{R})\cosh\left(\frac{2r}{R}\right)\\\
-12(\frac{r}{R})\log(-1+e^{\frac{2r}{R}})-6\text{PolyLog}_{2}\left(e^{2(\frac{r}{R})}\right)\Big{]},$
(43)

where $\int_{0}^{\frac{r}{R}}\frac{\sinh^{D-1}\left(y\right)}{y}\,dy$ was kept
in integral form, but can be computed exactly for any $D\geq 2$.

One can see that $\mathcal{GSC}_{2}(||\mathcal{S}_{p}^{r}||)$ is independent
of the dimension of the underlying space, for both positive and negative
constantly curved spaces. This renders
$\mathcal{GSC}_{2}(||\mathcal{S}_{p}^{r}||)$ ideal for making comparisons of
$\mathcal{GSC}_{i}$ of general spaces. For the sake of completeness, we
include a plot for $\mathcal{GSC}_{2}(||\mathcal{S}_{p}^{r}||)$ as a function
of $\frac{r}{R}$ in Fig. 9.

## References

  * [1] R Loll, G Fabiano, D Frattulillo and F Wagner  “Quantum gravity in 30 questions”  In _arXiv preprint arXiv:2206.06762_ , 2022
  * [2] R Loll  “Quantum gravity from causal dynamical triangulations: a review”  In _Classical and Quantum Gravity_ 37.1 IOP Publishing, 2019, pp. 013002  arXiv:1905.08669
  * [3] R. Loll  “Quantum Curvature as Key to the Quantum Universe”, 2023  arXiv:2306.13782 [gr-qc]
  * [4] T. Regge  “General relativity without coordinates”  In _Il Nuovo Cimento (1955-1965)_ 19.3, 1961, pp. 558–571  DOI: 10.1007/BF02733251
  * [5] Yann Ollivier  “Ricci curvature of Markov chains on metric spaces”  In _Journal of Functional Analysis_ 256.3, 2009, pp. 810–864  DOI: https://doi.org/10.1016/j.jfa.2008.11.001
  * [6] N. Klitgaard and R. Loll  “Introducing quantum Ricci curvature”  In _Physical Review D_ 97.4 American Physical Society (APS), 2018  DOI: 10.1103/physrevd.97.046008
  * [7] R. Loll  “Quantum Curvature as Key to the Quantum Universe”  In _arXiv e-prints_ , 2023, pp. arXiv:2306.13782  DOI: 10.48550/arXiv.2306.13782
  * [8] Karl-Theodor Sturm  “On the geometry of metric measure spaces”  In _Acta Mathematica_ 196.1 Institut Mittag-Leffler, 2006, pp. 65–131  DOI: 10.1007/s11511-006-0002-8
  * [9] Alfred Gray  “The volume of a small geodesic ball of a Riemannian manifold.”  In _Michigan Mathematical Journal_ 20.4 University of Michigan, Department of Mathematics, 1974, pp. 329–344
  * [10] Alexander Grigor’yan, Jiaxin Hu and Ka-Sing Lau  “Heat Kernels on Metric Measure Spaces”  In _Geometry and Analysis of Fractals_ Berlin, Heidelberg: Springer Berlin Heidelberg, 2014, pp. 147–207
  * [11] Gregory F Lawler  “Random walk and the heat equation”  American Mathematical Soc., 2010
  * [12] Dmitri V Vassilevich  “Heat kernel expansion: user’s manual”  In _Physics reports_ 388.5-6 Elsevier, 2003, pp. 279–360
  * [13] N. Klitgaard and R. Loll  “Implementing quantum Ricci curvature”  In _Physical Review D_ 97.10 American Physical Society (APS), 2018  DOI: 10.1103/physrevd.97.106017
  * [14] Der-Tsai Lee and Bruce J Schachter  “Two algorithms for constructing a Delaunay triangulation”  In _International Journal of Computer & Information Sciences_ 9.3 Springer, 1980, pp. 219–242
  * [15] Tong Wang  “Poisson-Disk Sampling: Theory and Applications”  In _Encyclopedia of Computer Graphics and Games_ Cham: Springer International Publishing, 2020, pp. 1–8  DOI: 10.1007/978-3-319-08234-9_398-1
  * [16] J. Ambjørn, J. Jurkiewicz and Y. Watabiki  “On the fractal structure of two-dimensional quantum gravity”  In _Nuclear Physics B_ 454.1-2 Elsevier BV, 1995, pp. 313–342  arXiv:hep-lat/9507014
  * [17] Jan Ambjørn, Andrzej Görlich, Jerzy Jurkiewicz and Renate Loll  “Quantum gravity via causal dynamical triangulations”  In _Springer handbook of spacetime_ Springer, 2014, pp. 723–741
  * [18] Jan Ambjørn, Lisa Glaser, Yuki Sato and Yoshiyuki Watabiki  “2d CDT is 2d Hořava–Lifshitz quantum gravity”  In _Physics Letters B_ 722.1-3 Elsevier, 2013, pp. 172–175
  * [19] J. Ambjørn, A. Görlich, J. Jurkiewicz and R. Loll  “Nonperturbative quantum gravity”  In _Physics Reports_ 519.4-5 Elsevier BV, 2012, pp. 127–210  DOI: 10.1016/j.physrep.2012.03.007
  * [20] Jesse Duin  “Curvature correlations in Quantum Gravity” To appear, 2023  URL: https://www.ru.nl/highenergyphysics/theses/master-theses/
  * [21] J. Ambjørn, K.. Anagnostopoulos and R. Loll  “New perspective on matter coupling in 2D quantum gravity”  In _Physical Review D_ 60.10 American Physical Society (APS), 1999  arXiv:hep-th/9904012
  * [22] J. Ambjørn, K. Anagnostopoulos and R. Loll  “Crossing the c=1 barrier in 2D Lorentzian quantum gravity”  In _Physical Review D_ 61.4 American Physical Society (APS), 2000  arXiv:hep-lat/9909129
  * [23] J. Brunekreef and R. Loll  “Quantum flatness in two-dimensional quantum gravity”  In _Physical Review D_ 104.12 American Physical Society (APS), 2021  DOI: 10.1103/physrevd.104.126024