Generalised spectral dimensions in non-perturbative quantum gravity

M. Reitza, D. Németha, D. Rajbhandaria, A. Görlicha,b, J. Gizbert-Studnickia,b

aInstitute of Theoretical Physics, Jagiellonian University,
Łojasiewicza 11, Kraków, PL 30-348, Poland.
bMark Kac Center for Complex Systems Research, Jagiellonian University,
Łojasiewicza 11, Kraków, PL 30-348, Poland.

email<EMAIL_ADDRESS>
###### Abstract

The seemingly universal phenomenon of scale-dependent effective dimensions in
non-perturbative theories of quantum gravity has been shown to be a potential
source of quantum gravity phenomenology. The scale-dependent effective
dimension from quantum gravity has only been considered for scalar fields. It
is however possible that the non-manifold like structures, that are expected
to appear near the Planck scale, have an effective dimension that depends on
the type of field under consideration. To investigate this question, we have
studied the spectral dimension associated to the Laplace-Beltrami operator
generalised to $k$-form fields on spatial slices of the non-perturbative model
of quantum gravity known as Causal Dynamical Triangulations. We have found
that the two-form, tensor and dual scalar spectral dimensions exhibit a flow
between two scales at which an effective dimension appears. However, the one-
form and vector spectral dimensions show only a single effective dimension.
The fact that the one-form and vector spectral dimension do not show a flow of
the effective dimension can potentially be related to the absence of a
dispersion relation for the electromagnetic field, but dynamically generated
instead of as an assumption.

###### Contents

  1. 1 Introduction
  2. 2 Overview of CDT
  3. 3 Tensor Diffusion
    1. 3.1 Continuum tensor diffusion
    2. 3.2 Discrete tensor diffusion
    3. 3.3 Two- and three-dimensional regular geometries
  4. 4 Generalised spectral dimensions in non-perturbative quantum gravity
    1. 4.1 Spatial Hausdorff dimension
  5. 5 Discussion and outlook
  6. A Analytical calculation of generalised spectral dimensions on the flat two-torus

##  1 Introduction

An important tasks of quantum gravity research is to find observables that
show possible deviations from classical gravity. Especially in view of the
recent availability of gravitational wave data, the search for new potentially
observable signatures of quantum gravity has a high priority. An example of
deviation from classical gravity coming from quantum gravity is the
possibility of a scale-dependent effective dimension of spacetime. This
running of the spectral dimension has been discovered to be present in many
different theories of quantum gravity [1, 2, 3, 4].

The interpretation of this seemingly universal quantum gravity phenomenon is
that spacetime is not well described by a manifold of a fixed dimension near
the Planck scale. Although it is not entirely clear how a running spectral
dimension would manifest itself in real-world observations, there have been
attempts to formulate model-independent signatures of the running spectral
dimension in gravitational wave physics [5, 6, 7]. While showing that there is
a potential for an observable effect in gravitational wave observations, the
analysis of signatures of the running spectral dimension in gravitational wave
physics relies on a number of assumptions. It is assumed that only the scalar
mode of the gravitational waves is sensitive to the running spectral dimension
and that other modes are unaffected. It is furthermore assumed that the
electromagnetic field, which is a vector field, is also unaffected.

The spectral dimension is a quantity related to the spectrum of the Laplacian,
which is a second-order operator on the space of scalar functions. A
generalisation of the Laplace operator is the Laplace-Beltrami111Sometimes
$\Delta^{(k)}$ is called the Laplace–de Rham operator. operator
$\Delta^{(k)}$, defined as a second order differential operator on the space
of $k$-forms. The Laplace-Beltrami operator $\Delta^{(k)}$ also defines a
generalised spectral dimension $D^{(k)}$. For a differentiable Riemannian
manifold $\mathcal{M}$, the spectral dimension is identical for all $k$.
However, because a single differentiable manifold is most likely not the
correct description of spacetime when quantum gravitational effects are
relevant, it is an important question whether the generalised spectral
dimension $D^{(k)}$ depends on $k$ in quantum gravity. The Laplace-Beltrami
operator $\Delta^{(k)}$ is also related to propagation of $k$-form fields. A
$k$-dependent spectral dimension $D^{(k)}$ can therefore signal that different
fields and gravitational wave modes have a different scale dependence. As far
as we are aware222A framework to investigate $\Delta^{(k)}$ for all $k$ in
discrete models of quantum gravity was discussed in [8], but only the dual
scalar case was considered in detail., the only cases that have been studied
in quantum gravity are those associated to scalar and dual scalar fields. The
goal of this work is to investigate whether the generalised spectral dimension
$D^{(k)}$ depends on $k$ in quantum gravity and whether it is scale dependent
for all $k$. This work is a step towards explicitly testing the assumptions
made in deriving signatures of the running spectral dimension in gravitational
wave physics and improving our knowledge on how quantum gravity can
potentially manifest itself.

The running of the spectral dimension was first discovered in Causal Dynamical
Triangulations (CDT) [9], a non-perturbative approach to quantum gravity that
is defined in terms of a lattice regularisation of the gravitational path-
integral [10, 11, 12]. One of the advantages of CDT is that it is particularly
suited for numerical simulations. Therefore, it is possible to calculate
observables that are not easily accessible in other frameworks. CDT is defined
in terms of piecewise flat manifolds or triangulations $\mathcal{T}$, a
specific type of simplicial complexes. Using well known tools from the study
of simplicial complexes [13], it is relatively straightforward to define the
Laplace-Beltrami operator $\Delta^{(k)}$ in CDT. We can therefore use the
numerical tools of CDT to study the $D^{(k)}$ in a non-perturbative
formulation of quantum gravity. In section 2 we will review those aspects of
CDT that are important for the remainder of the text. Section 3.1 contains a
discussion of the Laplace-Beltrami operator and the associated generalised
spectral dimension. Section 3.2 explains how the Laplace-Beltrami operator and
the generalised spectral dimension can be generalised to simplicial complexes.
Section 4 contains the measurements of the generalised spectral dimension in
CDT. Finally, we will summarise our results in section 5 and give an
interpretation in terms of quantum gravity phenomenology.

##  2 Overview of CDT

CDT is a non-perturbative model of quantum gravity [10, 11, 12], based on a
lattice regularisation of the formal gravitational path integral $Z$,

$Z=\int\mathcal{D}[g_{\mu\nu}]e^{iS_{EH}[g_{\mu\nu}]}.$ (1)

The regularisation procedure consists of a discretisation of spacetime and
replacing the Einstein-Hilbert action $S_{EH}$ with the (Lorentzian) Regge
action $S^{(L)}_{R}$ [14]. Formal integration over metric manifolds is
replaced by a sum over discrete spacetimes $T\in\mathcal{T}^{(L)}_{CDT}$,

$Z=\int\mathcal{D}[g_{\mu\nu}]e^{iS_{EH}[g_{\mu\nu}]}\longrightarrow\sum_{T\in\mathcal{T}^{(L)}_{CDT}}\frac{1}{C_{T}}e^{iS_{R}^{(L)}[\mathcal{T}]}.$
(2)

The factor $C_{T}$ is the order of the automorphism group of $T$. The set of
CDT geometries $\mathcal{T}^{(L)}_{CDT}$ consists of spacetimes that are
$n$-dimensional simplicial Lorentzian manifolds333These simplicial manifolds,
or triangulations, consist of a few types of identical $n$-dimensional
Lorentzian simplices glued together along $(n-1)$-dimensional faces.,
determined by a lattice spacing $a$, that admit a globally hyperbolic
foliation. Each leaf of the foliation is a $(n-1)$-dimensional simplicial
manifold (see figure 1) of fixed topology and is labelled by a discrete time
step $t$. The foliation ensures a causal evolution of spatial hypersurfaces.
The lattice regularisation can in principle be removed by a simultaneous limit
of the number of simplices $N\rightarrow\infty$ and of the lattice spacing
$a\rightarrow 0$.

Figure 1: The figure shows a CDT spacetime $T$ foliated by a family of space-
like hypersurfaces $(\sum_{t})_{t\in\mathbb{Z}}$

The foliation also makes a well-defined Wick rotation possible. The Wick
rotation transforms the Lorentzian action $S^{(L)}_{R}$ into the Euclidean
action $S_{R}$ and the set of Lorentzian triangulations
$\mathcal{T}^{(L)}_{CDT}$ into the set of Euclidean triangulations
$\mathcal{T}_{CDT}$,

$Z=\sum_{T\in\mathcal{T}_{CDT}}\frac{1}{C_{T}}e^{-S_{R}[T]},$ (3)

allowing for a statistical treatment of the model. In 3+1 dimensions, after
the Wick rotation the Euclidean Regge action $S_{R}$ is given by

$S_{R}=-(\kappa_{0}+6\Delta)N_{0}+\kappa_{4}(N_{41}+N_{32})+\Delta N_{41}.$
(4)

$N_{0}$, $N_{41}$ and $N_{32}$ are the global parameters related to the total
number of vertices, $\\{4,1\\}$-type of simplices and $\\{3,2\\}$-type of
simplices, respectively. The number $i$ for an $\\{i,j\\}$-type of simplex
denotes the number of vertices of a simplex at time $t$ and the number $j$
denotes the number of vertices at time $t\pm 1$. The bare couplings that
appear in $S_{R}$ are the (inverse) bare gravitational constant $\kappa_{0}$,
the bare cosmological constant $\kappa_{4}$ and the asymmetry parameter
$\Delta$. The coupling $\Delta$ determines the difference in proper length
between space-like and time-like links of the simplices. The expectation value
of an observable $\mathcal{O}$ is given by,

$\langle\mathcal{O}\rangle=\frac{1}{{Z}}\sum_{T}\frac{1}{{C}_{T}}\mathcal{O}(T)e^{-S_{R}[T]}.$
(5)

In the absence of an exact calculation of $\langle\mathcal{O}\rangle$, one can
sample the CDT configuration space with Monte Carlo simulations and
approximate $\langle\mathcal{O}\rangle$ by averaging $\mathcal{O}$ over the
resulting ensemble of geometries generated with the Boltzmann weight
$e^{-S_{R}[T]}$. The Monte Carlo simulations are performed for a range of
target volumes $\bar{N}_{41}$, controlled by the additional volume fixing term
$S_{fix}=\epsilon(N_{41}-\bar{N}_{41})^{2}$ added to the Regge action444In
order to force the volume $N_{41}$ of the triangulation to fluctuate around
the target $\bar{N}_{41}$ it is also necessary to tune the cosmological
coupling constant
$\kappa_{4}\to\kappa_{4}^{c}(\kappa_{0},\Delta,\bar{N}_{41})$. This way one
trades the $\kappa_{4}$ coupling constant with the target volume
$\bar{N}_{41}$.. Subsequently, the average of $\mathcal{O}$ can be
extrapolated to the infinite-volume limit using standard finite-volume scaling
analysis. The theory space of CDT can be investigated by varying the bare
coupling constants. The phase diagram of CDT (see Fig. 2) has a rich structure
[15].

