11institutetext: Dario Benedetti 22institutetext: CPHT, CNRS, École
polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France.
22email<EMAIL_ADDRESS>
# Landau Theory of Causal Dynamical Triangulations

Dario Benedetti

###### Abstract

Understanding the continuum limit of a theory of discrete random geometries is
a beautiful but difficult challenge. In this optic, we review here the
insights that can be obtained for Causal Dynamical Triangulations (CDT) by
employing the Landau approach to critical phenomena. In particular,
concentrating on the cases of two and three dimensions, we will make the case
that the configuration of the volume of spatial slices effectively plays the
role of an order parameter, helping us to understand the phase structure of
CDT. Moreover, consistency with numerical simulations of CDT provides hints
that the effective field theory of the model lives in the space of theories
invariant under foliation-preserving diffeomorphisms. Among such theories,
Hořava-Lifshitz gravity has the special status of being a perturbatively
renormalizable theory, while General Relativity sits in a subspace with
enhanced symmetry. In order to reach either of them, one would likely need to
fine tune some of the parameters in the CDT action, or additional ones from
some generalization thereof.

###### Keywords:

Quantum Gravity, Random Geometry, Lattice Models, Phase Diagrams, Causal
Dynamical Triangulations, Hořava-Lifshitz Gravity

Note: This is a contribution to the _Handbook of Quantum Gravity_ which is
expected to be published in 2023. It will appear as a chapter in the section
dedicated to _Causal Dynamical Triangulations_.

##  1 Introduction

Lattice regularization is a common approach to taming in a nonperturbative way
the ultraviolet divergences of a quantum field theory. For many quantum field
theories it is relatively straightforward to formulate a corresponding lattice
field theory, but showing that a nontrivial continuum limit can be obtained is
generally very difficult.111For example, only recently is has been rigorously
proven that the continumm limit of the $\phi^{4}$ theory in four dimensions is
trivial (i.e. non-interacting) Aizenman:2019yuo . The standard physical
picture, based on the Wilsonian renormalization group, is that such a limit
should exist if the lattice model has a second-order phase transition that
requires the tuning of the interaction coupling in order to be reached. In
such a scenario, one would construct the continuum limit as a scaling limit in
which the coupling is tuned to the phase transition in a coordinated way with
the tuning of the lattice spacing to zero. Lacking an exact solution for the
given model, such a scaling limit is a challenging task, especially when the
universality class of the phase transition is described by an interacting
theory. Luckily, even simple calculations and approximations can often help in
establishing a correct physical picture. In particular, the Landau theory of
critical phenomena provides very often a qualitatively (if not quantitatively)
correct description of the phase transitions of a model.

In gravity, even the first step of formulating a lattice theory is
nontrivial.222For the second step, that of the continuum limit, one hope would
be to find a realization of Weinberg’s asymptotic safety scenario
Weinberg:1980gg ; Niedermaier:2006wt ; Bonanno:2020bil . Indeed, in general
relativity the gravitational force is described in terms of the geometry of
spacetime, whose dynamics is governed by background independent laws, so we
should not rely on a fixed lattice background. One proposal to construct a
candidate lattice theory of Euclidean quantum gravity took shape in the 1980’s
in the context of noncritical bosonic string theory, and it is commonly known
as dynamical triangulations (DT) Ambjorn:1997di . It is based on Regge’s
coordinate-free approach to discrete gravity Regge:1961px , in which
continuous manifolds are replaced by piecewise flat manifolds (triangulations
and their higher-dimensional generalizations, simplicial manifolds). However,
unlike in Regge calculus, in DT one is prescribed to sum over all possible
triangulations, with all edge-lengths fixed to the same value $a$, thus making
DT also intrinsically background independent, in the sense that there is no
fixed structure at all and no identifiable points. Originally DT was
constructed for two-dimensional quantum gravity David:1984tx ; Ambjorn:1985az
; Kazakov:1985ea , in which case it is related to matrix models
DiFrancesco:1993nw , and it had an exciting success in reproducing continuum
results in the case of conformal charge $c<1$, thus proving that this way of
discretizing the functional integral over geometries makes sense.333Even from
a rigorous mathematical perspective, as reviewed in another chapter of the
Handbook of Quantum Gravity Budd:2022zry . Later, it was generalized to three
and four dimensions, where however the results were more disappointing: no
second-order phase transition could be found deBakker:1996zx ; Bialas:1996wu .
Moreover, the two phases of the model, separated by a first-order transition,
were characterized by very degenerate geometries that did not approach a
classical limit at large scales Ambjorn:1995dj .444DT were revived about a
decade ago, on the analytical side by the discovery of the large-$N$ limit of
tensor models Bonzom:2011zz ; Gurau:book , prompting the hope that analytical
control would show the way to improving DT, and on the numerical side by the
exploration of some generalized models with hints of a new (”crinkled”) phase
Laiho:2011ya  (see also Benedetti:2011nn  for a meeting point of the two
developments). Unfortunately, so far tensor models have not been able to solve
the problem of escaping the branched polymer phase Gurau:2013cbh ;
Bonzom:2016dwy , while the the crinckled phase of the generalized models
turned out to be again rather unphysical, and still no second-order phase
transition is in sight at finite coupling Ambjorn:2013eha ; Coumbe:2014nea ;
Laiho:2016nlp .

The state of affairs was significantly improved with the introduction of the
model of causal dynamical triangulations (CDT) by Ambjorn and Loll (initially
in two dimensions Ambjorn:1998xu , and later in three and four dimensions,
together with Jurkiewicz Ambjorn:2001cv ; Ambjorn:2005qt ). CDT modifies DT
only by the requirement that the ensemble of triangulations be restricted with
the requirement that each triangulation carries a non-singular foliation, i.e.
a clear $(d+1)$-dimensional decomposition with $d$-dimensional slices (or
leaves) of fixed topology, subsequent slices being always at fixed distance
from each other (see Ambjorn:2001cv  for the precise construction, or the two-
dimensional example in Fig. 1 for an intuitive understanding). Notice in
particular that the foliation induces a natural notion of time as the
direction perpendicular to the foliation, and a natural choice of time
coordinate $t$, such that all the vertices on a given slice are at the same
value of $t$. With such a simple modification, originally motivated by the
requirement of having always nonsingular light cones in a Lorentzian version
of the model,555In this review we will only discuss models in Euclidean
signature, as these are under better control, and in particular they are
suitable for numerical simulations. The Lorentzian and Euclidean models can be
easily Wick-rotated to each other at the discrete level. Given that Wick
rotation is notoriously problematic in general relativity Visser:2017atf ;
Baldazzi:2018mtl , the fact that in CDT (before any continuum limit) it poses
instead no problems is a signal that perhaps the chosen ensemble of geometries
is more restricted than one would have imagined. the main problems of DT
seemed to have been solved: a new phase of the model emerged with some
classical behavior at large scales Ambjorn:2005qt , separated from another
phase by a second-order phase transition Ambjorn:2012ij  (see the reviews
Ambjorn:2012jv ; Loll:2019rdj  for a proper summary of results and a complete
list references).

Figure 1: Example of a triangulated portion of spacetime in the two-
dimensional CDT model. The flattened picture captures the way triangles are
glued together, but does not represent faithfully intrinsic distances, as edge
lengths should be all equal.

We should at this stage take a step back, and reflect upon the way phase
diagrams are found in these models, which are effectively statistical models
of random geometry. As for any statistical model, the standard way to chart a
phase diagram is to identify suitable order parameters taking different values
in the different phases. Such order parameters should necessarily correspond
to expectation values of observables, meaning quantities which are independent
of (unobservable) gauge choices. Here one stumbles in the problem of
observables in quantum gravity (e.g. Rovelli:1990ph ; Giddings:2005id ): as in
any diffeomorphism invariant and background independent approach to quantum
gravity, observables in DT are necessarily nonlocal. Luckily, this is not an
immediate problem for the phase diagram, as order parameters are often
nonlocal (e.g. the average magnetization in the Ising model), at least for the
identification of homogenous phases, and indeed the two phases of DT can be
distinguished by a global observable: the average linear size of the
“universe”. In particular, studying the scaling of it with the total volume of
the universe, one can define an effective Hausdorff dimension $d_{H}$ of the
averaged universe. It turns out that for topological dimension $d=3$ and $4$,
varying the lattice Newton constant, the effective dimension jumps from
$d_{H}\sim 2$ (branched polymer phase) to $d_{H}\sim\infty$ (crumpled phase)
Ambjorn:1995dj . Phases with the same Hausdorff dimensions were also found in
CDT, but in addition a new phase was identified as well, with $d_{H}\sim d$
Ambjorn:2005qt . Clearly, this was a great advance, providing a zeroth order
test of classical behavior being possible in these models, but much more is
needed in order to reconstruct the possible continuum limit. For the latter,
global observables are not very instructive, as for example it is hard to
imagine how to construct and interpret an effective action for the Hausdorff
dimension.

In CDT, thanks to the foliation, more observables are available, of a slightly
more local nature: observables that are local in time but non-local in space.
The easiest observable of this type is the volume666In the two and three-
dimensional cases, it would be more appropriate to call it length and area,
respectively, but for simplicity we stick to a general $d$-dimensional
terminology. of spatial slices $N_{d-1}(t)$, simply counting the number of
$(d-1)$-simplices in the spatial slice at time $t$. Armed with such
observable, one can characterize the different phases of CDT in terms of
volume profiles, and even derive an effective action describing the dynamics
of the spatial volume Ambjorn:2004pw . The latter will be our main focus in
the following.

Given the premises above, a natural question arises: are the foliation and its
associated observables compatible with a fully diffeomorphism invariant and
background independent continuum limit? Originally, the foliation was thought
only as a way of selecting less pathological geometries in a partially gauge-
fixed (proper time gauge) version of discrete quantum gravity (see
Dasgupta:2001ue  for a continuum point of view on this). However, the
existence of observables that are local in time raises two problems: first,
they would be just a gauge artifact, and should not be true observables of the
theory; second, a coarse-grained effective action depending on them would only
be compatible with a gauge-fixed general-covariant action for very special
values of the couplings. The last point is extremely important, and, phrased
differently, means that the most general and natural theory space in which a
renormalization group flow of CDT should take place is that of theories
invariant under foliation-preserving diffeomorphisms (FPD), and invariance
under full (spacetime) diffeomorphisms would only be recovered on a
submanifold of theory space.777We are being cavalier about the fact that any
classical theory can be written in a fully diffeomorphism-invariant way. What
we mean here is a theory that can be written in such a way without introducing
other fields than the metric. FPD-invariant theories were introduced by Hořava
Horava:2008ih ; Horava:2009uw  in order to construct a gravitational theory
that could be at the same time perturbatively renormalizable and unitary. This
is achieved by introducing in the action higher derivatives in space, but not
in time, which is possible thanks to the reduced diffeomorphism invariance.
Such theories, of which several versions have been introduced and studied (see
Wang:2017brl ; Steinwachs:2020jkj  for reviews), are commonly known as Hořava-
Lifshitz (HL) gravity, as they are the gravitational analog of theories with a
Lifshitz critical point. On the other hand, the space of FPD-invariant
effective theories is more general and contains in particular also
nonrenormalizable ones.

First possible hints of a relation between CDT and HL-type gravity theories
were discussed in Horava:2009if ; Benedetti:2009ge ; Ambjorn:2010hu , relying
mostly on the presence of a foliation, on a qualitative resemblance between
the phase diagram of CDT with the generic Lifshitz phase diagram, and on
results concerning the spectral dimension (another useful notion of effective
dimension). Other dynamical arguments have subsequently been added Budd:2011zm
; Ambjorn:2013joa ; Benedetti:2014dra ; Benedetti:2016rwo , and a longer list
of arguments will be reviewed in Sec. 5, but of course understanding the
continuum limit of higher-dimensional background-independent statistical
models like CDT is highly nontrivial, and therefore the precise relation is
still open, in particular in four dimensions.