Figure 2: The phase diagram of CDT with its four phases. Phase $A$ is the
branched polymer phase, $B$ is the collapsed phase, $C_{b}$ the bifurcation
and $C$ the de Sitter phase. The purple dot at coordinate
$(\kappa_{0},\Delta)=(4.0,0.2)$ represents the location where the
configurations were sampled.

Out of four phases with distinct geometric properties, the most interesting
one is phase $C$, also called the de Sitter phase, where on large scales
(compared to the size of the triangulations) one observes an emerging four-
dimensional geometry [16]. In was shown that in phase $C$ the volume profile
obtained by integrating out all degrees of freedom except the scale factor is
consistent with classical solutions of General Relativity and quantum
fluctuations of the scale factor are well described by the minisuperspace-type
effective action [17, 18]. This result required an introduction of appropriate
observables which can be measured and averaged over in the Monte Carlo
simulations. One such observable was the (dual) scalar spectral dimension
which, as already mentioned, showed a non-trivial scale dependence, changing
from 4 in large scales to approximately 2 in short scales [9]. This result
suggests that even though the large scale behaviour of the quantum geometry
looks (semi-)classical, the short scale features can be very far from any
behaviour of classical geometries and most likely cannot be described in terms
of smooth differentiable manifolds. It was indeed shown that the leaves of the
CDT foliation (the spatial slices) have very non-trivial fractal geometry [16,
19]. The four-dimensional geometry is also non-smooth on short scales [16,
20]. We therefore expect that by performing numerical simulations in phase $C$
and studying observables sensitive to short scales one can investigate
features of the UV limit of quantum gravity, while observables sensitive to
large scales will at the same time show features of the IR limit. The scale-
dependent generalized spectral dimensions $D^{(k)}$ can therefore be used to
simultaneously investigate both regimes.

##  3 Tensor Diffusion

###  3.1 Continuum tensor diffusion

The heat equation governs diffusions and random walks on a general
$n$-dimensional Riemannian manifold $(\mathcal{M},g)$. We will only consider
the case where $\mathcal{M}$ is closed. The heat equation,

$\frac{\partial}{\partial\tau}K(x,x_{0};\tau)=-\Delta K(x,x_{0};\tau),$ (6)

describes the evolution of the heat kernel $K(x,x_{0};\tau)$, which represents
the probability density at position $x\in\mathcal{M}$ of finding a particle
after a random walk over a diffusion time $\tau$. The initial condition for a
particle which originated at position $x_{0}$ at $\tau=0$ is given by a delta
function,

$K(x,x_{0};\tau=0)=\delta^{n}(x-x_{0}).$ (7)

The total probability is preserved over time,

$\int dx\sqrt{g}K(x,x_{0};\tau)=1.$ (8)

The average return probability $P(\tau)$ of the random walk returning to the
origin after time $\tau$, averaged over the manifold $\mathcal{M}$, can also
be defined by the heat kernel,

$P(\tau)=\frac{1}{V}\int dx\sqrt{g}K(x,x;\tau),$ (9)

where $V=\int dx\sqrt{g}$ is the volume of $\mathcal{M}$. The return
probability can be expressed exactly for all $\tau$ in terms of the
eigenvalues $\lambda_{i}$ of the Laplacian $\Delta$,

$P(\tau)=\frac{1}{V}\sum_{i=0}e^{-\tau\lambda_{i}},$ (10)

which is identical to the trace of the heat kernel. The asymptotic expansion
of $P(\tau)$ for small $\tau$ is known [21],

$P(\tau)\sim\tau^{-\frac{n}{2}}\sum_{i=0}A_{i}\tau^{i},$ (11)

where $A_{i}$ are generally complicated integrated functions of the metric
$g$. In equation (10) we assume that the spectrum is discrete, which is valid
if $\mathcal{M}$ is a closed manifold. For an open manifold, the sum can be
replaced by an integration.

Analogous expressions to Eqs. (10) and (11) can be found for a generalised
heat equation based on the Laplace-Beltrami operator
$\Delta^{(k)}=d\delta+\delta d$, defined on $k$-forms. Here $d$ is the
exterior derivative and $\delta=(-1)^{n(k-1)+1}\star\\!d\star$ is the co-
differential. The $k$-form heat equation is,

$\frac{\partial}{\partial\tau}\omega(x,x_{0};\tau)=-\Delta^{(k)}\omega(x,x_{0};\tau),$
(12)

where $\omega(x,x_{0};\tau)$ is a $k$-form valued heat kernel. The analogue of
the average return probability $P^{(k)}(\tau)$ for $k$-forms, $0\leq k\leq n$,
is identical to the trace of the $k$-form heat kernel, as was the case for the
average return probability in equation (10). It can also be expressed in terms
of the eigenvalues $\lambda^{(k)}_{i}$ of $\Delta^{(k)}$,

$P^{(k)}(\tau)=\frac{1}{V}\sum_{i=0}e^{-\tau\lambda^{(k)}_{i}}.$ (13)

The asymptotic expansion of the function $P^{(k)}(\tau)$ has also essentially
the same form,

$P^{(k)}(\tau)\sim\tau^{-\frac{n}{2}}\sum_{i=0}B_{i}(\Delta^{(k)})\tau^{i},$
(14)

where again the $B_{i}(\Delta^{(k)})$ are in general complicated functions of
the metric $g$.

The dimension of the differentiable manifold $n$ can be extracted from the
asymptotic expansion,

$n=-2\lim_{\tau\rightarrow 0}\frac{d\log P^{(k)}_{\tau}}{d\log\tau}.$ (15)

The expression in equation (15) can be generalised by dropping the limit
$\tau\rightarrow 0$. The resulting object is the running spectral dimension
$D^{(k)}(\tau)$, which is a function of $\tau$,

$D^{(k)}(\tau)=-2\frac{d\log P^{(k)}_{\tau}}{d\log\tau}.$ (16)

For later use, we will comment here on the effect of a global rescaling of the
Laplacian $\Delta^{(k)}$ by a factor $b$, $\Delta^{(k)}\rightarrow
b\Delta^{(k)}$. The rescaling of $\Delta^{(k)}$ is equivalent to a rescaling
of the diffusion time $\tau$, which will leave the spectral dimension
invariant. This equivalence is apparent from the change of the return
probability under the rescaling,

$P(\tau)\rightarrow P^{\prime}(\tau)=\frac{1}{V}\sum_{i}e^{-\tau
b\lambda_{i}}=P(\tau^{\prime}),\quad\tau^{\prime}=b\tau.$ (17)

From Eqs. (13) and (16) we can see that the running spectral dimenision
$D^{(k)}(\tau)$ goes to zero for large $\tau$ if $\Delta^{(k)}$ has at least
one zero eigenvalue555The number of zero-valued eigenvalues of $\Delta^{(k)}$
is a topological property of $\mathcal{M}$ and is given by the Betti numbers
$b_{i},\ i\in\\{0,...,n\\}$, of $\mathcal{M}$. If $\mathcal{M}$ consists of a
single connected component, the Betti numbers $b_{0}$ and $b_{n}$ are equal to
$1$. The Betti numbers also adhere to the symmetry relation $b_{i}=b_{n-i}$..
At diffusion time $\tau$, the eigenvalues $\lambda_{i}\leq\tau^{-1}$ dominate
the behaviour of the return probability $P^{(k)}(\tau)$. The smallest non-zero
eigenvalue $\lambda_{1}$ is bounded from below, $\lambda_{1}>r^{-2}$, where
$r\sim V^{1/n}$ is the diameter of $\mathcal{M}$ [22]. We conclude that, for
timescales,

$\tau>V^{2/n},$ (18)

the zero mode of $\Delta^{(k)}$ becomes dominant and the diffusion process is
strongly affected by the finite size of $\mathcal{M}$.

The scalar case, corresponding to $k=0$, has been extensively
studied666Depending on the theory of quantum gravity, often the dual scalar
case is studied corresponding to $k=n$. in various approaches to quantum
gravity [2, 1, 3, 4] and is generally interpreted as a scale-dependent
effective dimension. The goal of this work is to generalise the studies of
scale-dependent effective dimensions in quantum gravity to general $k$-forms.
In the next section we will therefore discuss the discrete analogue of the
running spectral dimension $D^{(k)}(\tau)$ for $k$-forms.

###  3.2 Discrete tensor diffusion

The Laplace-Beltrami operator $\Delta^{(k)}$ can be generalised to an
equilateral simplicial complex $T$ [23]. To make a clear distinction we will
write $L^{(k)}$ for the discrete analogue of $\Delta^{(k)}$. The operator
$L^{(k)}$ is defined777See [24] for an introduction to the topic of discrete
differential forms. on the space of discrete $k$-form fields $\Omega^{k}(T)$,
which are associated to the $k$-simplices $\sigma^{k}$ that are contained in
$T$ and can be represented as a real symmetric matrix of rank $N_{k}$, the
number of $k$-simplices in $T$. The operator $L^{(k)}$ can be decomposed in
two terms, just as $\Delta^{(k)}$, and is given by,

$L^{(k)}=L^{(k)}_{+}+L^{(k)}_{-},\quad
L^{(k)}_{+}=(I^{k+1}_{k})^{T}I^{k+1}_{k},\quad
L^{(k)}_{-}=I^{k}_{k-1}(I^{k}_{k-1})^{T},$ (19)

where $I^{k+1}_{k}$ is the incidence matrix between the $k$-dimensional sub-
simplices $\sigma^{k}$ and $\sigma^{k+1}$ in a simplicial complex $T$,

$(I^{k+1}_{k})_{ij}=\begin{cases}\ \ 1&\textrm{if}\
\sigma^{k}_{i}\prec\sigma^{k+1}_{j}\quad\textrm{with same orientation},\\\
-1&\textrm{if}\ \sigma^{k}_{i}\prec\sigma^{k+1}_{j}\quad\textrm{with opposite
orientation},\\\ \ \ 0&\textrm{if}\ \sigma^{k}_{i}\ \not{\prec}\
\sigma^{k+1}_{j}.\end{cases}$ (20)

The resulting operator $L^{(k)}$ is real and semi-positive because of its
structure in terms of $I^{k+1}_{k}$ and its transpose. The eigenvectors
corresponding to non-zero eigenvalues of $L^{(k)}$ are the union of the
eigenvectors corresponding to non-zero eigenvalues of $L^{(k)}_{+}$ and
$L^{(k)}_{-}$, which are orthogonal. The results that are relevant in the
remainder of this text are invariant under changes of the orientation. For
definiteness, we will, however, give an explicit prescription for choosing an
orientation for all simplices in $T$. We label all vertices in $T$ with a
unique integer. All $k$-simplices $\sigma^{k}$ can then be represented by
their $k+1$ vertices $\sigma_{i}^{0}$, i.e.
$\sigma^{k}=[\sigma^{0}_{1},\ldots,\sigma^{0}_{k+1}]$. The orientation of
$\sigma^{k}$ is determined by the order of its vertices. We choose to order
them in increasing order of their labels, i.e.
$\sigma^{0}_{1}<\ldots<\sigma^{0}_{k+1}$. A $(k+1)$-simplex
$\sigma^{k+1}=[\sigma^{0}_{1},\ldots,\sigma^{0}_{k+2}]$ that contains the
$k$-simplex
$\sigma^{k}=[\sigma^{0}_{1},\ldots,\not{\sigma}^{0}_{i},\ldots,\sigma^{0}_{k+2}]$
are of the same orientation if $(-1)^{(i-1)}=1$ and of opposite orientation if
$(-1)^{(i-1)}=-1$. The vertex $\not{\sigma}^{0}_{i}$ denotes the vertex that
is part of $\sigma^{k+1}$, but not part of $\sigma^{k}$.