In any case, the fact that FPD-invariant effective theories are conceivable
theories of dynamical geometry raises the following question: if CDT was
describing a partially gauge-fixed but otherwise fully covariant theory, how
would we construct a nonperturbative quantization of FPD-invariant theories in
the spirit of dynamical triangulations, if not by introducing a foliation as
in CDT? Most likely, the answer is that for FPD-invariant theories we should
indeed use the same ensemble of triangulations as in CDT, in general with a
different action, with more parameters (as for example in Anderson:2011bj ).
And if a fully diffeomorphism-invariant continuum limit could ever be reached
from such generalized CDT models, this should be found within a subspace of
their full theory space, i.e. by a careful fine tuning of the couplings. Could
we be so lucky that the standard CDT already sits in such a subspace? This
might seem plausible, at least in two and three dimensions, where the CDT
action is just a Regge discretization of the Einstein-Hilbert action, with
cosmological and Newton constant.888Less so in four dimensions, where the CDT
action has a third coupling, not appearing in the Einstein-Hilbert action.
However, this is a minor point with respect to what we want to stress here.
Consider the example of a scalar model on a regular lattice. Typically, an
anisotropic action will lead to an anisotropic continuum limit, while discrete
rotation symmetry will result in full rotation invariance in the continuum
limit (see for example Borji:2020snd ). That is, in this case, starting from
an action which is a simple discretization of a rotation-invariant action,
leads to the expected result, or in other words, the fine tuning is trivial.
But rotation invariance is a global symmetry. Things are more complicated for
gauge symmetries, as their breaking leads to new degrees of freedom: for
example, HL gravity has a new scalar degree of freedom besides the massless
spin-2 graviton. It is a well known fact that in the case that the gauge
invariance of a quantum field theory is broken by a lattice or by a momentum
cutoff, in order to restore it in the continuum limit, and thus recover the
desired target theory, one needs a nontrivial fine tuning of all the possible
breaking terms in the action, as for example extensively discussed in some
lattice approach to chirality Testa:1997ia , or in constructive Magnen:1992wv
and functional renormalization group Ellwanger:1994iz ; Gies:2006wv
approaches to Yang-Mills theory. It seems reasonable to expect that recovering
full diffeomorphism invariance in CDT would require some fine tuning along
those lines.

Let us recap the key elements of the discussion above. DT models provide a
diffeomorphism invariant and background independent discretization of a
quantum theory of dynamical geometry, but unfortunately they do not have a
second order phase transition and they are dominated by very degenerate
geometries. In order to solve these problems, in CDT one reduces the ensemble
of triangulations by introducing a built-in regular foliation. However, the
foliation being a sort of background structure, this likely breaks part of the
diffeomorphism invariance of the target theory (general relativity), and in
order to restore that, one would presumably need to fine tune the model, as
well as add new counterterms to the action. The latter part of this summary is
so far conjectural, because, due to the complexity of the models, which are
mostly studied by Monte Carlo simulations, we cannot rule out in all certainty
that the standard CDT action would be a lucky choice leading to a full
diffeomorphism invariant theory in the continuum.

We will argue below that in two dimensions it is quite clear that indeed in
the continuum limit CDT is described by an FPD-invariant theory, that in three
dimensions there is some evidence supporting the same conclusion, and that
even in four dimensions there are some indications in the same direction. To
that end, we will focus on the effective dynamics of the volume of the spatial
slices, which can be seen as an application of Landau theory to CDT. In
particular, we will argue that by a deeper study of the effective action
governing such observable, one can in principle distinguish between a gauge-
fixed general-covariant action and an FPD-invariant one.

The rest of the review, mainly based on the results of Benedetti:2014dra ;
Benedetti:2016rwo , is organized as follows. In section 2, we review the main
ideas of the Landau theory of critical phenomena, the challenges of applying
it to a background independent model, and how instead a Landau free energy is
straightforwardly obtained for two-dimensional CDT, in a direct top-down
approach. Armed with that, one recognizes in the continuum limit the two-
dimensional version of HL gravity. In section 3, we review the setup and
numerical results for the dynamics of the spatial volumes in three-dimensional
CDT. In section 4, we review a bottom-up proposal (based on numerical data and
on the intuitions we tried to motivate in this introduction) for the Landau
free energy associated to such spatial volumes, viewed as (time-dependent)
order parameter of the model. We also review the numerical simulations of the
coarse-grained model based on such Landau free energy, showing that
qualitative features of the CDT phase diagram are reproduced. An extremization
analysis of the continuum limit of the Landau free energy is reviewed in
section 4.3, providing in particular an analytical understanding of the
condensation phenomenon observed in the semiclassical phase of CDT, and one
more indication that the correct effective action of the model sits in the
space of FPD-invariant theories. Lastly, we conclude in section 5 with a short
summary, and a discussion of our current understanding of the relation between
CDT and HL-like gravity theories.

##  2 Landau Theory and CDT

The goal of Landau theory (e.g. Goldenfeld:1992qy ) is to provide a
theoretical framework for understanding and predicting phase transitions in
statistical systems. In this respect, it has represented an important landmark
in statistical physics, successfully explaining many phenomena. It is also
well-known to have strong limitations, predicting wrong scaling exponents
below critical dimensions, and this led to the development of the
renormalization group. However, in general Landau theory remains a valid tool
for a first analysis, at least at the qualitative level, for example for
understanding the structure of the phase diagram of a given model.

The central aspect of Landau theory is that it postulates that phase diagrams
can be explained in terms of order parameters, and that these are effectively
governed by a coarse-grained free energy functional, known as Landau free
energy, which depends parametrically on the coupling constants of the system.
The minima of the Landau free energy determine the thermodynamically favored
configuration of the order parameters, and different favored configurations
correspond to different phases. Typically, there will be a _disordered phase_
, with vanishing order parameters and unbroken symmetries, and one or more
_ordered phases_ , with non-vanishing order parameters breaking some or all of
the symmetries of the system. Therefore, given a statistical model, it is
crucial to identify the order parameters and construct the corresponding
Landau free energy.

There are two main approaches for constructing the Landau free energy: a
phenomenological (bottom-up) approach, writing it as an expansion in the order
parameters, constrained by the symmetries, as in effective field theories; and
a top-down approach, deriving the Landau free energy by explicitly coarse-
graining the microscopic model. The latter is of course seldomly applicable in
practice, but it provides a very instructive way of thinking about the Landau
theory, and it is hence worth of being briefly recalled (we essentially follow
Goldenfeld:1992qy  for this).

Consider a spin system with Hamiltonian $H(\\{s\\})$, where
$\\{s\\}=\\{s_{i}\mid i\in\mathcal{L}\\}$ is a configuration of spin variables
$s_{i}$ on the $N$ sites of the lattice $\mathcal{L}$. The partition function
is

$Z_{\rm spin}=\sum_{\\{s\\}}e^{-\beta H(\\{s\\})}\,.$ (1)

Next, partition the lattice into blocks $\mathcal{B}_{r}$, labeled by $r$,
which can be thought of as a site of the coarse grained lattice
$\mathcal{L}^{\prime}$. Each block contains $N(r)$ sites, such that
$\sum_{r}N(r)=N$. We define the Landau free energy $L(\\{m\\})$, where
$\\{m\\}=\\{m(r)\mid r\in\mathcal{L}^{\prime}\\}$, as the Gibbs free energy
for the system, constrained to be in a configuration compatible with a local
magnetisation configuration specified by $\\{m\\}$, i.e.:

$e^{-L(\\{m\\})}=\sum_{\\{s_{i}\mid
i\in\mathcal{L},\,\frac{1}{N(r)}\sum_{j\in\mathcal{B}_{r}}s_{j}=m(r)\\}}e^{-\beta
H(\\{s\\})}\,.$ (2)

From the definitions, it follows straightforwardly that the partition function
can be expressed in terms of the Landau free energy as

$Z_{\rm spin}=\sum_{\\{m\\}}e^{-L(\\{m\\})}\,.$ (3)

It is clear that in this construction, $L(\\{m\\})$ can also be thought as a
Wilsonian effective Hamiltonian. This is in particular the case because the
local magnetisation $\\{m\\}$ is a similar (albeit coarse grained) type of
variable as the original spin variables $\\{s\\}$, in the sense that we went
from some spin variables to some other spin variables (perhaps with a
different set of values, due to the averaging). The idea is however more
general, and depending on the model and the relevant order parameter, one can
define different coarse-graining procedures and end up with very different
variables.

Landau theory and quantum gravity. Applying a similar construction to quantum
gravity is of course marred with difficulties, in particular concerning the
identification of order parameters, as observables are notoriously problematic
in quantum gravity. Diffeomorphism invariance and background independence
constrain observables to be non-local quantities, which are not very suitable
for reconstructing a local effective action. Moreover, coarse graining
presents several practical and conceptual obstacles, due to the absence of a
background structure. Nevertheless, inventive approaches to the problem have
been devised, with coarse graining procedures introduced for constructing a
renormalization group flow in DT Johnston:1994qi ; Thorleifsson:1995ki ;
Ambjorn:1996hu ; Renken:1996kf ; Henson:2009fy  and in spin foams
Markopoulou:2002ja ; Oeckl:2002ia ; Bahr:2012qj ; Dittrich:2013uqe ;
Steinhaus:2020lgb . More to the point of interest here, mean field models of
DT have also been proposed Renken:1997na ; Bialas:1998ci . The latter involved
no coarse graining, they were rather in the spirit of defining approximated
Hamiltonians (or actions, in the parlance of Euclidean QG) much like what is
done when replacing the Ising model Hamiltonian with one in which spin
interacts not just with its neighbour, but with an average of all the other
spins (the mean field). Again this procedure, which is straightforward in the
Ising case, is not at all obvious in DT, and in fact the mean field models for
DT were postulated, rather than derived from the model itself.

Once more, in CDT things are (slightly) different thanks to the foliation.
Small-step blocking is still a very clumsy procedure, but we do have a natural
partitioning of the lattice into large blocks: the leaves of the foliations.
We can therefore define a Landau free energy by summing over all
triangulations that on each slice lead to a prescribed value for a nonlocal
_slice observable_. As anticipated, the choice of slice observable on which we
will concentrate is the volume of the slice, which is essentially the only one
studied so far, but it would be interesting to explore other options. As a way
of clarifying this set of ideas, and of introducing the more difficult higher-
dimensional cases, we start by analyzing the two-dimensional CDT model in this
perspective.

###  2.1 CDT in a nutshell

The CDT approach exactly parallels the usual lattice field theory with one
fundamental difference: the fixed lattice is replaced by an ensemble of random
triangulations. This is required by the fact that gravity is a theory of
dynamical geometry, with no background spacetime fixed a priori.

More concretely, one defines an ensemble of “triangulations” to work with, a
triangulation being defined by a simplicial manifold, i.e. a collection of
$d$-dimensional flat simplices (the generalization of triangles and
tetrahedra) glued along their $(d-1)$-dimensional faces and such that the
neighbourhood of any vertex is homeomorphic to a $d$-dimensional ball. A
dynamical triangulation is one in which all the simplices are taken to be
equilateral. In the simulations we usually work with dynamical triangulations
having a fixed number of $d$-simplices $N$, which we will denote
$\mathcal{T}_{N}$. The ensemble of such triangulations $\\{\mathcal{T}_{N}\\}$
is obtained by gluing the $N$ simplices in all possible ways allowed by the
simplicial manifold condition, and respecting a chosen topology. Furthermore,
to avoid the sick behaviour that was found in the old models of dynamical
triangulations, CDT models have one further restriction on the ensemble: only
triangulations with a global time foliation, with respect to which no spatial
topology change occurs, are allowed. For more details on the geometrical
meaning of this restriction and on its implementation see Ambjorn:2001cv .

Once the ensemble is specified one can construct the partition function
(Euclidean version of the path integral) as

$Z=\sum_{N}\sum_{\mathcal{T}_{N}}\tfrac{1}{C(\mathcal{T}_{N})}\,e^{-S(\mathcal{T}_{N})}\,,$
(4)

where $S(\mathcal{T}_{N})$ is the bare action, and $C(\mathcal{T}_{N})$ is the
order of the automorphism group of $\mathcal{T}_{N}$, a symmetry factor
naturally appearing when summing over unlabeled triangulations (e.g.
Ambjorn:2012jv ). Since we wish to recover general relativity in the classical
limit, it is customary to use as a bare action the Einstein-Hilbert action
adapted to a simplicial manifold, which is known as the Regge action. On a
dynamical triangulation, the Regge action takes the very simple form

$S(\mathcal{T}_{N})=\kappa_{d}N-\kappa_{d-2}N_{d-2}\,,$ (5)

where $\kappa_{d}$ and $\kappa_{d-2}$ are two coupling constants depending on
the cosmological and Newton’s constant appearing in the Regge action, and
$N_{d-2}$ is the number of $(d-2)$-dimensional subsimplices (also called bones
or hinges).

In principle one could use a different action, with more parameters, but at
this stage this would only complicate the analysis of the results, and in a
minimalist attitude such a generalization of the CDT models is usually
postponed till the moment (if ever) at which the model itself will ask for
such an extension. For example in $3+1$ dimensions a new parameter has been
introduced in the action, without which no physically interesting region would
exist in the phase diagram Ambjorn:2005qt . Furthermore, we need to remember
that, for fixed topology, only $d/2$ (for $d$ even) or $(d+1)/2$ (for $d$ odd)
among the values $\\{N_{0},N_{1},...N_{d-1},N_{d}=N\\}$ are independent.
Hence, for $d=3$ and $d=4$ only two of such variables are independent, and as
a consequence, if we want to keep the action linear in $N_{j}$, we only have
two coupling constants. The counting changes in CDT because we can distinguish
subsimplices whose vertices are all on one slice from those having vertices on
two adjacent slices, and additional variables can be introduced in order to
keep track of that. However, new topological relations are found too. The
counting for CDT was carried out in Ambjorn:2001cv , and one has that for
$d=4$ there are 10 variables and 7 constraints, leaving 3 independent
variables, a fact that was used in Ambjorn:2005qt  to introduce the new
parameter. In $d=3$ the situation is instead unchanged with respect to the DT
case, as there are 5 constraints for 7 variables, and hence again only 2
independent variables. For this reason it is not possible to introduce in
three dimensions the analogue of the new parameter used in four dimensions.

###  2.2 Balls-in-boxes models

Before continuing our discussion of CDT, it is useful to introduce the so-
called balls-in-boxes (BIB) models, as they will play the role of effective
models of CDT.

The statistical models known as balls-in-boxes (or zero-range process in the
nonequilibrium version) are an interesting and versatile class of models which
have been extensively studied in the statistical mechanics literature (see for
example Evans-review ). They have also been used as mean field models of DT
Bialas:1996eh ; Bialas:1997qs ; Bialas:1998ci , and later, also as effective
models for the spatial volume dynamics of CDT in 3+1 dimensions Bogacz:2012sa
. We briefly review in this section their definition and some of their
relevant properties.

A BIB model is defined as a one-dimensional lattice with $T$ sites (boxes) to
each of which is associated an integer number $m_{i}\geq m_{\rm min}\geq 0$
(the number of balls in box $i\in\\{1,2,...,T\\}$). The total number of balls
is fixed to be $M$. The canonical partition function of the statistical model
is given as

$\begin{split}Z_{\rm BIB}(T,M)&=\sum_{m_{1}=m_{\rm
min}}^{M}...\sum_{m_{T}=m_{\rm
min}}^{M}\delta_{M,\sum_{i}m_{i}}\prod_{i=1}^{T}g(m_{i},m_{i+1})\\\
&=\sum_{\\{m_{j}\\}}e^{-S(\\{m_{j}\\})}\delta_{M,\sum_{i}m_{i}}\,,\end{split}$
(6)

with $m_{T+1}=m_{1}$. The last expression highlights the interpretation of
such models as discretised one-dimensional path integrals, subjected to the
constraint

$\sum_{i=1}^{T}m_{i}=M\,.$ (7)

The weight function $g(m,n)$ (or the action $S(\\{m_{j}\\})$) defines the
particular model. In the standard BIB models there is no nearest-neighbour
interaction, meaning that $g(m_{i},m_{i+1})=g(m_{i})$; the model above is a
generalisation studied in Evans-prl ; Waclaw:2009zz .

Often, in particular for an analytical approach, it is useful to work in the
grand canonical ensemble, for which the partition function reads

$\begin{split}Z_{\rm BIB-gc}(T,z)&=\sum_{M}Z_{\rm BIB}(T,M)\,z^{M}\\\
&=\operatorname{Tr}\,\hat{T}^{T}\,,\end{split}$ (8)

where we introduced the transfer matrix

$\hat{T}_{m,n}=z^{(m+n)/2}g(m,n)\,.$ (9)

In the light of this relation, $g(m,n)$ is sometimes referred to as reduced
transfer matrix.

One interesting feature of these models is that they can exhibit a
condensation phenomenon. In the original BIB models, with
$g(m_{i},m_{i+1})=g(m_{i})$, this means that for certain values of the
parameters the model enters into a phase dominated by configurations
completely localised at one (random) site. The mechanism behind such
condensation remains similar in the more general models, but the nearest-
neighbour interaction allows the condensate to spread over a region whose
width scales with a power of the total volume $M$. As we will see, it is this
type of condensation which provides the basis for an explanation of the
droplet configuration in CDT based on the much simpler BIB models.

###  2.3 Landau theory of two-dimensional CDT

In two dimensions, only one of the $N_{k}$ variables is independent, e.g. the
number of triangles $N_{2}$. This is clear also from a continuum perspective,
as the curvature term is topological, hence at fixed topology the Einstein-
Hilbert action reduces to a simple volume term. The two-dimensional CDT model
Ambjorn:1998xu  at fixed topology has thus only one coupling, the bare
cosmological constant, while Newton’s constant plays no role. The
corresponding partition function has therefore the interpretation of
generating function for the number of triangulations of the type in figure 1
with fixed number of triangles $N$ and some given boundary conditions.
Typically one fixes also the number of time slices $T$, which in fact can be
seen as a new free variable, appearing thanks to the foliation.

The two-dimensional model of CDT can be solved exactly by various means
Ambjorn:1998xu ; DiFrancesco:1999em ; DiFrancesco:2000nn ; DiFrancesco:2001xur
, so it would seem pointless to look for a Landau theory. However, the latter
appears very naturally in the process of solving the model, and it is
therefore a useful example of top-down approach to a Landau free energy in a
quantum gravity model. In this case, the partition function for causal
triangulations with fixed $N$ and $T$ (denoted ($\mathcal{T}_{N,T}$) can be
written as

$\begin{split}Z_{\rm 2d-CDT}(N,T)&=\sum_{\mathcal{T}_{N,T}}1\\\
&=\sum_{l_{1}=1}^{N}...\sum_{l_{T}=1}^{N}\delta_{N_{2},2\sum_{i}l_{i}}\prod_{i=1}^{T}g(l_{i},l_{i+1})\\\
&\equiv\sum_{\\{l\\}}e^{-L(\\{l\\};N,T)}\,,\end{split}$ (10)

with $l_{T+1}=l_{1}$ (periodic boundary conditions in the time direction), and

$g(l_{i},l_{i+1})=\frac{(l_{i}+l_{i+1})!}{l_{i}!\,l_{i+1}!}\,,$ (11)

counting the number of triangulations of a strip with boundary lengths $l_{i}$
and $l_{i+1}$, with open boundary conditions.999It basically counts the number
of ways we can place $l_{i+1}$ balls in $l_{i}+1$ boxes, so that in this case
the reduced transfer matrix is itself a BIB model (with trivial reduced
transfer matrix).

In the second line of (10) we recognize a BIB model (compare to (6)), while in
the last line we defined the function $L(\\{l\\};N,T)$, which we interpret as
a Landau free energy. Indeed, comparing to (2) and (3), we see that in the
definition of $L(\\{l\\};N,T)$ we are summing over triangulations with a block
constraint, the blocks here being the spatial slices of each triangulation,
and the constraint fixing the number of edges in each block.101010Such a
nonlocal observable on a codimension-one slice is reminiscent of similar
quantities considered in anisotropic systems, e.g. the Ising model with
anisotropic competing next-to-nearest-neighbor interactions Murtazaev:2013 .

Notice that the Landau free energy $L(\\{l\\};N,T)$ contains a global
constraint, but we can get rid of it by switching to the grandcanonical
ensemble. That is, by summing over $N$ with a Boltzmann weight
$e^{-\kappa_{2}N}$, we obtain:

$L_{\rm
gc}(\\{l\\};T,\kappa_{2})=2\kappa_{2}\sum_{i}l_{i}-\sum_{i}\ln\left(g(l_{i},l_{i+1})\right)\,.$
(12)

It should be stressed that the second term is a purely entropic contribution,
with no lattice coupling constant.

Continuum limit. The model is exactly solvable Ambjorn:1998xu ;
DiFrancesco:1999em ; DiFrancesco:2000nn ; DiFrancesco:2001xur , and a
continuum limit can be taken on the solution. It is however instructive to see
what we can learn by first taking a continuum approximation, and then
minimizing the continuum Landau free energy. Using Stirling’s formula one
finds that, for large $l_{i}$ and $l_{i+1}$, and small
$(l_{i+1}-l_{i})/(l_{i}+l_{i+1})$,

$g(l_{i},l_{i+1})\sim
2^{l_{i}+l_{i+1}}e^{-\frac{(l_{i+1}-l_{i})^{2}}{l_{i}+l_{i+1}}}\,.$ (13)

It is easy to take the continuum limit of the exponent in (13), thus obtaining

$S_{\rm
2d-HL}[\ell]=\int_{-\frac{\tau}{2}}^{\frac{\tau}{2}}dt\,\frac{\dot{\ell}(t)^{2}}{4\ell(t)}\,.$
(14)

Such action111111Here and in the following we will generally refer to the
continuum limit of the Landau free energy as an (effective) action. can be
interpreted as an action for the length $\ell(t)$ of the slices, describing
two-dimensional HL gravity in proper-time gauge, as noticed in Ambjorn:2013joa
(see also the dedicated chapter of the Handbook of Quantum Gravity
Sato:2022ory ).

It is well known (see for example Mattei:2005cm  and references therein) that
in a path integral quantisation of gravity in the proper-time gauge we loose
the Hamiltonian constraint, unless we integrate over the total proper time.
Such integration is not performed in CDT when computing finite-time
propagators (by the definition of such observables), or when doing simulations
with periodic boundary conditions in time, for obvious practical reasons.
However, it should be stressed that in principle this only means that the
integral over time is postponed to a later stage. Indeed, in 1+1 dimensions
this integral can be carried out exactly, leading to a solution of the quantum
Hamiltonian constraint, or Wheeler-DeWitt equation Ambjorn:1998xu . What we
want to stress here is that in order to make contact between continuum models
and CDT results with fixed total time, one should not try to impose the
Hamiltonian constraint in the former. Bearing this in mind, we can try to see
what a semiclassical analysis of (14) tells us.

Solving the equations of motion of (14) with the constraint
$V_{2}=\int_{-\frac{\tau}{2}}^{\frac{\tau}{2}}dt\ell(t)$, and with periodic
boundary conditions in time, one finds either constant or oscillating
solutions. The constant solution is in fact unique, due to the volume
constraint: $\ell(t)=V_{2}/\tau$. The oscillating solutions form a discrete
set, due to the periodicity condition. Plugging these solutions into the
action (14), since $\ell(t)\geq 0$ and $\dot{\ell}(t)\neq 0$ (except at
isolated points) for the oscillating solution, it is obvious that the constant
solution has the least action (it evaluates to zero) and therefore it must
dominate the path integral. Note that the Hamiltonian constraint would fix the
amplitude independently of the total time $\tau$, and therefore we would have
periodic solutions only for special values of $\tau$. We could consider also
droplet configurations of the kind that we will introduce in higher
dimensions, but the constant solution would still dominate over them, as the
action (14) is non-negative, and equal to zero if and only if
$\dot{\ell}(t)=0$ for all $t$.