Besides making a choice for a discrete analogue of $\Delta^{(k)}$, we will
have to make a choice how to treat the diffusion time $\tau$ to define a
discrete analogue of the heat equation in Eq. (12). For numerical reasons we
will choose a discrete time variable $\tau$ and write $\tau=st$, where $s$ is
a positive real number and $t$ is an integer time. The discrete heat equation
will be defined on a $\tau$-dependent discrete $k$-form
$v(\tau)\in\Omega^{k}(T)$. The state $v(\tau)$ at time $\tau$ can be
represented by a vector of dimension $N_{k}$. Replacing the time derivative in
Eq. (12) by a finite difference equation in $t$ and replacing $\Delta^{(k)}$
by $L^{(k)}$, the discrete heat equation is given by a finite difference
equation in terms of the vector $v(\tau)$,

$v(\tau+s)=(\mathbb{I}-sL^{(k)})v(\tau),$ (21)

where $\mathbb{I}$ is the unit matrix of dimension $N_{k}$. The equation for
$v(\tau)$ can be solved iteratively. The iterations will be numerically stable
if $s^{-1}\leq 2\lambda_{max}$, where $\lambda_{max}$ is the largest
eigenvalue of $L^{(k)}$. Iteratively acting with $A\equiv\mathbb{I}-sL^{(k)}$,
the state $v(\tau)$ is given by,

$v(\tau)=A^{t}v(0),\quad\tau=st.$ (22)

The vector $v(0)$ at $\tau=0$ can be decomposed in the orthonormal eigenbasis
$v_{m}$ of $L^{(k)}$,

$v(0)=\sum_{m}c_{m}v_{m},\quad L^{(k)}v_{m}=\lambda_{m}v_{m}.$ (23)

We have chosen888In previous studies of spectral dimensions in CDT [16],
$s^{-1}$ was chosen to be equal to the number of neighbours of the simplices.
These studies where conducted for the case $k=n$, so the number of neighbours
was the same for every simplex. The discretisation artefact in [16] of the
running spectral dimensions for even and odd diffusion times for small $t$ is
no longer present with the choice $s^{-1}=\lambda_{max}$.
$s^{-1}=\lambda_{max}$, such that the eigenvalues of $A$ lie between $0$ and
$1$. The eigenvalues of $A$ equal to $1$ correspond to the zero modes of
$L^{(k)}$. The diffusion will therefore converge to the subspace of zero modes
of $L^{(k)}$ for large $\tau$,

$\lim_{\tau\to\infty}v(\tau)=\sum_{m=1}^{m_{0}}(v(0)\cdot v_{m})v_{m},$ (24)

where $m_{0}$ is the number of zero modes and $v_{m},\ m\in\\{1,...,m_{0}\\}$,
is the orthonormal set of eigenvectors that spans the subspace of zero modes
of $L^{(k)}$. The inner product $v(\tau)\cdot v_{m}$ is a conserved quantity
for $m\in\\{1,...,m_{0}\\}$.

We can now define the discrete return probability $P_{i}^{(k)}(\tau)$ for a
simplex $\sigma^{k}_{i}$. The discrete return probability is obtained by
taking the orthonormal position basis vectors $e_{i}$ as the initial state,
i.e. $v(0)=e_{i}$ and projecting $v(\tau)=A^{t}v(0)$ on the same initial state
$e_{i}$,

$P_{i}^{(k)}(\tau)=(A^{t}e_{i})\cdot e_{i}\,.$ (25)

The basis vectors $e_{i}$ are the analogue of the delta function distributions
of equation (7) and are associated with the $k$-simplex $\sigma^{k}_{i}$. The
average return probability $P^{(k)}$ is then given by

$P^{(k)}(\tau)=\frac{1}{N_{k}}\sum_{i}P_{i}^{(k)}(\tau)=\frac{1}{N_{k}}\mathrm{Tr}\
A^{t},\quad\tau=st.$ (26)

The trace of the operator $A^{t}$ can be expressed in terms of its
eigenvalues, so we conclude that,

$P^{(k)}(\tau)=\frac{1}{N_{k}}\sum_{m,n}(1-s\lambda^{m}_{n})^{\tau/s}.$ (27)

We now have an explicit $\tau$-dependent expression for the discrete spectral
dimension $D^{(k)}$ for $k$-forms,

$D^{(k)}(\tau)=-2\tau\frac{\Delta\log
P^{(k)}(\tau)}{\Delta\tau}=-2(\tau/s)\log(P^{(k)}(\tau+s)/P^{(k)}(\tau)),$
(28)

where $\Delta$, in this equation, is the finite difference operator for a
discrete diffusion time step $\tau$. If we want that the discrete diffusion
time step is approximately equal to a continuum diffusion time we require,

$|\sum_{i}(1-\lambda_{i})^{\tau}-\sum_{i}e^{-\tau\lambda_{i}}|\ll 1,\
\mathrm{for},\ \tau\gg\log N_{k}.$ (29)

We can therefore expect that the artefacts of discretising the diffusion time
are small for $\tau\gg\log N_{k}$ and we expect that we can measure
regularisation invariant properties of expectation values of the spectral
dimension $\langle D^{(k)}\rangle$ for these timescales. The largest
eigenvalues of $L^{(k)}$ are associated to length scales close to the lattice
spacing and are considered to be regularisation artefacts in the context of
the gravitational path-integral. For longer diffusion times $\tau$ the process
will explore a larger portion of the triangulation $T$. We will interpret the
spectral dimension $D^{(k)}$ at larger $\tau$ as the effective dimension at a
length scale $\ell_{\tau}$ associated to the timescale $\tau$. In the
continuum case, a normalisation of $\Delta^{(k)}$ corresponds to a rescaling
of the diffusion time $\tau$ with respect to the volume of the geometry $V$,
as was shown in equation (17). To compare the timescale of the diffusion on
different geometries of the same volume in the discrete case, we therefore
have to take the scale $s$ of the diffusion time $\tau=st$ into account. When
comparing the spectral dimensions $D^{(k)}(\tau)$ for different geometries of
the same volume, we will therefore normalise the diffusion time $\tau$ by the
factor $s$.

It is warranted to associate an effective dimension to timescale $\tau$ if the
average return probability $P^{(k)}(\tau)$ (approximately) scales with a
constant power. This will be the case if we have a timescale of the spectral
dimension $D^{k}(\tau)$ at which the derivative is (approximately) zero. We
will call such a regime of $D^{k}(\tau)$ a plateau. At such a timescale,
$D^{k}(\tau)$ is constant up to linear order in some small expansion parameter
$\epsilon$, so we can write,

$D^{k}(\tau)=c+\mathcal{O}(\epsilon).$ (30)

For a plateau around the value $c$, $P^{(k)}(\tau)$ scales like,

$P^{(k)}(\tau)\sim\tau^{-c/2}+\mathcal{O}(\epsilon).$ (31)

We therefore conclude that we can interpret the presence of a plateau in the
spectral dimension at a timescale $\tau$ as signalling an effective dimension
equal to $c$ in the triangulation $T$ as long as $D^{k}(\tau)$ changes slowly
in comparison to $c$.

We can now compute the discrete spectral dimension $D^{(k)}$ by computing the
averaged return probability with the eigenvalues of $L^{(k)}$, which is
equivalent to computing the trace of $A^{t}$. For numerical studies of large
geometries, it is necessary to approximate the averaged return probability
$P^{(k)}(\tau)$. This can be done by sampling the return probability
$P^{(k)}_{i}(\tau)$ given by equation (25) for a subset of initial
$k$-simplices $\sigma^{k}_{i}$. For the cases $k=0$ and $k=n$, the return
probability $P^{(k)}_{i}(\tau)$ can be interpreted in terms of the probability
that a random walk that starts at $\sigma^{k}_{i}$ returns to $\sigma^{k}_{i}$
after a time $\tau$. Equivalently, for $k=0$ and $k=n$, $P^{(k)}_{i}(\tau)$
can be interpreted in terms of the amplitude of a scalar field at
$\sigma^{k}_{i}$ that initially has a delta function distribution and is
subjected to a diffusion process. For other values of $k$, we will interpret
$P^{(k)}_{i}(\tau)$ in analogy to the scalar cases.

For a differentiable manifold, $k$-form fields and $k$-tensor fields are dual
to each other. As is often the case for generalisations of continuum objects
to discrete manifolds, there is no unique choice for the definition of
discrete $k$-form fields and discrete $k$-tensor fields. A specific definition
for both objects has been studied in detail in [25]. The $k$-form fields were
already discussed above and are associated to the $k$-simplices. The
$k$-tensor fields $X^{k}\in\mathfrak{X}^{k}(T)$ however, are elements of
$\otimes^{k}\mathbb{R}^{n}$. Noting that the tangent space is not well defined
on the boundary of a $n$-simplex, it is unnatural to define $X^{k}$ on the
$k$-simplex $\sigma^{k}$ for the cases $0<k<n$. Instead, it will be defined on
the interior of the $n$-simplices, where there exists a unique definition of a
tangent space. In [25] it was discussed how defining discrete vector fields on
the interior of simplices has certain advantageous properties. It is important
to note that discrete $k$-tensor fields defined in this fashion do not admit a
duality relation to the discrete $k$-form fields, because the number of
$k$-simplices for $0<k<n$ is different from the number of $n$-simplices for
simplicial manifolds $T$. However, there exists a surjective function999This
definition is similar to what is known as a Whitney form [26], [27], which has
been shown to have good convergence properties to continuum tensor fields.
$\sharp:\Omega^{k}(T)\rightarrow\mathfrak{X}^{k}(T)$,

$(\alpha^{k})^{\sharp}(\sigma^{n}_{i})=\sum_{\sigma^{k}_{j}\prec\sigma_{i}^{n}}\alpha^{k}(\sigma^{k}_{j})\vec{\sigma}_{j}^{k},$
(32)

where $(\alpha^{k})^{\sharp}(\sigma^{n}_{i})$ is the value (in
$\otimes^{k}\mathbb{R}^{n}$) of the discrete vector field
$X^{k}=(\alpha^{k})^{\sharp}$ in the simplex $\sigma^{n}_{i}$, the sum runs
over all simplices $\sigma^{k}_{j}$ contained in the simplex $\sigma_{i}^{n}$,
$\alpha^{k}(\sigma^{k}_{j})$ is the value (in $\mathbb{R}$) of the discrete
$k$-form evaluated on the simplex $\sigma^{k}_{j}$ and $\vec{\sigma}_{j}^{k}$
is the $k$-tensor associated to the simplex $\sigma^{k}_{j}$. We can also
define the injective function
$\flat:\mathfrak{X}^{k}(T)\rightarrow\Omega^{k}(T)$,

$(X^{k})^{\flat}(\sigma^{k}_{j})=\sum_{\sigma^{n}_{i}\succ\sigma_{j}^{k}}X^{k}(\sigma_{i}^{n})\cdot\vec{\sigma}^{k}_{j},$
(33)

where $(X^{k})^{\flat}(\sigma^{k}_{j})$ is the value (in $\mathbb{R}$) of the
$k$-form $\alpha^{k}=(X^{k})^{\flat}$ evaluated on $\sigma^{k}_{j}$, the sum
runs over all simplices $\sigma^{n}_{i}$ which share the simplex
$\sigma_{j}^{k}$, $X^{k}(\sigma^{n}_{i})$ is the value (in
$\otimes^{k}\mathbb{R}^{n}$) of the discrete vector field $X^{k}$ in the
simplex $\sigma^{n}_{i}$ and $\cdot$ denotes the standard inner product
defined on the tangent space of the flat interior of $\sigma_{i}^{n}$. The two
functions $\sharp$ and $\flat$ connect the two spaces $\Omega^{k}(T)$ and
$\mathfrak{X}^{k}(T)$. The sums in these equations run over the $k$-simplices
$\sigma^{k}_{j}$ contained in the $n$-simplices $\sigma^{n}_{i}$ and the
$n$-simplices $\sigma^{n}_{i}$ that contain the $k$-simplices
$\sigma^{k}_{j}$, respectively. The vector $\vec{\sigma}^{k}_{j}$ and the dot
product are defined with respect to the tangent space of the flat interior of
$\sigma^{n}_{i}$. For the case $k=1$, $\mathfrak{X}^{k}(T)$ defines what is
known as discrete vector fields, for which an example is given in figure 3.
Various studies show that discrete vector fields possess more favourable
properties than discrete one-form fields when the goal is to approximate
continuum vector fields or one-form fields [24, 28, 25]. In view of these
deliberations, the question arises if a discrete notion of diffusion can be
defined both in terms of discrete $k$-form fields and discrete $k$-tensor
fields. The case of discrete $k$-form fields is given by equation (25). For
diffusion of discrete $k$-tensor fields, we propose the following definition
of vector field return probability $\vec{P}_{i}^{(k)}$. We consider a discrete
unit-length vector field $X_{i}(0)$ which is zero everywhere except in one
$n$-simplex $\sigma^{n}_{i}$. The vector field is then mapped to a discrete
$k$-form field $f_{i}=(X_{i})^{\flat}$ with the flat operator $\flat$. The
resulting $k$-form field $f_{i}$ is multiplied $\tau$ times by the generator
of diffusion $A$ and mapped back to the space of discrete vector fields with
the sharp operator $\sharp$. The resulting discrete vector field
$(A^{t}f_{i})^{\sharp}$ is then projected on the initial $n$-simplex
$\sigma^{n}_{i}$, which gives a single vector $\vec{\xi}(\tau)$ for which we
compute the norm $||\vec{\xi}(\tau)||$. This defines the $k$-tensor return
probability $\vec{P}_{i}^{(k)}$,

$\vec{P}_{i}^{(k)}(\tau)=||\vec{\xi}(\tau)||,\quad\vec{\xi}(\tau)=(A^{t}f_{i})^{\sharp}(\sigma^{n}_{i}),\quad
f_{i}=(X_{i})^{\flat},\quad\tau=st.$ (34)

where the index $i$ refers to the $n$-dimensional simplex $\sigma^{n}_{i}$.
With the $k$-tensor return probability $\vec{P}_{i}^{(k)}(\tau)$ we define the
$k$-tensor spectral dimension $\vec{D}^{(k)}$,

$\vec{D}^{(k)}(\tau)=-2\tau\log(\vec{P}^{(k)}(\tau+s)/\vec{P}^{(k)}(\tau)),\quad\vec{P}^{(k)}(\tau)=\frac{1}{N_{n}}\sum^{N_{n}}_{i}\vec{P_{i}}^{(k)}(\tau).$
(35)

In Fig. 4 we illustrate both the one-form diffusion $P^{(1)}(\tau)$ and vector
diffusion $\vec{P}^{(1)}(\tau)$ for the relatively simple case of the flat
two-torus.

Figure 3: The figure shows the four vertices $\sigma_{i}^{0}$ of a tetrahedron
$\sigma^{3}$ with its link vectors $\vec{\sigma}_{i}^{1}$. Given a one-form
field $\alpha^{1}$, the corresponding vector field evaluated on $\sigma^{3}$
is given by
$\vec{\xi}=\alpha^{\sharp}(\sigma^{3})=\sum^{6}_{i=1}\alpha(\sigma^{1}_{i})\vec{\sigma}^{1}_{i}$.

Figure 4: The figure shows the diffusion process of a discrete one-form field
$\alpha^{1}$ and a discrete vector field $X^{1}$, both governed by the one-
form heat equation on a flat two-torus consisting of $30\times 30$ vertices.
The links are coloured according to $-1$ times the log of norm of the one-form
field $\alpha^{1}(\sigma^{1}_{i})$ evaluated on the link $\sigma^{1}_{i}$ and
the triangles are coloured according to $-1$ times the log of the norm of the
vector field $||\xi(\sigma^{2}_{i})||$ evaluated on the triangle
$\sigma^{2}_{i}$, both as a function of $\tau$. The four images refer to the
diffusion steps $t=0$ (top-left), $t=3$ (top-right), $t=5$ (bottom-left),
$t=9$ (bottom-right), $\tau=st$. The initial state of $X^{1}$ is concentrated
on the triangle indicated by the three orange dots. The initial state of
$\alpha^{1}$ is concentrated on one of the links of the triangle indicated by
the three orange dots. At the locus of their respective initial states, the
norm of $\alpha^{1}$ is equal to $P^{1}(\tau)$ and the norm of $X^{1}$ is
equal to $\vec{P}(\tau)$.

###  3.3 Two- and three-dimensional regular geometries

To interpret our results on generalised spectral dimensions in spatial slices
of four-dimensional CDT, it is worthwhile to first discuss generalised
spectral dimensions in a simplified setting. It is a difficult task to
disentangle physically relevant information from discretisation effects
appearing at small diffusion times and finite-size effects, which appear at
large diffusion times. Both discretisation effects and finite size effects are
considered regularisation artefacts. As an aid to interpret the measurements,
it is useful to first investigate these effects on regular discrete
geometries.

We will consider a two-dimensional triangulation of the flat torus, which can
be triangulated exactly. We will use Eqs. (28) and (35) to calculate the
spectral dimensions for $k\in\\{0,1,2\\}$. Fig. 5 shows the four different
spectral dimensions, the scalar spectral dimension $D^{(0)}(\tau)$, the one-
form spectral dimension $D^{(1)}(\tau)$, the two-form spectral dimension
$D^{(2)}(\tau)$ and the vector spectral dimension $\vec{D}^{(1)}(\tau)$, that
appear for the two-dimensional flat torus. The figure to the left shows the
spectral dimension curves obtained with numerical methods. The calculation For
$D^{(0)}(\tau)$, $D^{(1)}(\tau)$ and $D^{(2)}(\tau)$, we see significant
deviations from the expected value of $2$ for small values of $\tau$. The
deviation from $2$ is smallest for $\vec{D}^{(1)}(\tau)$. These results
suggests that $\vec{D}^{(1)}(\tau)$ is less sensitive to discretisation
effects.

An extended plateau at a value $2\pm 0.002$ is present for each of the
spectral dimensions. We therefore see that plateaus in the spectral dimension
indeed indicate a range of time scales at which the effective dimension can be
separated from discretisation artefacts and finite-size effects. The timescale
at which the spectral dimension goes to zero varies for the different values
of $k$. For different values of $k$, a discrete time step corresponds to a
varying fraction of the total volume of the triangulation. It is therefore to
be expected that the time scale at which finite-size effects become apparent
depends on $k$. The curves in the figure to the right serve as a point of
reference for the numerical results. These curves were obtained for
$D^{(1)}(\tau)$ from an analytical calculation which is presented in Appendix
A. The continuum calculation of the one-form spectral dimension was obtained
from the known spectrum of the continuum one-form Laplacian of the flat two-
dimensional torus, which can be found in [21]. From the analytical
calculations we can see that the peak for small diffusion time is indeed a
discretisation artefact, because it is absent for the continuum case and that
the plateau is the feature of the curve that is associated to the dimension of
the geometry.

Figure 5: The figure to the left shows the generalised spectral dimensions for
a triangulation of the flat two-dimensional torus consisting of $30\times 30$
vertices. The spectral dimensions $D^{(0)}$, $D^{(1)}$ and $D^{(2)}$ differ
significantly from 2 for small diffusion time $\tau$, due to discretisation
effects. These effects are less pronounced for the vector spectral dimension
$\vec{D}^{(1)}$, which has a value closer to $2$ for small $\tau$. An extended
plateau appears for each of the spectral dimensions at a value $2\pm 0.002$.
The figure to the right shows $D^{(1)}$ from the analytical calculation for
the flat torus presented in Appendix A and the continuum one-form spectral
dimension as a point of reference. It can be seen that the initial peak for
small diffusion times is an artefact of the discretisation and the plateau is
the feature that shows the dimension of the geometry.

The CDT geometries we have investigated are equilateral, three-dimensional
simplicial manifolds of toroidal topology. We will therefore discuss another
regular geometry as a point of reference. An approximately flat three-
dimensional torus can be constructed by gluing together square blocks of five
tetrahedra such as the one illustrated in Fig. 6 and identifying opposite
sides. The resulting geometry is only approximately flat, because flat three-
dimensional space cannot be exactly triangulated by equilateral tetrahedra.
The degrees of the vertices of the torus constructed by such square blocks are
not homogeneous, as was the case for the flat two-dimensional torus. With this
approximately flat torus we can investigate how such inhomogeneities impact
the diffusion processes we are interested in.