The dominance of the constant solution is in complete agreement with the Monte
Carlo snapshots from numerical simulations Ambjorn:1999gi , which, unlike the
higher-dimensional models, show no sign of translational symmetry breaking,
and average to a constant profile. This can also be seen from the analytical
solution Ambjorn:1998xu , which gives a constant $\langle\ell(t)\rangle\propto
1/\sqrt{\Lambda}$, where $\Lambda$ is the continuum cosmological constant in
the grand canonical ensemble.121212One might worry that in the case of
periodic boundary conditions a simple average will always lead to a
translational-invariant result. The usual way to deal to avoid blurring a
broken phase with an average over vacua is to introduce an explicit symmetry
breaking term in the model and check if the symmetry breaking persists when
this is continuously removed. For example, one could break explicitly the
translational symmetry by introducing initial and final boundaries of fixed
length, and check that for large $\tau$ the bulk is approximately constant,
but this would be a long and unnecessary parentheses here. As we will discuss
later, in numerical simulations there is a more practical way to effectively
select a single vacuum. In any case, one should also remember that in $1+1$
dimensions $\sqrt{\langle\ell^{2}\rangle-\langle\ell\rangle^{2}}\propto
1/\sqrt{\Lambda}$, i.e. fluctuations have the same magnitude as the average
configuration, thus hiding any possible classical behaviour of the latter.

The role of higher order terms in the Stirling approximation has been studied
in Ambjorn:2015gea , where it was shown that including enough matter fields
the leading order correction (a logarithmic potential term in the exponent of
(13)) becomes important and leads to a phase transition to a droplet phase,
similar to the one we will describe below.

Effective theory vs minisuperspace models. We should stress once more that
(14) is not a minisuperspace approximation for the quantization of the bare
theory. In fact, the bare action in two-dimensional CDT is trivial. Equation
(14) is instead obtained nonperturbatively as a purely entropic contribution
from the sum over all triangulations. This also shows as a proof of principle
that even if we start with a bare action of general relativity, the structure
of triangulations of our ensemble can determine a more general universality
class. Of course such a phenomenon should not come as a surprise: while it is
easy to define a diffeomorphism-invariant action (in the continuum), it is
highly nontrivial to define a regularized diffeomorphism-invariant measure for
the path integral, and therefore it is precisely there that a possible
breaking of gauge symmetries might occur.

##  3 Spatial volume dynamics in three-dimensional CDT

In this section we concentrate on the case $d=3$, for which very few
analytical results are known Ambjorn:2001br ; Benedetti:2007pp  because of the
difficulty in solving statistical models in dimensions higher than two.
Therefore, most of the current understanding of this case derives from Monte
Carlo simulations.

In the simulations one typically uses the topological constraints in order to
trade the variable $N_{1}$ for $N_{0}$, which is easier to keep track of, and
thus replace (5) by

$S(\mathcal{T}_{N})=\kappa_{3}N-\kappa_{0}N_{0}\,.$ (15)

Furthermore, as we mentioned, in the computer simulations we work at fixed
volume, and hence we replace (4) by the partition function for the canonical
ensemble,

$Z_{N}=\sum_{\mathcal{T}_{N}}\tfrac{1}{C(\mathcal{T}_{N})}\,e^{\kappa_{0}N_{0}}\,,$
(16)

where we have made use of the simple form of the action (15). Note that the
grand canonical partition function $Z$ is the discrete Laplace transform of
$Z_{N}$ with respect to $N$.

The expectation value of an observable $\mathcal{O}$ is calculated as

$\langle\mathcal{O}\rangle_{N}=\frac{1}{Z_{N}}\sum_{\mathcal{T}_{N}}\tfrac{1}{C(\mathcal{T}_{N})}\,e^{\kappa_{0}N_{0}}\mathcal{O}(\mathcal{T}_{N})\,,$
(17)

which is related to the grand canonical expectation value as a function of
$\kappa_{3}$ via

$\langle\mathcal{O}\rangle=\frac{1}{Z}\sum_{N}e^{-\kappa_{3}N}Z_{N}\langle\mathcal{O}\rangle_{N}\,.$
(18)

Simulations. Simulations were performed in Benedetti:2009ge ;
Benedetti:2014dra  using the Markov-chain Monte Carlo technique. An adaptation
of some previously existing code for the Monte Carlo simulations (used in
Ambjorn:2000dja ) was used for this purpose. The code generates a finite set
of sample configurations $\\{\mathcal{T}^{(1)},...,\mathcal{T}^{(M)}\\}$
according to the probability distribution
$\mathcal{P}(\mathcal{T})=\frac{1}{Z}e^{-S(\mathcal{T})}$. One then
approximates the expectation value of an observable by its arithmetic mean
across these samples:

$\langle\mathcal{O}\rangle\approx\frac{1}{M}\sum_{j=1}^{M}\mathcal{O}(\mathcal{T}^{(j)}).$
(19)

As usual in three-dimensional CDT simulations, the spacetime topology was
fixed to $S^{2}\times S^{1}$, i.e. spherical spatial sections and cyclical
time.131313Simulations with different boundary conditions in the time or
spatial directions have been performed in Cooperman:2013mma  and Budd:2013waa
, respectively, and the results are consistent with those reviewed here.
Values of $N$ up to a maximum of 200k$\,\equiv 2\times 10^{5}$ were studied,
although some errors for the larger values of $N$ are greater since less
configurations could be generated to be averaged over, within practical time
constraints. All simulations were carried out with coupling constant
$\kappa_{0}=5$, in the phase where previous evidence points to the emergence
of a well-behaved (three-dimensional) geometry. The total number of time-steps
was set to $T=96$.

###  3.1 The volume data

The observable we study here is the volume of the spatial slices, which in the
three-dimensional CDT model corresponds to the number of spatial triangles
$N_{2}(i)$ as a function of discrete time $i$. Because the triangulation is
connected we always have $N_{2}(i)>0$. Furthermore, because we restrict to
simplicial manifolds, the smallest triangulation of a two-sphere has four
triangles, giving

$N_{2}(i)\geq n_{\rm min}=4\,.$ (20)

Another possible observable is the volume of the spatial slices at half-
integer values of time, which amounts to a weighted sum of the number of (3,1)
and of (2,2) tetrahedra (see Fig. 2) between slices $i$ and $i+1$. We expect
that in the phase of extended geometry (where $N_{(3,1)}\sim 2N_{(2,2)}$) any
such differences in definitions of volume as a function of time should be
irrelevant in the continuum limit.

Figure 2: The three types of tetrahedral building blocks in three-dimensional
CDT, and their intersections with the $t+1/2$ plane.

Note that a triangle is always shared by two tetrahedra, so that, for the
number $N_{2}^{(s)}$ of spatial triangles, we have

$N_{2}^{(s)}=\sum_{i=1}^{T}N_{2}(i)=\frac{1}{2}N_{(3,1)}\,,$ (21)

which in the extended phase we expect to be roughly one third of the total
volume $N_{3}$. For the value of $\kappa_{0}$ we used, the distribution of
$N_{2}^{(s)}$ is very peaked (for $N_{3}=$100k the relative standard deviation
is only $0.2\%$, see Fig. 3), and we find that $N_{(2,2)}$ is just slightly
smaller than a third of the total volume.

Figure 3: The distribution of the total number of spatial triangles
$N_{2}^{(s)}$ (grouped in bins of size 10), for $N_{3}$=100k. The expectation
value is $\langle N_{2}^{(s)}\rangle=35026$, with standard deviation
$\sigma=85$.

In Fig. 4 we show the volume profile $N_{2}(i)$ from a snapshot of a Monte
Carlo simulation. One notices immediately a phenomenon of spontaneous
(translational) symmetry breaking: the Monte Carlo configuration shows a
condensation of the volume around a specific time. Averaging over Monte Carlo
configurations, the translational symmetry gets restored, but this is because
a blind average amounts to summing over the degenerate vacua of the broken
phase. As usual, in order to see spontaneous symmetry breaking we should
select only one vacuum. For example, in a spin system this is done by first
coupling the spins to an external magnetic field, which is then switched off.
In the CDT simulations, we select one vacuum by performing a recentering of
the Monte Carlo configurations. In practice, we have to find the center of
volume $t_{CV}(j)$ for each Monte Carlo configuration $j$, and we have to
shift time so that $t^{\prime}_{CV}(j)=T/2$ for every configuration in the new
time variable. We performed this operation following the method given in
Ambjorn:2008wc ; Gorlich:2011ga . Once the data are centered in this way, it
makes sense to study the average of $N_{2}(i)$. A plot of $\langle
N_{2}(i)\rangle$, together with fluctuations, is displayed in Fig. 5.

Figure 4: The spatial area $N_{2}(i)$ as a function of time, from a single
configuration taken at random in the data set of Monte Carlo simulations for
$N_{3}$=100k.  Figure 5: The mean spatial area as a function of discrete time,
for $N_{3}$=100k. The blue dots represent the mean area $\langle
N_{2}(i)\rangle$ of the volume data at time $i$, while the blue line is just
an interpolation curve. Error bars (estimated for example by the jackknife
method) are roughly of the size of the dot radii, or smaller, and thus we
chose to omit them for clarity (this is done throughout the paper). The dashed
red lines are interpolating curves for $\langle N_{2}(i)\rangle\pm\sigma/2$,
where $\sigma=\sqrt{\langle(N_{2}(i))^{2}\rangle-\langle N_{2}(i)\rangle^{2}}$
is the size of the fluctuations.

The volume profile has a characteristic extended part (typically referred to
as the blob or droplet), and a long flat tail (referred to as the stalk).
Within the latter, the spatial volume is very close to its kinematical
minimum, $\frac{1}{m}\sum_{i=1\ldots m}^{i\in{\rm stalk}}\langle
N_{2}(i)\rangle\equiv n_{s}\sim 10$, and it is independent of the total
volume. Most of the total volume is therefore concentrated in the blob. We
will discuss in the following sections how to explain this condensation
phenomenon, and which function best describes the volume profile.

###  3.2 The continuum limit

We conclude this section by discussing how the continuum limit $a\to 0$ is
investigated on the basis of the simulation data. All observables and
couplings in the simulations are given as dimensionless numbers, as such are
also all the quantities appearing in (15) and the definition of the model.
Length dimensions are introduced by multiplying the quantity of interest by
the appropriate power of the cutoff $a$, the length of spacelike
edges.141414Despite the Euclidean signature we use the words spacelike and
timelike to distinguish the orientation of edges with respect to the
foliation. For example, we can write $\tau=a\,\alpha\,T$ for the time
interval, where we have introduced a parameter $\alpha>0$ to allow a different
length of the timelike edges. Next, we can write
$V_{2}=\frac{\sqrt{3}}{4}a^{2}N_{2}$ for the volume of a slice (the numerical
prefactor being the area of an equilateral triangle of unit side),
$V_{3}=v_{(3,1)}(\alpha)a^{3}N_{(3,1)}+v_{(2,2)}(\alpha)a^{3}N_{(2,2)}$ for
the total volume, etc. Here, $v_{(3,1)}(\alpha)$ and $v_{(2,2)}(\alpha)$ stand
for the volume of the $(3,1)$ and $(2,2)$ simplices with spatial edges of
length one, and time edges of length $\alpha$ (they both coincide with the
equilateral tetrahedron for $\alpha=1$, with volume $v_{3}=1/6\sqrt{2}$, see
Ambjorn:2012jv ).

An essential part of the continuum limit procedure is to have $N\to\infty$ and
$a\to 0$ in such a way that the physical volume

$V\sim a^{d}N$ (22)

remains finite. This implies that when we multiply a dimensionless quantity by
$a^{n}$, in order to switch to its dimensional counterpart, in practice we
multiply it by $N^{-n/d}$. For example, in CDT we also demand that $\tau\sim
N^{-1/d}T$ stays finite.