Figure 6: An approximately flat equilateral three-dimensional torus can be
constructed from repeatedly gluing copies (along the dashed lines) of the
square block in this figure and identifying opposite sides. Every block
consists of five equilateral tetrahedra. The blocks are glued in a way that
links of different blocks coincide. All the links shown in the figure
represent links of equal length. Such a block is not embeddable in a flat
three-dimensional space.

On this approximately flat three-dimensional simplicial manifold we can
calculate the k-form spectral dimensions $D^{(k)}$ with $k\in\\{0,...,3\\}$
and the tensor spectral dimensions $\vec{D}^{(k)}$ for $k\in\\{1,2\\}$. Fig. 7
shows all the running spectral dimensions on the approximately flat torus for
diffusion timescales before the zero modes become dominant (except for the
scalar diffusion $D^{(0)}$). We note that all running spectral dimensions
show, besides short time-scale effects, a plateau approximately equal to three
as expected. Comparing the scalar spectral dimension $D^{(0)}$ to the dual
scalar spectral dimension $D^{(3)}$, we see that there is a strong effect from
the vertices of high degree, which cause the peak for small diffusion time.
For larger diffusion times there is however an overlap between $D^{(0)}$ and
$D^{(3)}$, showing that the inhomogeneities are washed out for larger
diffusion times. There are significantly fewer vertices than tetrahedra (5
times fewer for the geometry of Fig. 7), so the finite-size effects are
dominant much earlier for $D^{(0)}$, which diminishes the presence of the
plateau.

Inspecting the curves for $D^{(1)},D^{(2)},\vec{D}^{(1)}$ and $\vec{D}^{(2)}$
we note that all curves show a plateau close to a value of three, but that the
tensor spectral dimensions are less sensitive to small scale artefacts. The
main conclusion we have drawn from investigating two and three-dimensional
regular geometries, is that all spectral dimensions show a plateau at the
expected value equal to the topological dimension for larger diffusion times.
We have also shown that the various spectral dimensions are sensitive to
inhomogeneities and lattice artefacts to a varying degree. The dual scalar
spectral dimension $D^{(3)}$ and the $k$-tensor spectral dimensions
$\vec{D}^{(k)}$ are particularly insensitive to lattice artefacts. Section 4
contains the measurements of the various spectral dimensions of spatial slices
in the semi-classical phase $C$ of four-dimensional CDT.

Figure 7: The figures show the generalised spectral dimensions of an
approximately flat three-dimensional torus. The approximately flat three-
dimensional torus was constructed by gluing together blocks such as the one
illustrated in Fig. 6 and identifying opposite sides. The torus consists of
$4\cdot 10^{4}$ tetrahedra.

##  4 Generalised spectral dimensions in non-perturbative quantum gravity

To investigate the various spectral dimensions in the context of non-
perturbative quantum gravity we have generated CDT geometries with a Monte-
Carlo algorithm based on the action given in equation (4). We have chosen to
study CDT geometries of toroidal topology, because in this case all Betti
numbers are non-zero and all Laplace-Beltrami operators therefore have zero
modes. The parameters we have chosen are $\kappa_{0}=4.0,\Delta=0.2$ and the
number of time slices is fixed to four. These parameters lie within the semi-
classical phase $C$ of the CDT phase diagram (see Fig. 2) and have recently
been studied in detail for toroidal topology [29]. We have also chosen this
specific location in the CDT parameter space, because it is close to two
triple points where various phase transition lines meet. As evidenced by the
renormalization group studies [30, 31], the triple points are potential
candidates for a UV fixed point of CDT, however the existence of such a UV
fixed point has not yet been definitively proven. It was shown that the
effective lattice spacing decreases towards large $\kappa_{0}$ [32] (but
depends slightly on $\Delta$). Low $\kappa_{0}$ seems to be the region of the
parameter space related to the IR limit and high $\kappa_{0}$ (close to the
triple points) corresponds to the UV limit of the theory. Our choice enables
us therefore to investigate much finer effective lattice spacing and
consequently quantum gravity effects related to much shorter physical
distances than the _generic_ point $(\kappa_{0}=2.2,\Delta=0.6)$, where most
of CDT results in phase $C$ were so far obtained101010There exists no explicit
estimate of the effective lattice spacing for the case of CDT with toroidal
spatial topology discussed here, but in the case of spherical topology the
lattice spacing was estimated to change from around $2\ \ell_{pl}$ (Planck
lengths) for $\kappa_{0}=2.2$ to around $1\ \ell_{pl}$ for $\kappa_{0}=4.4$
[32].. At the same time we didn’t want to get too close to the phase
transition lines in order to be able to measure the IR properties of the
quantum geometries.

In order to extrapolate our results to the infinite volume limit, we have
generated ensembles of geometries for the following target volumes:
$\bar{N}_{41}=\\{10\mathrm{k},20\mathrm{k},40\mathrm{k},80\mathrm{k},160\mathrm{k},240\mathrm{k}\\}$.
With these ensembles, we estimate the expectation value of the average return
probabilities $\langle{P}_{i}^{(k)}(\tau)\rangle$ and
$\langle\vec{P}_{i}^{(k)}(\tau)\rangle$, as shown by equation (5), where we
take ${P}_{i}^{(k)}(\tau)$ and $\vec{P}_{i}^{(k)}(\tau)$ as the observable
$\mathcal{O}(\mathcal{T})$.

For all generated four-dimensional geometries, we consider the three-
dimensional spatial slices separately and calculate the average return
probabilities ${P}_{i}^{(k)}(\tau)$ and $\vec{P}_{i}^{(k)}(\tau)$. Due to the
finite computation time, the average return probability is calculated from a
subset of the $k$-simplices in the three-dimensional slices. The average
return probabilities are sampled by taking $2\times 10^{3}$ randomly chosen
$k$-simplices as the initial location of a diffusion process for each
geometry. This number of initial locations gives a good approximation of the
average return probability. Additional starting points give a contribution
which is smaller than the quantum fluctuations present in the ensembles of
geometries.

For the present case of CDT with toroidal topology, distinguishing between the
spatial slices of the geometries has no physical significance. We have,
therefore, estimated the expectation value of the average return probabilities
$\langle P_{i}^{(k)}(\tau)\rangle$ and $\langle\vec{P}_{i}^{(k)}(\tau)\rangle$
for target volume $\bar{N}_{41}$ by averaging over starting points and
configurations and also over slices. From $\langle P_{i}^{(k)}(\tau)\rangle$
and $\langle\vec{P}_{i}^{(k)}(\tau)\rangle$ we calculate the running spectral
dimensions $\langle D^{(k)}(\tau)\rangle$ and
$\langle\vec{D}^{(k)}(\tau)\rangle$ with,

$\langle D^{(k)}(\tau)\rangle=-2\tau\log(\langle
P^{(k)}(\tau+s)\rangle/\langle P^{(k)}(\tau)\rangle),$ (36)

and

$\langle\vec{D}^{(k)}(\tau)\rangle=-2\tau\log(\langle\vec{P}^{(k)}(\tau+s)\rangle/\langle\vec{P}^{(k)}(\tau)\rangle).$
(37)

We also keep track of the standard deviation of the spectral dimensions, which
include both statistical (from the average over starting points) as well as
quantum fluctuations (from the average over geometries). We show the spectral
dimensions for the scalar (Fig. 8), one-form (Fig. 10), vector (Fig. 10), two-
form (Fig. 11), tensor (Fig. 13) and dual scalar diffusion (Fig. 15) for
different target volumes $\bar{N}_{41}$.

To be able to compare the spectral dimension curves for different target
volumes, the diffusion time of the curves is normalised by
$\bar{N}_{41}^{\alpha}$, where $\bar{N}_{41}$ is the target volume discussed
in Sec. 2. The number of four-simplices in a CDT geometry is approximately
equal to $\bar{N}_{41}$ and the average number of tetrahedra in a spatial
slice is proportional to $\bar{N}_{41}$.

The exponent $\alpha$ is chosen such that the maximum of the curves for large
diffusion time are aligned for the same volume-normalised diffusion time
$\tau/\bar{N}_{41}^{\alpha}$. As was shown in equation (18), the exponent
$\alpha$ for a differentiable manifold $\mathcal{M}$ is related to the
dimension $n$ of $\mathcal{M}$. Although we could therefore potentially define
an alternative definition of an effective dimension through $\alpha$, we will
leave a detailed discussion of $\alpha$ for a future investigation. Each of
the spectral dimension curves are shown up to a time-scale at which the zero
modes become dominant and the curves monotonically go to zero. We find that
all spectral dimensions except $D^{(0)}$ exhibit a clear finite-volume
scaling, which signals the existence of an infinite volume limit for the
spectral dimension curves. Plateaus in the infinite volume curve signal the
appearance of an effective dimension for some range of scales, as was
illustrated with equation (31) and the examples of regular geometries in Figs.
5 and 7. The examples of regular geometries showed that plateaus take the form
of an extremum or inflection point of the spectral dimension curve surrounded
by a (small) range of diffusion time where the spectral dimension changes
slowly. From the finite volume scaling of the maximum of the plateaus, we can
extrapolate an effective dimension. The extrapolated values of the effective
dimensions derived from $\langle D^{(k)}\rangle$ and
$\langle\vec{D}^{(k)}\rangle$ are summarised in Tables 1 and 2, which can be
found at the end of this section.

For the scalar spectral dimension $D^{(0)}$ (Fig. 8), we see that a plateau
appears for larger volumes similar to those of the regular geometries
discussed in Sec. 3.3. The initial peak of the curves shrinks for larger
volumes and was also present for the regular geometries as a spurious effect.
We therefore conclude that the initial peak most likely is a lattice artefact.
However, for the target volumes $\bar{N}_{41}$ we have considered, the initial
peak and the plateau cannot be disentangled and we cannot extrapolate an
infinite-volume value for the plateau. From the maximum of the plateau of
$\langle D^{(0)}\rangle$ for the largest volume we could expect that an
infinite volume limit will be close to four. However, it remains to be seen if
including geometries of larger volume indeed makes it possible to extrapolate
an infinite volume limit more accurately. We also note that there is no other
timescale at which $\langle D^{(0)}\rangle$ exhibits a flattening of the
curve. The curves of $\langle D^{(0)}\rangle$ start at zero and, for larger
volumes, are increasingly steeper for small diffusion times.

Both the expectation value of the one-form spectral dimension $\langle
D^{(1)}\rangle$ (Fig. 10), and of the vector spectral dimension
$\langle\vec{D}^{(1)}\rangle$ (Fig. 10), show an increasingly steep initial
increase from zero to a plateau, implying that there exists no small-scale
spectral dimension. The plateau is more elongated for the vector spectral
dimension, indicating that finite-size effects are less dominant than for the
case of the one-form spectral dimension. We can extrapolate an effective
dimension for both $\langle D^{(1)}\rangle$ and $\langle\vec{D}^{(1)}\rangle$
from fitting the finite volume scaling of the maximum of the emerging plateaus
to a power law of the form $a+b*\bar{N}_{41}^{c}$.

The two-form spectral dimension $\langle D^{(2)}\rangle$ (Fig. 11) and the
tensor spectral dimension $\langle\vec{D}^{(2)}\rangle$ (Fig. 13) have more
structure than the scalar, one-form and vector spectral dimensions. For small
diffusion times $\langle D^{(2)}\rangle$ and $\langle\vec{D}^{(2)}\rangle$
exhibit a sharp peak that moves to $\tau/N^{\alpha}=0$ for larger volumes. The
volume scaling behaviour of the peak at small diffusion times indicates that
these peaks are lattice artefacts. Immediately after the peak the curves of
$\langle D^{(2)}\rangle$ and $\langle\vec{D}^{(2)}\rangle$ show a plateau at
small diffusion times with a clear finite volume scaling. We therefore
conclude that the infinite volume limit exhibits an effective dimension in the
UV for both the two-form and the tensor case. The values of the extrapolated
effective dimension for $\langle D^{(2)}\rangle$ and
$\langle\vec{D}^{(2)}\rangle$ agree within one standard deviation. For
significantly larger diffusion times, a second plateau appears for both
$\langle D^{(2)}\rangle$ and $\langle\vec{D}^{(2)}\rangle$ for which the
extrapolated effective dimensions also agree within one standard deviation.
The finite volume scaling and extrapolated effective dimension of the UV and
IR plateaus are given in Fig. 12 for $\langle D^{(2)}\rangle$ and in Fig. 14
for $\langle\vec{D}^{(2)}\rangle$.

The dual scalar spectral dimensions $\langle D^{(3)}\rangle$ (Fig. 15), like
the two-form and tensor spectral dimension, exhibits a sharp initial peak that
moves to $\tau/N^{\alpha}=0$ for larger volumes. Therefore, this initial peak
is interpreted as a lattice artefact as well. The subsequent plateau at small
diffusion times persists for large volumes and shows a clear finite volume
scaling so signals the presence of a UV effective dimension. For significantly
larger diffusion times a second plateau emerges which exhibits a finite volume
scaling as well. We can therefore also extrapolate an IR effective dimension
for $\langle D^{(3)}\rangle$ (Fig. 16). A summary of the results for the UV
and IR spectral dimensions we found for all $\langle D^{(k)}\rangle$ and
$\langle\vec{D}^{(k)}\rangle$ are given in Tables 1 and 2.

Type | Effective Spectral Dimension
---|---
Scalar $\langle D^{(0)}\rangle$ | $\sim 4$
One-form $\langle D^{(1)}\rangle$ | $4.98\pm 0.36$
Vector $\langle\vec{D}^{(1)}\rangle$ | $4.03\pm 0.45$
Table 1: The table contains the extrapolated infinite volume values of the effective spectral dimension related to the scalar, one-form, and vector diffusion. Type | UV | IR
---|---|---
Two-form $\langle D^{(2)}\rangle$ | $1.44\pm 0.23$ | $2.30\pm 0.64$
Tensor $\langle\vec{D}^{(2)}\rangle$ | $1.27\pm 0.26$ | $2.03\pm 0.11$
Dual scalar $\langle D^{(3)}\rangle$ | $1.47\pm 0.01$ | $2.66\pm 0.21$
Table 2: The table contains the extrapolated close range (UV) and long range
(IR) values of the effective spectral dimensions related to the two-form,
tensor and dual scalar diffusion. Figure 8: The upper figure shows a sequence
of curves with volume-normalised diffusion time $\tau/N^{\alpha}$ of the
expectation value of the scalar spectral dimension $\langle D^{(0)}\rangle$
for geometries of different volumes. The volume of a CDT geometry is
determined by the number of four-simplices, which is approximately equal to
$\bar{N}_{41}$. The two curves for $\bar{N}_{41}=160\mathrm{k}$ and
$240\mathrm{k}$ are shown with a bold line. The parameter $\alpha=1.90$ was
determined solely from the curves for which the plateaus are apparent. The
lower figure shows the standard deviation $SD$ of the curves in the upper
plot.

Figure 9: The upper left figure shows a sequence of curves with volume-
normalised diffusion time $\tau/N^{\alpha}$ of the expectation value of the
one-form spectral dimension $\langle D^{(1)}\rangle$ for geometries of
different volumes. The volume of a CDT geometry is determined by the number of
four-simplices, which is approximately equal to $\bar{N}_{41}$. The two curves
for $\bar{N}_{41}=160\mathrm{k},240\mathrm{k}$ are shown with a bold line. The
lower left figure shows the standard deviation $SD$ of the curves in the upper
left plot. The right plot shows the finite-volume scaling of the maxima of the
emerging plateaus of $\langle D^{(1)}\rangle$ as a function of $\bar{N}_{41}$.
The same plot shows a power law fit of the form $a+b*\bar{N}_{41}^{c}$, from
which we extrapolate an effective dimension
$\langle{D_{eff}^{(1)}}\rangle=4.98\pm 0.36$ in the infinite volume limit. The
curves are shown for $\alpha=1.36$.

Figure 10: The upper left figure shows a sequence of curves with volume-
normalised diffusion time $\tau/N^{\alpha}$ of the expectation value of the
vector spectral dimension $\langle\vec{D}^{(1)}\rangle$ for geometries of
different volumes. The volume of a CDT geometry is determined by the number of
four-simplices, which is approximately equal to $\bar{N}_{41}$. The two curves
for $\bar{N}_{41}=160\mathrm{k},240\mathrm{k}$ are shown with a bold line. The
lower left figure shows the standard deviation $SD$ of the curves in the upper
left plot. The right plot shows the finite volume scaling of the maxima of the
emerging plateaus of $\langle\vec{D}^{(1)}\rangle$ as a function of
$\bar{N}_{41}$. The same plot shows a power law fit of the form
$a+b*\bar{N}_{41}^{c}$, from which we extrapolate an effective dimension
$\langle{\vec{D}_{eff}^{(1)}}\rangle=4.03\pm 0.45$ in the infinite volume
limit. The curves are shown for $\alpha=1.65$.

Figure 11: The upper left figure shows a sequence of curves with volume-
normalised diffusion time $\tau/N^{\alpha}$ of the expectation value of the
two-form spectral dimension $\langle D^{(2)}\rangle$ for geometries of
different volumes. The volume of a CDT geometry is determined by the number of
four-simplices, which is approximately equal to $\bar{N}_{41}$. The two curves
for $\bar{N}_{41}=160\mathrm{k},240\mathrm{k}$ are shown with a bold line. The
lower-left figure shows the standard deviation $SD$ of the curves in the
upper-left plot. The upper right figure shows $\langle D^{(2)}\rangle$ for
small diffusion times, and the lower right figure shows the corresponding
standard deviation. The figures to the right show in detail how the lattice
artefact peaks move to the left for increasing $\bar{N}_{41}$ and how the
small diffusion time plateau emerges. The curves are shown for $\alpha=1.31$.

Figure 12: The left figure shows the finite volume scaling of the maxima of
the emerging plateaus of $\langle D^{(2)}\rangle$ at large diffusion time as a
function of $\bar{N}_{41}$. The same plot shows a power law fit of the form
$a+b*\bar{N}_{41}^{c}$, from which we extrapolate an IR effective dimension
$\langle D_{eff}^{(2)}\rangle=2.30\pm 0.64$ in the infinite volume limit. The
right figure shows the finite volume scaling of the minima of the emerging
plateaus of $\langle D^{(2)}\rangle$ at small diffusion time. From the
corresponding power law fit we extrapolate a UV effective dimension $\langle
D_{eff}^{(2)}\rangle=1.44\pm 0.23$.

Figure 13: The upper left figure shows a sequence of curves with volume-
normalised diffusion time $\tau/N^{\alpha}$ of the expectation value of the
tensor spectral dimension $\langle\vec{D}^{(2)}\rangle$ for geometries of
different volumes. The volume of a CDT geometry is determined by the number of
four-simplices, which is approximately equal to $\bar{N}_{41}$. The two curves
for $\bar{N}_{41}=160\mathrm{k},240\mathrm{k}$ are shown with a bold line. The
lower left figure shows the standard deviation $SD$ of the curves in the upper
left plot. The upper right figure shows $\langle\vec{D}^{(2)}\rangle$ for
small diffusion times, and the lower right figure shows the corresponding
standard deviation. The figures to the right show in detail how the lattice
artefact peaks move to the left for increasing $\bar{N}_{41}$ and how the
small diffusion time plateau emerges. The curves are shown for $\alpha=1.25$.

Figure 14: The left figure shows the finite volume scaling of the maxima of
the emerging plateaus of $\langle\vec{D}^{(2)}\rangle$ at large diffusion time
as a function of $\bar{N}_{41}$. The same plot shows a power law fit of the
form $a+b*\bar{N}_{41}^{c}$, from which we extrapolate an IR effective
dimension $\langle\vec{D}_{eff}^{(2)}\rangle=2.03\pm 0.11$ in the infinite
volume limit. The right figure shows the finite volume scaling of the minima
of the emerging plateaus of $\langle\vec{D}^{(2)}\rangle$ at small diffusion
time. From the corresponding power-law fit we extrapolate a UV effective
dimension $\langle\vec{D}_{eff}^{(2)}\rangle=1.27\pm 0.26$.

Figure 15: The upper left figure shows a sequence of curves with volume-
normalised diffusion time $\tau/N^{\alpha}$ of the expectation value of the
dual scalar spectral dimension $\langle D^{(3)}\rangle$ for geometries of
different volumes. The volume of a CDT geometry is determined by the number of
four-simplices, which is approximately equal to $\bar{N}_{41}$. The two curves
for $\bar{N}_{41}=160\mathrm{k},240\mathrm{k}$ are shown with a bold line. The
lower left figure shows the standard deviation $SD$ of the curves in the upper
left plot. The upper right figure shows $\langle D^{(3)}\rangle$ for small
diffusion times and the lower right figure shows the corresponding standard
deviation. The figures to the right show in detail how the lattice artefact
peaks move to the left for increasing $\bar{N}_{41}$ and how the small
diffusion time plateau emerges. The curves are shown for $\alpha=1.06$.

Figure 16: The left figure shows the finite volume scaling of the maxima of
the emerging plateaus of $\langle D^{(3)}\rangle$ at large diffusion time as a
function of $\bar{N}_{41}$. The same plot shows a power law fit of the form
$a+b*\bar{N}_{41}^{c}$, from which we extrapolate an IR effective dimension
$\langle D_{eff}^{(3)}\rangle=2.66\pm 0.21$ in the infinite volume limit. The
right figure shows the finite volume scaling of the minima of the emerging
plateaus of $\langle D^{(3)}\rangle$ at small diffusion time. From the
corresponding power law fit we extrapolate a UV effective dimension $\langle
D_{eff}^{(3)}\rangle=1.47\pm 0.01$.