In the thermodynamic limit we expect that, besides the total volume (22), also
other dimensional large-scale observables will become independent of the
cutoff $a$. Therefore, we expect to see scaling behavior when working with
simulations at sufficiently large $N$, i.e. we expect that an observable
$\langle\mathcal{O}(i)\rangle_{N}$, depending on the discrete time variable
$i=1,\ldots,T$, will satisfy for some fixed (continuum time) $t$:

$N^{-n/d}\langle\mathcal{O}(N^{1/d}t)\rangle_{N}=N^{\prime-n/d}\langle\mathcal{O}(N^{\prime
1/d}t)\rangle_{N^{\prime}}\,,$ (23)

where the value of the observable at non-integer time is to be intended from
an interpolating fit. The natural expectation would be that (22) and (23) hold
with $n$ given by the expected dimension of the observable, but this is not
guaranteed a priori and it is instead used as a check of the good classical
properties of the model, as we do in the following. Moreover, not all
quantities will show scaling: the dimensionless version of some quantity might
remain independent of $N$. The typical example is the correlation length
$l_{c}$, from
$\langle\mathcal{O}(i)\mathcal{O}(j)\rangle_{N}-\langle\mathcal{O}(i)\rangle_{N}\langle\mathcal{O}(j)\rangle_{N}\sim
e^{-|i-j|/l_{c}}$ (at large $|i-j|$); in order to obtain a finite dimensionful
correlation length $\ell_{c}=al_{c}\sim N^{-1/d}l_{c}$, one needs to tune the
couplings towards a critical point where $l_{c}$ diverges. We will not be
concerned with correlation lengths here, but we will see another example of
quantity that stays at the cutoff scale, i.e. the volume of the stalk in the
droplet phase.

Figure 6: The square root of the mean spatial area as a function of discrete
time (shifted so that the peak is at the origin). Both the mean area and the
time variable are rescaled with the appropriate power of the average total
volume $\tilde{N}_{2}^{(s)}=N_{2}^{(s)}-n_{s}T$, in order to display scaling.

Fig. 6 shows a scaling of the type (23). Here we have plotted $\langle
N_{2}(i)\rangle^{1/2}/\langle\tilde{N}_{2}^{(s)}\rangle^{1/3}$ as a function
of $i/\langle\tilde{N}_{2}^{(s)}\rangle^{1/3}$, for different data sets
corresponding to different total volumes $N$. Following the procedure used in
Ambjorn:2005qt , we used $\langle\tilde{N}_{2}^{(s)}\rangle=\langle
N_{2}^{(s)}\rangle-n_{s}T$, instead of $\langle N_{2}^{(s)}\rangle$ (or
$N_{3}$, which as we saw, is proportional to it) in the rescaling because we
know that the volume in the stalk does not scale. The plot clearly shows that
the superposition is extremely good inside the droplet, while in the stalk the
rescaled volume goes to zero for growing $N_{3}$.

Another dynamical quantity that goes to zero in the continuum is the size of
the fluctuations around the average, which we expect to be controlled by
Newton’s constant. Indeed, without any fine tuning of the dimensionless
couplings, all the couplings having positive (negative) length dimension will
go to zero (infinity) as $a\to 0$. As Newton’s constant has the dimension of
length for $d=3$, it is expected to go to zero in the naive continuum limit
(in four spacetime dimensions, where Newton’s constant has dimension of length
to the second power, this has been observed in Ambjorn:2008wc ). A finite
coupling in the continuum could perhaps be attained by fine tuning the bare
coupling to a second order phase transition.

##  4 Effective models

In this section we review how in four and (in more detail) three dimensions
the dynamics of spatial volumes in CDT can be effectively described by BIB
models. Minimization of the BIB model action, in accordance with its
interpretation as a Landau free energy, leads to a good qualitative
description of the full CDT phase diagram.

###  4.1 Four-dimensional CDT

In Bogacz:2012sa , Bogacz et al. studied a BIB model which is roughly a
discretised version of a minisuperspace model corresponding to four-
dimensional general relativity, and which was originally considered in
Ambjorn:2004pw  to explain the data from the simulations. It was found that
this very simple model can account for many of the observed features of CDT in
four dimensions, including its rich phase diagram. In particular, a droplet
phase was found, which has remarkable similarities to the extended phase of
CDT, although of course the comparison is limited to the behaviour of the
spatial volume against time.

The model is defined by the following reduced transfer matrix,

$g(m_{i},m_{i+1})=\exp\left(-c_{1}\frac{2(m_{i+1}-m_{i})^{2}}{m_{i}+m_{i+1}}-c_{2}\frac{m_{i}^{1/3}+m_{i+1}^{1/3}}{2}\right)\,.$
(24)

In the continuum, this corresponds to the following action for the volume
$V_{3}(t)$ of the 3-dimensional spatial slices,

$S_{\rm
4d-mini}[V_{3}]=\frac{1}{2G}\int_{-\frac{\tau}{2}}^{\frac{\tau}{2}}dt\,\left(c_{1}\frac{\dot{V}_{3}^{2}(t)}{V_{3}(t)}+c_{2}V_{3}^{1/3}(t)\right)\,,$
(25)

which is precisely of the type obtained from a minisuperspace reduction of
general relativity (where $c_{1}=1/\mathcal{N}$, and
$c_{2}=9(2\pi^{2})^{2/3}\mathcal{N}$, $\mathcal{N}$ being the lapse function).
This is the action that was conjectured from the very beginning by Ambjorn et
al. as an effective description for the extended part of the universe (or
blob) in their simulations Ambjorn:2004pw , a conjecture which was further
corroborated over the years Ambjorn:2005qt ; Ambjorn:2007jv ; Ambjorn:2008wc ;
Ambjorn:2011ph . The novelty in Bogacz:2012sa  was the suggestion that the
same effective action can explain much more than just the dynamics inside the
blob. Effectively, we can reinterpret the insight from Bogacz:2012sa  as the
observation that the BIB model with reduced transfer matrix (24) provides a
good guess for a Landau free energy, i.e. in a bottom-up approach.

Notice that in the minisuperspace action that one would derive from the
Einstein-Hilbert action the constant $c_{1}$ would be negative, while for a
meaningful use of (25) as an effective action, and from comparison to the
numerical data Ambjorn:2004pw , we must take it to be positive.151515The
choice looks of course irrelevant if all we have is this action, as we can
just change its overall sign, but this is not so if in the full Einstein-
Hilbert action we want also the spin-two modes to have the good sign. It would
thus seem that in CDT the conformal factor problem Gibbons:1978ac  affecting
Euclidean quantum gravity is solved, perhaps cured by nonperturbative
contributions from the functional measure, as suggested in Dasgupta:2001ue .
We will comment further on this below, arguing that in fact a simpler
explanation is possible.

Another important difference between the usual minisuperspace model of general
relativity and the BIB model, is that in the latter there is no analogue of
the lapse to be integrated in the partition function and neither there is a
summation/integration over $T$. As a consequence, there is no Hamiltonian
constraint to be imposed in the semiclassical analysis. Above, we emphasised
that the same situation should be expected in CDT, where the distance between
one spatial slice and the next is constant and the total time extension of the
universe is fixed in all simulations to date. This is also supported by the
strong evidence from numerical simulations that the BIB model is a good
effective description for CDT.

The equations of motion derived by varying (25) with respect to $V_{3}(t)$ are

$c_{1}\left(\left(\frac{\dot{V}_{3}}{V_{3}}\right)^{2}-2\frac{\ddot{V}_{3}}{V_{3}}\right)+\frac{c_{2}}{3}\frac{1}{V_{3}^{2/3}}-\Lambda=0\,,$
(26)

where the cosmological constant $\Lambda$ is introduced as a Lagrange
multiplier, to be fixed by imposing the volume constraint. If we were to
impose also the Hamiltonian constraint $H\equiv
c_{1}\dot{V}_{3}^{2}(t)/V_{3}(t)-c_{2}V_{3}^{1/3}(t)+\Lambda V_{3}(t)=0$, the
combined system of equations would reduce to a first order differential
equation (by deriving the Hamiltonian constraint with respect to time, and
eliminating $\ddot{V}_{3}(t)$ between the two equations). As such, its
solutions would have only one free integration constant, which could be fixed
for example by demanding that the maximum of $V_{3}(t)$ be at $t=0$. The
solution would then be the “$\cos^{3}$” solution discussed by Ambjorn et al.
in Ambjorn:2007jv ; Ambjorn:2008wc . However, without the Hamiltonian
constraint the equation remains second-order, and thus there is one more free
parameter, which allows us to perform a minimisation of the on-shell action.

For such a minimization one does not need to restrict to class $C^{2}(S^{1})$
functions, since for a well defined action (25) it is sufficient that
$V_{3}(t)\in C^{1}(S^{1})$. This allows the authors of Bogacz:2012sa  to
consider droplet configurations that are not classical solutions, but that are
expected on the basis of simulations and heuristic arguments.161616One should
bear in mind that this sort of analysis is not aimed at reproducing the
detailed profile of CDT at the junction between blob and stalk. In the
junction region of the droplet we expect the effect of subleading terms in the
action to be non-negligible. Furthermore, the time-interval mesh in the CDT
simulations is not fine enough to reveal much about the smoothness of the
average profile at such junction. Using
$V_{4}=\int_{-\frac{\tau}{2}}^{\frac{\tau}{2}}dtV_{3}(t)$ for the total volume
in the continuum, they obtain the following expression for the dominant
contribution to the path integral in a particular region of the the phase
diagram:171717As we will see in more detail for the three-dimensional case, a
number of parametric conditions must be satisfied for such configuration to
dominate.

$\bar{V}_{3}(t)=\begin{cases}\frac{3\omega V_{4}}{4}\cos^{3}(\omega
t)\,,&\text{for }t\in[-\frac{\pi}{2\omega},+\frac{\pi}{2\omega}]\,,\\\
0\,,&\text{for
}t\in[-\frac{\tau}{2},-\frac{\pi}{2\omega})\cup(+\frac{\pi}{2\omega},+\frac{\tau}{2}]\,,\end{cases}$
(27)

where

$\omega=\frac{\sqrt{2}}{3V_{4}^{1/4}}\left(\frac{c_{2}}{c_{1}}\right)^{3/8}\,.$
(28)

Notice that $V_{3}(t)=0$ obviously minimises the action (25) for positive
$c_{1}$ and $c_{2}$. However, alone it would fail to satisfy the volume
constraint, while a balance between the zero and the “$\cos^{3}$” solutions
wins the energy balance, resulting in a condensation. Interestingly, the blob
part of (27), together with (28), corresponds to the solution obtained by
imposing also the Hamiltonian constraint, but this seems a mere coincidence.

The crucial point to be made here is that the presence of a potential term in
(25) allows non-constant configurations such as (27) to dominate the path
integral. Thus, the conclusions derived in this case are qualitatively
different from those derived from (14) in the two-dimensional case.

The result is very interesting because it shows how the reduced model in four
dimensions not only reproduces the extended part of the universe, but also its
stalk, and it gives a prediction for their relative time extension.

It is also worth noticing that from the Einstein-Hilbert action the ratio
$c_{2}/c_{1}$ would be fixed, thus leaving us with no parameters for a fit to
the CDT simulations. Since the width of the universe (for fixed $V_{4}$)
depends on the bare coupling $\kappa_{0}$ Ambjorn:2008wc , this is another
indication that the effective action describing the volume dynamics is
different form what would be expected from general relativity. We will now
argue that such conclusion is even stronger in three dimensions.

###  4.2 Three-dimensional CDT

In three dimensions the minisuperspace reduction of the Einstein-Hilbert
action contains no potential term, i.e. the action for the spatial areas in
the proper-time gauge is exactly of the same form as (14), but with the length
$L(t)$ replaced by the area of the spatial slices $V_{2}(t)$. As in the two-
dimensional case, in the presence of a volume constraint $V_{3}=\int
dtV_{2}(t)$ one finds oscillating solutions of the equations of motion, with a
volume profile $V_{2}(t)\sim\cos^{2}(\omega t)$ which is compatible with the
one observed inside the blob of Fig. 5. For this reason, such action has been
suggested as an effective action for CDT Ambjorn:2000dja ; Ambjorn:2002nu
(see also Cooperman:2013mma ). However, as pointed out in Benedetti:2014dra ,
in the light of the results on BIB models for two and four-dimensional CDT,
the similarity of the general relativistic minisuperspace action in three
dimensions to (14) immediately raises the question of how such an effective
action could ever explain the important differences between two and three-
dimensional CDT. Given that the two-dimensional case reduces exactly to a BIB
model, and the four-dimensional case is well described (in its spatial volume
dynamics) by a BIB model, we would expect a similar model to give also a good
approximation for the three-dimensional model.