###  4.1 Spatial Hausdorff dimension

As a point of comparison for the results on the various spectral dimensions,
we have also considered another observable related to the effective dimension
of spacetime, the Hausdorff dimension, which has been studied extensively in
the context of non-perturbative quantum gravity [33]. The Hausdorff dimension
for a geometry $T$ is defined through the scaling of the volume $V(r)$ of
balls with the radius $r$. We use the tetrahedron adjacency matrix of each
spatial slice of $T$ to construct balls consisting of all tetrahedra within
the lattice distance $r$ from a central tetrahedron. Tetrahedra at a geodesic
distance $r$ from a central tetrahedron form discrete shells of these balls. A
shell volume $n(r)$ denotes the number of tetrahedra in a shell of radius $r$,
the total volume of all shells constitutes the number of tetrahedra in a slice
$N_{3}=\sum_{r}n(r)$. The average of $n(r)$ over centre points, slices and
geometries $T$ gives the expectation value $\langle n(r)\rangle$ of shell
volumes. The Hausdorff dimension can be obtained from the finite volume
scaling of $n(r)$. In order to compare shell volume profiles $n(r)$ for
various slice volumes $N_{3}$, we rescale the radius $r$ to
$N_{3}^{-\frac{1}{d}}r$ with respect to the Hausdorff dimension $d$ and
normalise the shell volumes $n(r)$ to $N_{3}^{\frac{1-d}{d}}n(r)$, so that an
integral over $r$ of rescaled shell volumes is equal to $1$. If there is a
common Hausdorff dimension $d=d_{H}$ for geometries of different sizes, the
rescaled curves will overlap. Fig. 17 shows the rescaled average curves
calculated for the for the same CDT geometries we have studied for the
spectral dimensions $\langle D^{(k)}\rangle$ and
$\langle\vec{D}^{(k)}\rangle$. The best fit for $d$ results in a Hausdorff
dimension $d_{H}=3.95\pm 0.18$, which is different than $3$, the dimension of
the tetrahedra that make up the slices.

Figure 17:  The figure shows the expectation values of the shell volumes
$N_{3}^{\frac{1-d}{d}}\langle n(r)\rangle$ as a function of radius
$N_{3}^{-\frac{1}{d}}r$ rescaled with respect to the Hausdorff dimension $d$.
The plots include error bars and different colours correspond to different
target volumes $\bar{N}_{41}$. The best overlap of rescaled shell volumes for
different target volumes is obtained for $d_{H}=3.95\pm 0.18$.

##  5 Discussion and outlook

The aim of this work was to investigate whether the universal phenomenon of
scale-dependent effective dimensions in non-perturbative theories of quantum
gravity depends on the type of field that probes the effective spacetime. To
investigate this question, we have studied the scaling of the trace of the
heat kernel of the Laplace-Beltrami operator defined on $k$-form fields on
spatial slices of the non-perturbative model of quantum gravity known as
Causal Dynamical Triangulations. From the scaling of the trace of the heat
kernel, we can derive the scale-dependent spectral dimension, which was
extensively studied in various theories of non-perturbative quantum gravity
for the cases $k=0$ and $k=n$. To our knowledge, this work is the first time
that cases $k\in\\{1,...,n-1\\}$ have been studied. As mentioned in the
introduction, one of the important tasks for quantum gravity research is to
find examples of quantum gravity phenomenology, especially in view of the
recent technological advances in gravitational wave observations and direct
imaging of black hole shadows. It has been argued, using a model-independent
framework, that the seemingly universal scale-dependent effective dimension
can have an effect on the relative propagation of a gravitational wave signal
and the corresponding electromagnetic signal [5, 6, 7]. However, it was
assumed that only the scalar mode of the gravitational waves were affected and
that the propagation of the electromagnetic signal was unaffected by the
running effective dimension. This study of the various spectral dimensions can
be a starting point to investigate these assumptions in a non-perturbative
theory of quantum gravity. We have found that the two-form, tensor and dual
scalar spectral dimensions exhibit two scales at which an effective dimension
appears, i.e. the curve has two regimes with an extremum. However, the one-
form and vector spectral dimension show only a single scale at which an
effective dimension emerges, i.e. the curve only shows one extremum. The fact
that the one-form and vector spectral dimensions do not show a dimensional
flow between two scales can potentially be related to the absence of a
dispersion relation for the electromagnetic field, but dynamically generated
instead of as an ad hoc assumption. It is, however, as of yet unclear how to
directly relate scale-dependent effective dimensions to dispersion relations
for the propagation of physical fields.

Before an attempt can be made to answer an ambitious phenomenological question
such as the one stated above, it will be advantageous to better understand the
dynamics of the various spectral dimensions. For example, it is clear that the
size and number of the geometries studied was too small to extrapolate an
infinite volume limit from the scalar spectral dimension $\langle
D^{(0)}\rangle$, because the lattice artefacts in $\langle D^{(0)}\rangle$
were strong in the measurements we made. In the simplicial geometries under
consideration, the number of vertices is small in comparison to the simplices
of higher dimension, so it is expected that lattice artefacts are harder to
control.

For small diffusion times, $\langle D^{(3)}\rangle$ shows a local minimum
after an initial peak that we consider to be a lattice artefact, just like the
initial peaks for the regular geometries in Figs. 5 and 7. For these regular
geometries, we know that the initial peaks are lattice artefacts, because we
know the dimension of the manifolds the regular geometries approximate. We
have identified the local minimum of $\langle D^{(3)}\rangle$ as the UV
plateau of $\langle D^{(3)}\rangle$. The value of the UV plateau is at
approximately $1.5$, which is in agreement with the value of the same spectral
dimension obtained for spatial slices of four-dimensional CDT with spherical
spatial topology [16]. We note here that the plateau of $\langle
D^{(3)}\rangle$ in [16] was obtained for the largest diffusion time of that
numerical study, while $\langle D^{(3)}\rangle$ in Fig. 15 takes the value
$1.5$ for small diffusion time. The size of the geometries and the maximal
diffusion time of the numerical study in [16] was however much smaller
(between $500$ and $2000$ tetrahedra), because of the computer resources that
were available at that time. The value obtained for the long range diffusion
time of $\langle D^{(3)}\rangle$ in [16] therefore should be compared to the
short range (up to approximately $\tau/\bar{N}_{41}=0.2$) value of $\langle
D^{(3)}\rangle$ of the $20k$ and $10k$ curves111111A spatial slice of a
typical geometry with $\bar{N}_{41}=20k$ with $4$ slices consists of
approximately $2500$ tetrahedra, because every tetrahedron is shared by two
four-dimensional simplices. The diffusion time of $200$ discrete steps
corresponds to approximately $\tau/\bar{N}_{41}=0.2$ in Fig. 15 of the $20k$
curve. in Fig. 15. Reobtaining the value $1.5$ for $\langle D^{(3)}\rangle$ is
an indication that results for the effective dimension at small scales are
universal, even though the choice of normalisation121212The choice of
normalisation was discussed in Sec. 3.2. of the Laplacian $L^{(3)}$ and the
topology of the geometries in the current work is different than the
normalisation and topology in [16]. An IR spectral dimension has not been
previously reported for spatial slices in CDT. Our measurements were performed
at a specific point $(\kappa_{0}=4.0,\Delta=0.2)$ in the CDT phase diagram, at
which we expect a much smaller effective lattice spacing than at the point
$(\kappa_{0}=2.2,\Delta=0.6)$ in phase $C$, where most previous spectral
dimension measurements were made. From the point of view of the conjectured RG
trajectories in the CDT phase diagram [30, 31], we expect that the effective
theory at $(\kappa_{0}=4.0,\Delta=0.2)$ captures more features of the deep
quantum regime and the potential UV fixed point than the effective theory at
$(\kappa_{0}=2.2,\Delta=0.6)$.

In a (semi-)classical regime, the various spectral dimensions should be
(approximately) equal for large diffusion times, as is the case for a
differentiable manifold. As presented in Tab. 2, our results show a potential
agreement, within our measurement accuracy, between the IR values of the
effective dimension of two-form $\langle D^{(2)}\rangle$, tensor
$\langle\vec{D}^{(2)}\rangle$ and dual scalar $\langle D^{(3)}\rangle$. The UV
values ($\approx 1.5$) of $\langle D^{(2)}\rangle$,
$\langle\vec{D}^{(2)}\rangle$ and $\langle D^{(3)}\rangle$ agree within error
bars as well. On the contrary, we observe completely different behaviour for
the one-form $\langle D^{(1)}\rangle$ and vector $\langle\vec{D}^{(1)}\rangle$
spectral dimensions (Tab. 1), which show only a single effective scale. We
also note that only the vector effective dimension agrees with the Hausdorff
dimension ($d_{H}=4$) and that none of the spectral dimensions is consistent
with the topological dimension of the spatial slices, which is equal to $3$.
All in all, we have established that the effective geometry of CDT in the deep
quantum regime exhibits a dynamically generated effective dimension that shows
significantly qualitative differences for different $k$-form fields. Exact
numerical values in the UV regime are however difficult to establish due to
lattice and discretisation artefacts. We also conclude that the ensembles of
spatial slices we have investigated do not represent an effective geometry
which is close to a classical manifold. This is to be expected as previous
studies of spatial slices also showed that their geometry is non-classical
[16, 19].

It is difficult to compare the physical scales at which the effective
dimensions appear for different $k$. One reason for this is that the discrete
Laplace-Beltrami operator, constructed from the unweighted incidence matrices
of the $k$-simplices, is not defined in a manner consistent between different
values of $k$. The fraction of the geometry that is probed with a discrete
diffusion step from some starting point is therefore determined by the
connectivity matrix for the $k$-simplex. It is non-trivial how to relate the
connectivity matrices for different $k$, so it is also difficult to relate the
diffusion time for different $k$. It will be interesting to explore to which
extent an alternative choice of weights for the incidence matrix makes it
possible to compare scales at which effective dimensions appear for different
values of $k$. A potential candidate for such a choice of weights is given by
the framework of Discrete Exterior Calculus (DEC) [24]. Using DEC, we can
choose weights for the incidence matrices such that a diffusion step is
weighted by the associated dual volume of the $k$-simplices.