Of course an important difference between two and three dimensions is in the
scaling of dimensionful quantities. Most importantly, Newton’s constant is
dimensionless for $d=2$ while it has dimension of length for $d=3$. As a
consequence, in the (naive, not fine-tuned) continuum limit, Newton’s constant
(and with it the fluctuations around the average volume of the slices) scales
to zero in the latter case, while it stays constant in the former. In two
dimensions, as there are no other scales besides the cosmological one, the
size of the fluctuations in the continuum limit is as large as the expectation
value, thus blurring any classical behaviour. On the contrary, in three
dimensions the fluctuations go to zero and the classical (mean field)
behaviour should dominate.

Regardless of how the fluctuations behave, it turns out that the action
derived from the Einstein-Hilbert action fails in reproducing the CDT results
in an important way. Repeating the analysis of Bogacz et al. for three
dimensions we simply have to set $c_{2}=0$ in the previous subsection, or just
recall what we have said about the two-dimensional case. From (28) we then
find that $\omega=0$, and the width of the droplet diverges. However, (28)
does not hold in this case, as it would violate the condition
$\pi/\omega<\tau$ which is to be assumed in (27). On the other hand we have
already explained what happens in the two-dimensional case, and going to three
dimensions we simply have to replace $V_{2}\to V_{3}$ and $\ell(t)\to
V_{2}(t)$. The droplet solution which minimises the action is obtained for
$\omega=\pi/\tau$. However, it is easily checked that in such a case the
action is strictly positive, while for $V_{2}(t)=V_{3}/\tau$ the action
vanishes. We conclude that a BIB model derived from the three-dimensional
Einstein-Hilbert action would predict a constant average profile for the two-
dimensional volumes. This is also supported by the numerical simulation of
Bogacz:2012sa , as for $c_{2}=0$ and $c_{1}>0$ the model defined by (24) lies
in the correlated fluid phase, not the droplet phase.

We are left with the challenge of explaining the droplet condensation of
three-dimensional CDT as a condensation of a BIB model. This also provides us
with an extraordinary opportunity to test corrections to the effective action
of Ambjorn:2000dja ; Ambjorn:2002nu . In higher dimensions, such corrections
are expected to be subdominant with respect to the the linear spatial
curvature term coming from the Einstein-Hilbert action (the one multiplied by
$c_{2}$ in (25)). Fortunately, however, the $d=3$ case is an exception to
this, because this linear curvature term results in an irrelevant constant
term for the potential, and hence drops out of the story, making higher order
corrections relevant.

In view of that, and inspired by the reasonings outlined in the introduction,
as well as by the fact that there is presently no evidence for the presence of
higher-order time derivatives in the effective action Ambjorn:2011ph , in
Benedetti:2014dra  we proposed an alternative effective theory for the spatial
areas, one derived from HL gravity.

Starting point: One sets off with the most generic, projectable, $z=2$ action
for Hořava-Lifshitz gravity in three dimensions Horava:2008ih :

$S_{\rm 3d-HL}=\frac{1}{16\pi
G}\int\textrm{d}t\,\textrm{d}^{2}x\;\mathcal{N}\sqrt{g}\left\\{(K_{ij}K^{ij}-\lambda
K^{2})+2\Lambda-bR-\gamma R^{2}\right\\}\,.$ (29)

As discussed in the introduction, HL gravity describes a class of metric
theories supporting a preferred foliation. Thus, the space-time diffeomorphism
symmetry is broken down to foliation-preserving diffeomorphisms.181818Such
foliation-preserving diffeomorphisms are described by time-reparameterizations
and spatial diffeomorphisms of the form:  $t\rightarrow
t+\xi^{0}(t)\qquad\textrm{and}\qquad x^{i}\rightarrow x^{i}+\xi^{i}(t,x)\,,$
(30) where $(t,x^{i})$ are co-ordinates in an atlas of charts adapted to the
foliation.

In (29), the action is presented in terms of ADM variables: $\mathcal{N}$ is
the lapse function, $g$ is the determinant of the spatial metric, $R$ is its
Ricci scalar, $K_{ij}$ is the extrinsic curvature associated to the leaves of
the foliation, while $K$ is its trace.

The action contains the familiar parameter pair $(G,\Lambda)$, corresponding
to Newton’s constant and the cosmological constant, respectively. The
parameter $\sigma=\pm 1$ neatly encapsulates some metric signature
information. Meanwhile $\lambda$, $b$ and $\gamma$ characterize the deviation
from full diffeomorphism invariance. Indeed, for $\lambda=b=1$ and $\gamma=0$,
one recovers the Einstein-Hilbert Lagrangian in Euclidean signature:
$2\Lambda-\mathcal{R}$, where $\mathcal{R}$ is the spacetime Ricci
tensor.191919Notice that we have changed the overall sign of the action with
respect to Benedetti:2014dra ; Benedetti:2016rwo , because in retrospective we
find the present choice more clear. Ultimately the two ways of presenting this
motivational steps are equivalent as they lead to the same effective action
(thanks also to the freedom we have in choosing $\lambda$ in HL gravity).

The theory is said to be projectable if one imposes at the outset that the
lapse function is spatially constant: $\mathcal{N}=\mathcal{N}(t)$.

The $z$-exponent refers to half the maximal order of spatial derivatives
appearing in the action. Thus for $z=2$, it may contain at most four spatial
derivatives, permitting the inclusion of the $R^{2}$ term.

We consider spacetimes with topology $S^{2}\times S^{1}$, that is, spherical
spatial slices and a compactified time. To implement this, we impose periodic
boundary conditions in time, with period $\tau$.

Mini-superspace reduction: We perform a mini-superspace reduction to constant
lapse, vanishing shift vector (hidden so far in the extrinsic curvature) and
spatial metric $g_{ij}=\phi^{2}(t)\hat{g}_{ij}$, where $\hat{g}_{ij}$ is the
standard metric on the unit sphere. And in order to compare to CDT in the
canonical ensemble, we replace the cosmological term as:

$\frac{2\Lambda}{16\pi
G}\int\textrm{d}t\,\textrm{d}^{2}x\;\mathcal{N}\sqrt{g}\to\frac{2\Lambda}{16\pi
G}\left(V_{3}-\int\textrm{d}t\,\textrm{d}^{2}x\;\mathcal{N}\sqrt{g}\right)\,,$
(31)

and treat $\Lambda$ as a Lagrange multiplier.

With a redefinition of parameters,202020We define:
$\kappa^{2}=\frac{NG}{1-2\lambda},\qquad\omega^{2}=\frac{N^{2}\Lambda}{1-2\lambda}\,,\qquad
b^{\prime}=-\frac{\mathcal{N}^{2}b}{1-2\lambda}\,,\qquad\xi=\frac{2\mathcal{N}^{2}\gamma}{1-2\lambda}\,.$
(32) the action becomes:

$S_{\textrm{3d-HL-
mini}}[\phi]=\frac{1}{2\kappa^{2}}\int_{-\frac{\tau}{2}}^{\frac{\tau}{2}}\textrm{d}t\left\\{\dot{\phi}^{2}-\omega^{2}\phi^{2}+b^{\prime}-\frac{\xi}{\phi^{2}}\right\\}\,.$
(33)

The remaining field $\phi(t)$ is a time-dependent scale factor determining the
area of the spatial slice at time $t$: $V_{2}(t)=4\pi\phi^{2}(t)$. The
constant $b^{\prime}$ term is clearly irrelevant for the problem of
minimization of the action, and this can be traced back to the topological
nature of the $R$ term in (29) for the projectable case.

The good sign of the action (a positive definite kinetic term) corresponds to
$\kappa^{2}>0$, i.e. $\lambda<1/2$. However, in order to satisfy the periodic
boundary conditions and to get real oscillating solutions, it turns out that
one needs to take $\omega^{2}>0$ and $\xi>0$ (that is, an $R^{2}$ term with
the bad sign), thus leading to a potential which is unbounded from below. Both
sources of unboundedness are cured by (again CDT-inspired) constraints on the
configuration space:

  1. 1.

The unbounded $\phi^{2}$ term is tamed by the the fact that we are at fixed
total volume, i.e. in the canonical ensemble. As the original $\Lambda$ above,
$\omega^{2}$ should be treated as a Lagrange multiplier. The Euler-Lagrange
equations for $\phi$ are unchanged, while variation with respect to the
Lagrange multiplier imposes the constraint:

$\mathcal{V}\equiv 4\pi
N\int_{-\frac{\tau}{2}}^{\frac{\tau}{2}}\textrm{d}t\,\phi^{2}(t)-V_{3}=0\,,$
(34)

that is, it fixes the 3-volume, and thus there is no unboundedness problem
from the sign of the $\omega^{2}$ term.

  2. 2.

The potential unboundedness stemming from the $1/\phi^{2}$ term is instead
avoided by imposing a minimal spatial volume constraint at the outset, such
as:212121Note that the compatibility of (34) with (35) requires
$\epsilon<\sqrt{V_{3}/(4\pi N\tau)}$.

$\phi(t)>\epsilon\,,\quad\forall t\,.$ (35)

This can be thought as a regularization of the theory, and as discussed in
Benedetti:2014dra , there are possible scaling limits to safely let
$\epsilon\to 0$.

Alternatively, we could think of the $1/\phi^{2}$ term in (33) as a truncation
of the large-$\phi$ expansion of a potential that is not singular at $\phi=0$,
e.g. $1/(\epsilon^{2}+\phi^{2})$.

Note that the constant lapse $\mathcal{N}$ has been neatly hidden away in
(33). This is rather innocuous in a classical setting. But in principle, it is
a degree of freedom that should be integrated over in a quantum regime leading
to the imposition of a Hamiltonian constraint. Following Benedetti:2014dra ,
we shall not consider this scenario, rather setting $\mathcal{N}=1$ from here
on. As we discussed above, that is appropriate when wishing to compare to CDT
with fixed total time.

In order to rewrite, in analogy to the two and four-dimensional cases, the
effective action in terms of area of the slices, we can simply change
variables to:

$V_{2}=4\pi\phi^{2}\,,$ (36)

which comes from having assumed the slices to be 2-spheres of radius $\phi$.
Leaving aside the constant term and the harmonic term, which is part of the
volume constraint, one can rewrite (33) as:

$\tilde{S}_{\textrm{3d-HL-
mini}}[\phi]\equiv\frac{1}{2\kappa^{2}}\int_{-\frac{\tau}{2}}^{\frac{\tau}{2}}\textrm{d}t\left\\{\dot{\phi}^{2}-\frac{\xi}{\phi^{2}}\right\\}=\frac{1}{2\kappa^{2}}\int_{-\frac{\tau}{2}}^{\frac{\tau}{2}}\textrm{d}t\left\\{\frac{1}{16\pi}\frac{\dot{V_{2}}^{2}}{V_{2}}-\frac{4\pi\xi}{V_{2}}\right\\}\,\,.$
(37)

BIB model and its phases: The action (37) is the form of the action that can
most naturally be translated into a discrete BIB model of the type (6). The
obvious discretization amounts to choosing the following reduced transfer
matrix:

$g(m_{j},m_{j+1})=\exp\left\\{-\frac{2(m_{j+1}-m_{j})^{2}}{m_{j+1}+m_{j}}b_{1}+\frac{2}{m_{j+1}+m_{j}}b_{2}\right\\}\,,$
(38)

where $b_{1}$ and $b_{2}$ are parameters that can be related to the continuous
ones, and the factors 2 remind us that we have chosen to write in the
denominators the arithmetic mean of $m_{j}$ and $m_{j+1}$ (other choices, even
non-symmetric, are possible, but are expected to be irrelevant in the
continuum limit Bogacz:2012sa ).

Figure 7:  Phase diagram for the BIB model with reduced transfer matrix (38),
with $T=80$ and $M=4000$: droplet (red triangles), localized (yellow squares),
antiferromagnetic (blue pentagons), correlated fluid (green hexagons), and
uncorrelated fluid (origin).

We performed Monte Carlo simulations of this BIB model in Benedetti:2016rwo ,
and we found evidence for five phases, whose location in the $(b_{1},b_{2})$
plane is illustrated in figure 7. Typical configurations for each phase are
presented in figure 8, and they are characterized as follows (see
Benedetti:2016rwo  for more details):

  * •

Droplet phase: This phase occurs for $b_{1}>0$, $b_{2}>b_{2,crit}(b_{1})$. For
a typical configuration, the majority of volume condenses onto a contiguous
subset of time slices of width greater than one. The remainder of the volume
forms a stalk with each constituent time slice possessing minimal volume.

  * •

Localized phase: This phase exists for $b_{1}<0$, $b_{2}>b_{2,crit}(b_{1})$.
For a typical configuration, all volume resides on one time slice, while all
other times possess minimal volume.

  * •

Antiferromagnetic fluid phase: This phase exists for $b_{1}<0$,
$b_{2}<b_{2,crit}(b_{1})$. A typical configuration consists of alternating
peaks and troughs, where the troughs possess minimal volume.

  * •

Correlated fluid phase: This phase exists for $b_{1}>0$,
$b_{2}<b_{2,crit}(b_{1})$. A typical configuration consists of the volume
approximately evenly allotted to the time slices, with relatively small
fluctuations about the mean volume.

  * •

Uncorrelated fluid phase: This phase exists near the origin, i.e. for
$b_{1}\sim b_{2}\sim 0$, and probably only at the origin in the thermodynamic
limit. It is a phase dominated by the entropy of configurations rather than by
the action, which indeed vanishes at the origin, where the model is easily
solved Bogacz:2012sa .

Figure 8:  Typical configurations for the various phases (left-to-right):
droplet, localized, antiferromagnetic, correlated fluid, uncorrelated fluid.

Figure 9:  Mean value $\langle m_{i}\rangle$ as a function of $i$, for non-
centred samples in the correlated phase and centred samples in the droplet
phase.

Two of the phases above are akin to the two phases appearing in three-
dimensional CDT, namely the droplet and the correlated phase. Clearly the
former (see its averaged profile in Fig. 9) should correspond to the droplet
configuration of Fig. 5, and we will get back to their comparison below.
Notice also that although the typical configuration in the correlated phase
shows relatively large fluctuation at our value of $M$ the ensemble average in
figure 9 shows that they neatly average to a constant configuration. The
antiferromagnetic and localized phases appear at negative kinetic term, and
for this reason we will not discuss them further.

###  4.3 Analysis of the model

We can reverse the derivation of the BIB model and go back to the continuum,
in order to look for the configurations that minimize the Landau free energy.
First off, we notice that a three-dimensional interpretation of the BIB model
is justified by the scaling observed in the droplet phase, see Fig. 10, which
is of the same type as the scaling in the three-dimensional CDT model, see
Fig. 6.

Figure 10:  Scaling of averaged droplet with respect to volume $M$ (between 4k
and 64k) at fixed $T=80$. We notice that rescaling both $\sqrt{\langle
m_{i}\rangle}$ and $i$ with $\sqrt[3]{M}$, we obtain a droplet whose shape is
essentially independent of $M$.

Let us introduce a lattice spacing $a$, and define a continuous time variable
$t=aj$, with period $\tau=aT$. Furthermore, we define a continuous volume
variable $V_{3}=v_{3}a^{3}M$, as well as interpret the number of balls in a
given box $j$ as the two-dimensional volume of a slice at time $t$:
$V_{2}=v_{2}a^{2}m$. Here, $v_{2}$ and $v_{3}$ are two arbitrary constants,
that could be related to the details of the three-dimensional building blocks
of CDT.

In the limit $a\to 0$, with $\tau$ and $V_{3}$ fixed, the exponent in (38)
becomes the action

$S[V_{2}]=\frac{1}{a}\int_{0}^{\tau}dt\left\\{\frac{b_{1}}{v_{2}}\frac{\dot{V_{2}}^{2}}{V_{2}}-a^{2}b_{2}v_{2}\frac{1}{V_{2}}\right\\}\,,$
(39)

while the constraint on the total number of balls becomes the following volume
constraint:

$\int_{0}^{\tau}dtV_{2}=\frac{v_{2}}{v_{3}}V_{3}\,.$ (40)

It is then natural to take $v_{2}=v_{3}\equiv v$.

The presence of the $1/a$ factor in front of the action (39) suggests that in
the continuum limit the minimization of the action should provide a trustable
approach to understanding the phase diagram of the BIB model. Note that it
would be wrong to discard the potential term because of the $a^{2}$
multiplying it, because $V_{2}$ itself can be of order $a^{2}$.

A comparison of the actions (37) and (39) leads to the identifications:

$\frac{1}{32\pi\kappa^{2}}=\frac{b_{1}}{av}\,,\;\;\;\frac{2\pi\xi}{\kappa^{2}}=ab_{2}v\,,$
(41)

or:

$\frac{b_{2}}{b_{1}}=\frac{64\pi^{2}\xi}{v^{2}a^{2}}\,.$ (42)

One can eliminate any reference to the cutoff scale $a$ by looking only at
dimensionless ratios of the parameters of the model. For example, using
$va^{2}m_{min}=4\pi\epsilon^{2}$, one can write:

$\frac{v\,b_{2}}{b_{1}m_{min}}=\frac{16\pi\xi}{\epsilon^{2}}\,.$ (43)

In this way we can translate predictions from the continuum to the discrete.

For the continuum analysis, we will stick to the action in the $\phi$
parametrization, written as in (33) for deriving the equations of motion, and
as in (37) for evaluating it on-shell (the $\omega^{2}$ term being part of the
constraint it does not contribute to the on-shell action). The equations of
motion derived from (33) are of Ermakov-Pinney type, i.e.:

$\ddot{\phi}+\omega^{2}\phi-\frac{\xi}{\phi^{3}}=0\,,$ (44)

and their solution is:

$\phi_{0}(t)=\frac{1}{\omega A}\sqrt{(\omega^{2}A^{4}-\xi)\cos^{2}(\omega
t+\psi)+\xi}\,,$ (45)

where $A$ and $\psi$ are integration constants. Shifting the maximum of the
curve to $t=0$ fixes $\psi=0$, and the maximal value $\phi_{0}(0)=A$ is then
fixed by initial conditions. Meanwhile, the constraint $\mathcal{V}$ suffices
to determine $\omega$ in terms of $\\{V_{3},A,\xi\\}$. However, notice that
these solutions oscillate with period $\frac{\pi}{\omega}$, and therefore in
order to satisfy periodic boundary conditions at $t=\pm\tau/2$, $\omega$ must
also satisfy $\frac{\pi}{\omega}=\frac{\tau}{n}$ for some positive integer
$n$. As a consequence, the space of solutions forms a discrete set.

These solutions never reach zero for $\xi>0$. This has ramifications for the
space-time topology. In particular, it indicates the non-occurence of
potential conical singularities, rather there is a minimal throat, at which
the space-time bounces.

Besides the oscillating solutions, there is also a constant solution:

$\bar{\phi}_{0}(t)=\sqrt{\frac{V_{3}}{4\pi\tau}}\,,$ (46)

which is a special case of equation (45) with $A=\xi^{1/4}/\sqrt{\omega}$, and
$\omega=4\pi\tau\sqrt{\xi}/V_{3}$ fixed by the volume constraint. This
solution has special significance, since in comparison to the other local
extrema, it has least action:222222The on-shell action evaluates to:
$\begin{array}[]{rcl}\tilde{S}_{\textrm{3d-HL-
mini}}[\phi_{0}]&=&\dfrac{n\pi}{8\kappa^{2}}\left(\dfrac{nV_{3}}{N\tau^{2}}-8\sqrt{\xi}\right)\,,\\\\[14.22636pt]
\tilde{S}_{\textrm{3d-HL-mini}}[\bar{\phi}_{0}]&=&-\dfrac{2\pi
N\tau^{2}\xi}{\kappa^{2}V_{3}}\;.\end{array}$ (47)

$\tilde{S}_{\textrm{3d-HL-mini}}[\bar{\phi}_{0}]\leq\tilde{S}_{\textrm{3d-HL-
mini}}[\phi_{0}]\,.$ (48)

However, a standard second derivative test for constrained local extrema
reveals that even the constant solution might not be a stable minimum for
$\xi>0$ sufficiently large. Indeed the Hessian of (33) reads

$\mathcal{H}(t,t^{\prime})=\left(-\partial_{t}^{2}-\omega^{2}-\frac{3\xi}{\phi(t)^{4}}\right)\delta(t-t^{\prime})\,,$
(49)

and on the constant configuration it is diagonalized by the Fourier basis; as
the tangent space to the constraint surface (34) at the constant solution is
spanned by all the non-constant Fourier modes, we simply find that the
condition for such a solution to be a minimum is

$\left(\frac{2\pi
n}{\tau}\right)^{2}-\omega^{2}-\frac{3\xi}{\bar{\phi}_{0}^{4}}=\left(\frac{2\pi
n}{\tau}\right)^{2}-\frac{4\xi(4\pi\tau)^{2}}{V_{3}{}^{2}}>0\,,$ (50)

which is violated (at least by the $n=1$ mode) as soon as
$\xi>V_{3}{}^{2}/(16\tau^{4})$.

Keeping in mind that droplet-stalk configurations are of interest to us, we
notice that the solutions (45) and (46) above capture fairly well the bulk of
the droplet and the stalk, respectively. Therefore, with a touch of naiveté,
let us graft the two together, and construct a configuration of the form:

$\tilde{\phi}(t)=\left\\{\begin{array}[]{lcl}\dfrac{1}{\omega
A}\sqrt{(\omega^{2}A^{4}-\xi)\cos^{2}(\omega
t)+\xi}&\textrm{for}&t\in[-\frac{\pi}{2\omega},+\frac{\pi}{2\omega}]\,,\\\\[8.5359pt]
\dfrac{\sqrt{\xi}}{\omega
A}&\textrm{for}&t\in[-\frac{\tau}{2},-\frac{\pi}{2\omega})\cup(+\frac{\pi}{2\omega},+\frac{\tau}{2}]\,,\\\
\end{array}\right.$ (51)

where one period of a configuration of the form (45) is confined to a
subinterval of length $\pi/\omega<\tau$, while a constant configuration is
attached in the remainder.

Such configurations are unusual for at least two reasons. First, they do not
in general correspond to solutions of the Euler-Lagrange equations, except in
the degenerate cases
$A^{2}=(V_{3}+\sqrt{V_{3}^{2}-16\tau^{4}\xi})/(4\pi\tau)$, for which
$\omega=\pi/\tau$, i.e. there is no stalk, and
$A^{2}=\xi^{1/2}/\omega=V_{3}/(4\pi\tau)$, for which
$\tilde{\phi}=\bar{\phi}_{0}$, i.e. there is no droplet. Second, they are only
$C^{1}([-\tau/2,\tau/2])$, the second derivative being discontinuous at
$t=\pm\frac{\pi}{2\omega}$. Nevertheless, we will now argue that they are
relevant for the analysis of the model.

Local extrema vs absolute minima: The partition function of the system, in the
continuum, reads:

$Z_{\textrm{3d-HL-
mini}}=\int_{\phi(t)>\epsilon}\mathcal{D}\phi(t)\,\delta(\mathcal{V})\,\exp\left\\{-\frac{1}{2\kappa^{2}}\int_{-\frac{\tau}{2}}^{\frac{\tau}{2}}\textrm{d}t\left[\dot{\phi}^{2}-\frac{\xi}{\phi^{2}}\right]\right\\}\,.$
(52)

In the limit $\kappa\to 0$ we expect the partition function (and the
observables) to be dominated by those configurations that minimize the action.
Under the assumption that the space of field configurations is such that the
action is at least differentiable (in the functional sense), there are two
possible ways minima can arise: local extrema (i.e. solutions of the equations
of motion) or minima lying on the boundary of configuration space. In the case
that local extrema provide the dominant configurations for the path integral,
this would imply dominance of the constant solution, because as we have seen
this has the least action among all the solutions. On the other hand, if a
configuration lying on the boundary of configuration space provides an
absolute minimum, then we can have a non-constant profile. Such a possibility
arises quite naturally in situations where an action unbounded from below is
tamed by constraints. In our case, as we have already pointed out, the
unboundedness of the action (37) at $\xi>0$ is tamed by the constraint (35),
which introduces a boundary in configuration space. Therefore, absolute minima
that are not local extrema can be expected. Indeed, it was argued in
Benedetti:2014dra  that this is precisely what gives rise to the droplet
phase, with a profile of the form (51) dominating the partition function.