Another interesting avenue to explore would be to investigate all spectral
dimensions for a sequence of points in the phase diagram. Such a study can
indicate if the RG flows towards the IR of the effective dimensions for large
diffusion time converge. If we find such an RG trajectory, we can also
potentially answer the question whether the $k$-form spectral dimension
$\langle D^{(k)}\rangle$ or the $k$-tensor spectral dimension
$\langle\vec{D}^{(k)}\rangle$ better captures the physical properties of the
effective dimension. In the present study $\langle D^{(k)}\rangle$ and
$\langle\vec{D}^{(k)}\rangle$ clearly have different properties, while they
are approximately equal for the regular geometries. We found that lattice
artefacts for small diffusion time are weaker for
$\langle\vec{D}^{(k)}\rangle$ and that the corresponding $k$-tensor fields
have faster convergence properties. However, these properties are not
sufficient to rule out the relevance of the effective dimension derived from
$\langle D^{(k)}\rangle$. Therefore, we have chosen to present the results for
both $\langle D^{(k)}\rangle$ and $\langle\vec{D}^{(k)}\rangle$ in this text.

It is also interesting to consider the four-dimensional geometries for which
the dual scalar spectral dimension has been studied extensively in the
literature. In the four-dimensional case, the dual scalar131313In the case of
four dimensional CDT, the dual scalar Laplacian is determined by the
connectivity of the four-dimensional simplices. spectral dimension has been
shown to be equal to four for large diffusion times. Traditionally, this
result has been interpreted as an indication that CDT shows some
(semi-)classical properties in phase $C$. It would be interesting to see to
what extent the other spectral dimensions support this interpretation. The
numerical challenges are greater for the four-dimensional geometries, but are
in essence unchanged from what was presented in this text. For four-
dimensional geometries there will also be one additional spectral dimension,
because $k$ will take values in $\\{0,...,4\\}$.

## Acknowledgements

The research in this work was supported by the National Science Centre,
Poland, under grant no. 2019/33/B/ST2/00589. We would also like to express our
deep gratitude to Prof. Jerzy Jurkiewicz (from the Institute of Theoretical
Physics of the Jagiellonian University in Krakow) who collaborated with us in
this research; unfortunately he unexpectedly passed away on 30th November
2021, before we finished this work.

##  Appendix A Analytical calculation of generalised spectral dimensions on
the flat two-torus

The following scheme describes a triangulation of the flat two-dimensional
torus. We repeat the elementary square in both $x$ and $y$ direction, such
that every vertex will be of degree six (see figure 18) and by pairwise
identifying the sides of the triangulation we obtain a finite lattice with
periods $L_{x}$ and $L_{y}$.

Figure 18: The figure shows a triangulation of the flat torus. The elementary
cell consisting of three oriented edges is coloured gray. The red and green
boundaries are identified, respectively.

The link Laplacian $\Delta^{(1)}$ consists of two components as can be seen
from Eq. (19). The component of $\Delta^{(1)}$ that corresponds to the vertex-
edge incidence matrix $I^{1}_{0}$ is denoted by $\mathbf{F}$ and the component
of $\Delta^{(1)}$ that corresponds to the edge-triangle incidence matrix
$I^{2}_{1}$ is denoted by $\mathbf{G}$. The matrices $\mathbf{F}$ and
$\mathbf{G}$ can be given in the coordinate basis of the vertices
parameterised by two integers ${k,l}$. We can equivalently describe the
matrices $\mathbf{F}$and $\mathbf{G}$ in terms of the Fourier basis
$\hat{\delta}_{kl}$ parameterised by two wave numbers ${p,q}$ such that

$\displaystyle\hat{\delta}_{kl}(p,q)=\mathcal{N}e^{i(kp+lq)},$ (38)

where $\mathcal{N}$ is the normalisation factor. As a consequence of the
periodicity, the wave numbers $p$ and $q$ take the values
$p_{m}=\frac{2\pi}{L_{x}}m$ and $q_{n}=\frac{2\pi}{L_{y}}n$ where
$m=0,1,\ldots,L_{x}-1$ and $n=0,1,\ldots,L_{y}-1$.

It can be shown that the Fourier transform of the matrices $\mathbf{F}$ and
$\mathbf{G}$ are block-diagonal in each ${p,q}$ sector. Each block has the
form

$\displaystyle\mathbf{F}(p,q)=\begin{pmatrix}\mathbf{F}_{aa}&\mathbf{F}_{ab}&\mathbf{F}_{ac}\\\
\mathbf{F}_{ba}&\mathbf{F}_{bb}&\mathbf{F}_{bc}\\\
\mathbf{F}_{ca}&\mathbf{F}_{cb}&\mathbf{F}_{cc}\end{pmatrix},\ \ \ \
\mathbf{G}(p,q)=\begin{pmatrix}\mathbf{G}_{aa}&\mathbf{G}_{ab}&\mathbf{G}_{ac}\\\
\mathbf{G}_{ba}&\mathbf{G}_{bb}&\mathbf{G}_{bc}\\\
\mathbf{G}_{ca}&\mathbf{G}_{cb}&\mathbf{G}_{cc}\end{pmatrix}.$ (39)

A non-zero contribution to $\mathbf{F}$ can only occur for two edges which are
shared by a vertex. Two neighbouring edges which are shifted relatively by a
single positive or negative step in the parameters $k,l$ are, in Fourier
space, related by a factor $e^{\pm ip}$ and $e^{\pm iq}$, respectively. From
the neighbourhood relations of the triangulation of the flat torus and these
factors, we obtain the following components of $\mathbf{F}$,

$\displaystyle\mathbf{F}_{aa}$
$\displaystyle=2-e^{ip}-e^{-ip}=4\sin^{2}\left(\frac{p}{2}\right)$ (40)
$\displaystyle\mathbf{F}_{bb}$
$\displaystyle=2-e^{i(p+q)}-e^{-i(p+q)}=4\sin^{2}\left(\frac{p+q}{2}\right)$
(41) $\displaystyle\mathbf{F}_{cc}$
$\displaystyle=2-e^{iq}-e^{-iq}=4\sin^{2}\left(\frac{q}{2}\right)$ (42)
$\displaystyle\mathbf{F}_{ab}$
$\displaystyle=1-e^{-ip}+e^{iq}-e^{i(p+q)}=4e^{iq/2}\sin\left(\frac{p}{2}\right)\sin\left(\frac{p+q}{2}\right)$
(43) $\displaystyle\mathbf{F}_{ac}$
$\displaystyle=1-e^{-ip}-e^{-iq}+e^{i(p-q)}=4e^{i(p-q)/2}\sin\left(\frac{p}{2}\right)\sin\left(\frac{q}{2}\right)$
(44) $\displaystyle\mathbf{F}_{bc}$
$\displaystyle=1+e^{-ip}-e^{iq}-e^{i(p+q)}=4e^{-ip/2}\sin\left(\frac{q}{2}\right)\sin\left(\frac{p+q}{2}\right)$
(45) $\displaystyle\mathbf{F}_{ba}$
$\displaystyle=\mathbf{F}^{*}_{ab},\mathbf{F}_{ca}=\mathbf{F}^{*}_{ac},\mathbf{F}_{cb}=\mathbf{F}^{*}_{bc}.$
(46)

where $F*$ is the conjugate transpose of $F$. We derive the matrix
$\mathbf{G}$ as a function of $p$ and $q$ in a similar fashion. A non-zero
contribution of $\mathbf{G}$ can only occur if two edges are shared by a
triangle. Using the same factors as before, we find the following components
of $\mathbf{G}$,

$\displaystyle\mathbf{G}_{aa}$
$\displaystyle=\mathbf{G}_{bb}=\mathbf{G}_{cc}=2$ (47)
$\displaystyle\mathbf{G}_{ab}$
$\displaystyle=-1-e^{iq}=-2e^{iq/2}\cos\left(\frac{q}{2}\right)$ (48)
$\displaystyle\mathbf{G}_{ac}$
$\displaystyle=e^{-ip}+e^{iq}=2e^{-i(p-q)/2}\cos\left(\frac{p+q}{2}\right)$
(49) $\displaystyle\mathbf{G}_{bc}$
$\displaystyle=-1-e^{-ip}=-2e^{-ip/2}\cos\left(\frac{p}{2}\right)$ (50)
$\displaystyle\mathbf{G}_{ba}$
$\displaystyle=\mathbf{G}^{*}_{ab},\mathbf{G}_{ca}=\mathbf{G}^{*}_{ac},\mathbf{G}_{cb}=\mathbf{G}^{*}_{bc}.$
(51)

The eigenvectors of $\mathbf{F}$ and $\mathbf{G}$ are orthogonal. The
eigenvalues of $\Delta^{(1)}$ are therefore either an eigenvalue
$\lambda^{\mathbf{F}}(p,q)$ of $\mathbf{F}$ or an eigenvalue
$\lambda^{\mathbf{G}}(p,q)$ of $\mathbf{G}$,

$\displaystyle\lambda_{1}^{\mathbf{F}}(p,q)$ $\displaystyle=0$ (52)
$\displaystyle\lambda_{2}^{\mathbf{F}}(p,q)$ $\displaystyle=0$ (53)
$\displaystyle\lambda_{3}^{\mathbf{F}}(p,q)$
$\displaystyle=4\left(\sin^{2}\left(\frac{p}{2}\right)+\sin^{2}\left(\frac{p+q}{2}\right)+\sin^{2}\left(\frac{q}{2}\right)\right)$
(54)

and

$\displaystyle\lambda_{1}^{\mathbf{G}}(p,q)$
$\displaystyle=3-\sqrt{9-4\left(\sin^{2}\left(\frac{p}{2}\right)+\sin^{2}\left(\frac{p+q}{2}\right)+\sin^{2}\left(\frac{q}{2}\right)\right)}$
(55) $\displaystyle\lambda_{2}^{\mathbf{G}}(p,q)$
$\displaystyle=3+\sqrt{9-4\left(\sin^{2}\left(\frac{p}{2}\right)+\sin^{2}\left(\frac{p+q}{2}\right)+\sin^{2}\left(\frac{q}{2}\right)\right)}$
(56) $\displaystyle\lambda_{3}^{\mathbf{G}}(p,q)$ $\displaystyle=0.$ (57)

The eigenvalues $\lambda^{\mathbf{F}}(p,q)$ and $\lambda^{\mathbf{G}}(p,q)$
are given as a function of $p$ and $q$. We note that
$\lambda_{i}^{\mathbf{F}}(p,q)$ is equal to $0$ if
$\lambda_{i}^{\mathbf{G}}(p,q)$ is not and vice versa. The eigenvalues
$\lambda_{i}(p,q)$ of $\Delta^{(1)}$ can therefore be written as
$\lambda_{i}(p,q)=\lambda_{i}^{\mathbf{F}}(p,q)+\lambda_{i}^{\mathbf{G}}(p,q)$,
for $i\in\\{1,...,3\\}$. Using Eq. (27), we obtain the spectral dimension
$D^{(1)}$ from the eigenvalues $\lambda_{i}(p,q)$. Fig. 5 shows the curve of
$D^{(1)}$ as a function of $\tau$ derived from the analytical calculation of
the spectrum of $\Delta^{(1)}$ for a triangulation of a flat two-dimensional
torus of size $L_{x}\times L_{y}$.

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