The rationale behind (51) is based on a balance between kinetic and potential
terms in the action (created by their relative sign difference). If one looks
at the kinetic term alone (at positive $b_{1}$, i.e. $\kappa^{2}$), then it is
obvious that the minimizing configuration is a constant one,
$\dot{\phi}(t)=0$. The volume constraint then fixes the value of $\phi(t)$ to
(46). On the other hand, if one looks only at the potential term, then
minimization would seem to push to a configuration with $\phi(t)=0$, leading
to a singular value of the action which would clearly dominate over any other
configuration (as long as $\dot{\phi}(t)$ stays finite). This is of course the
instability caused by the unboundedness of the action, and as already
repeatedly stressed it is cured by the constraint (35), as a consequence of
which the potential actually favors the configuration $\phi(t)=\epsilon$. In
fact, the volume constraint forbids a solution with $\phi(t)=\epsilon$
everywhere, but clearly if it were just for the potential term a delta-
function configuration
$V_{2}(t)=(V_{3}-4\pi\epsilon^{2}\tau)\,\delta(t-t_{0})+4\pi\epsilon^{2}$
would do the job,232323Our choice of working with the variable $\phi$ rather
than $V_{2}$ would not be the best in this case due to the need of taking the
square root of the delta function. i.e. a localized configuration as seen in
the top-right of Fig. 8, which indeed is found in particular at $b_{2}>0$ and
$b_{1}=0$.

Therefore, for $\epsilon$ small enough, one sees a competition between the
kinetic term, which favors the constant configuration, and the potential,
which favors the delta-function configuration. What (51) envisages is a
situation where the latter essentially wins, with the constant part being
fixed at the minimal slice volume for the optimal choice of $A$, but with the
kinetic term smoothing out the delta function, to a shape determined by local
minimization of the action.242424Notice that for negative $b_{1}$ the kinetic
term looses its smoothing effect and we might expect the delta function
configuration to dominate. This is indeed what we find in the simulations.

Phase transition: The discussion above was intended to provide a simple
intuition of why a configuration such as (51) might dominate the path
integral, but of course one has to explicitly check whether that is the case.
The presence of several parameters and the fact that (34) leads to a cubic
equation for $\omega$ complicate things, and in Benedetti:2014dra  only a
perturbative analysis for small $\epsilon$ and $\xi$ was presented.

Indeed plugging (51) into (34) we find:

$V_{3}=4\pi\left(\frac{\pi(\xi+A^{4}\omega^{2})}{2A^{2}\omega^{3}}+\frac{\xi}{A^{2}\omega^{2}}\left(\tau-\frac{\pi}{\omega}\right)\right)\,,$
(53)

which, multiplied by $\omega^{3}$ (non-zero otherwise we would have the
constant solution again), gives us a cubic equation for $\omega$. It is
convenient to rewrite the latter in terms of $\epsilon$ rather than $A$,
taking $A=\sqrt{\xi}/(\epsilon\omega)$, because we expect the constant part to
reach its minimal allowed value (from the considerations above, and from the
perturbative analysis of Benedetti:2014dra ). We arrive at:

$(4\pi\tau\epsilon^{4}-V_{3}\epsilon^{2})\omega^{3}-2\pi^{2}\epsilon^{4}\omega^{2}+2\pi^{2}\xi=0\,.$
(54)

Although solvable, the general solution to such an equation is not very
enlightening due to the presence of several parameters. We can gain some
insight with some further assumptions on the nature of the roots. The
discriminant of the cubic equation is:

$-108\pi^{4}\xi^{2}\epsilon^{4}\left(V_{3}-4\pi\tau\epsilon^{2}\right)^{2}+64\pi^{8}\xi\epsilon^{12}\,,$
(55)

and it has the following two roots when viewed as a function of $\xi$:

$\xi_{1}=0\,,\;\;\;\text{and}\;\;\;\xi_{2}=\frac{16\pi^{4}\epsilon^{8}}{27\left(V_{3}-4\pi\tau\epsilon^{2}\right)^{2}}>0\,.$
(56)

For small $\epsilon$ and large $V_{3}$, the positive root takes a very small
value, and therefore we concentrate on the case $\xi>\xi_{2}$. In such case,
the discriminant is negative and therefore the cubic equation has only one
real root. Then we can use the representation of the real root in terms of
hyperbolic functions, writing:

$\omega_{0}=2\sqrt{\frac{-p}{3}}\cosh\left(\frac{1}{3}{\rm
arcosh}\left(\frac{3q}{2p}\sqrt{-\frac{3}{p}}\right)\right)-\frac{a_{2}}{3a_{3}}\,,$
(57)

where:

$p=\frac{3a_{3}a_{1}-a_{2}^{2}}{3a_{3}^{2}}=-\frac{4\pi^{4}\epsilon^{4}}{3\left(V_{3}-4\pi\tau\epsilon^{2}\right)^{2}}\,,$
(58)
$q=\frac{27a_{3}^{2}a_{0}-9a_{3}a_{2}a_{1}+2a_{2}^{3}}{27a_{3}^{3}}=\frac{16\pi^{6}\epsilon^{8}-54\pi^{2}\xi\left(V_{3}-4\pi\tau\epsilon^{2}\right)^{2}}{27\epsilon^{2}\left(V_{3}-4\pi\tau\epsilon^{2}\right)^{3}}\,,$
(59)

and $a_{n}$ is the coefficient of $\omega^{n}$ in the cubic equation (54).
Furthermore, in (57) we have assumed $p<0$, $q<0$ and $4p^{3}+27q^{2}>0$,
which are all valid for large enough $V_{3}$.

Lastly, we take (51) with $A=\sqrt{\xi}/(\epsilon\omega)$ and replace $\omega$
by the solution (57), to obtain a profile which is a function of $\xi$,
$V_{3}$, $\tau$ and $\epsilon$. Denoting such configuration as
$\hat{\phi}_{0}(t)$, we are interested in studying:

$\mathcal{S}\equiv\tilde{S}_{\textrm{3d-HL-
mini}}[\bar{\phi}_{0}]-\tilde{S}_{\textrm{3d-HL-mini}}[\hat{\phi}_{0}]$ (60)

as a function of $\xi$, at fixed $\kappa^{2}$, $V_{3}$, $\tau$ and $\epsilon$,
in order to check if and when the droplet configuration (51) dominates (i.e.
$\mathcal{S}>0$).

More conveniently, we can re-express $\mathcal{S}$ in terms of the discrete
BIB dimensionless parameters, eliminating $\xi$, $V_{3}$ and $\tau$ by means
of the relations:

$\frac{16\pi\xi}{\epsilon^{2}}=\frac{vb_{2}}{b_{1}m_{min}}\,,$ (61)
$\frac{V_{3}}{(4\pi\epsilon^{2})^{3/2}}=\frac{M}{m_{min}^{3/2}v^{1/2}}\,,$
(62) $\frac{\tau}{(4\pi\epsilon^{2})^{1/2}}=\frac{T}{(m_{min}v)^{1/2}}\,.$
(63)

One then finds that both $\epsilon$ and $\sqrt{v}$ factor out, so that
$\mathcal{S}^{\prime}\equiv\kappa^{2}\mathcal{S}/(\epsilon\sqrt{v})$ depends
only on the remaining discrete parameters. In particular, for $m_{min}=1$,
$\mathcal{S}^{\prime}$ depends only on $M$, $T$ and the ratio $b_{2}/b_{1}$.

We thus come to our main conclusion: for fixed $M$ and $T$, the reasoning
based on the minimization of the action predicts that there is a phase
transition between a droplet ($\mathcal{S}>0$) and a correlated fluid phase
($\mathcal{S}<0$), with the boundary between the two phases being given by a
straight line in the $(b_{1},b_{2})$ plane. In fact, for fixed $M$ and $T$,
$\mathcal{S}^{\prime}$ is only a function of the ratio $\alpha\equiv
b_{2}/b_{1}$, and the point at which $\mathcal{S}^{\prime}(\alpha_{c})=0$
corresponds to the phase transition. Unfortunately, such an equation cannot be
solved in a closed form, and we are limited to a numerical evaluation of
$\alpha_{c}$. For example, at $M=4000$ and $T=80$ , we find $\alpha_{c}=1.5$,
while at $M=16000$ and $T=80$ , we find $\alpha_{c}=7.1$. We can also check
numerically that at large volume $\alpha_{c}\propto M/T^{3}$. However, the
quantitative predictions about the location of the phase transition should not
be taken too seriously, because near the phase transition we expect the
fluctuations to become important and affect the transition point.

The above results show that the droplet configuration (51) wins over the
constant one in a certain range of parameters. What we have not shown is that
there are no other solutions that win over both in that same range of
parameters. In fact it is plausible that the actual dominant configuration is
a smooth version of (51), i.e. one in which bulk and stalk are joined
smoothly. However, we believe that the mechanism described above for our
ansatz correctly captures the essential physical features of the problem, and
this feeling is corroborated by the numerical simulations that have confirmed
the existence of a droplet phase in the BIB model (6) with reduced transfer
matrix (38).

##  5 Conclusions

In this review we have argued that the spatial volume profile in CDT can be
interpreted as a sort of coarse grained order parameter, and that its
associated Landau free energy can help us to understand the CDT phased diagram
and the dynamics leading to it. At the same time it can also teach us
something about the continuum limit, thanks to the partially local nature of
such observable. Armed with such observable, and with the Landau theory point
of view in mind, we have made the case that the continuum limit of CDT is most
likely described by an FPD-invariant effective field theory, as in HL-type
theories.

In particular, we have first reviewed the fact that in two-dimensional CDT the
Landau free energy for the spatial length is obtained exactly and in the
continuum limit it coincides with the two-dimensional HL gravity model. We
have then argued that in order to explain the spacetime condensation
phenomenon of three-dimensional CDT, reviewed in Sec. 3.1, a Landau free
energy obtained as the discretization of a minisuperspace reduction of the
Einstein-Hilbert action is not sufficient, as it would instead predict the
dominance of a constant profile. By studying instead a Landau free energy
inspired by HL gravity, proposed in Benedetti:2014dra , we have shown that a
continuum semiclassical analysis leads to a condensation, captured by the
dominant profile (51), compatible with the one observed in CDT. Such a
semiclassical analysis is corroborated by the Monte Carlo simulations
Benedetti:2016rwo  of the effective BIB model, whose main results are
summarized by the figures 7 and 11. The former shows the rich phase diagram of
the BIB model proposed in Benedetti:2014dra . In particular, note the presence
of a droplet phase, which however would be absent were one to set $b_{2}=0$,
as in the BIB model related to minisuperspace reduction of general relativity.
Figure 11 shows instead the excellent correspondence between the droplet phase
of the BIB model and the CDT volume profile. Thus, we have compelling evidence
that the effective dynamics of the CDT spatial slices is well captured by the
BIB model (6)-(38), which in turn can be seen as a discretized path integral
for a minusuperspace reduction of the Hořava-Lifshitz model (29).

Figure 11:  Comparison of the mean volume profile from the BIB model and
three-dimensional CDT.

As we mentioned along the course of this review, there are a number of other
reasons in support of a link between CDT and HL gravity (or more generally,
FPD-invariant effective field theories). It is worth to collect them here in
order to summarize the status of this relation:

  * •

Presence of a foliation: This is on one hand a trivial observation, as it is a
defining property of both CDT and HL gravity, and on the other a subtle one,
as understanding the fate of the CDT foliation in the conitnuum limit is
nontrivial. In particular, it is nontrivial because we have not yet identified
a proper tuning towards the second-order phase transition that might lead to a
continuum limit with propagating degrees of freedom (see Ambjorn:2014gsa ;
Ambjorn:2020rcn  for a first attempt). As a counterargument to the importance
of the foliation, a generalized CDT model with a local causality constraint
but no built-in foliation was introduced in Jordan:2013awa ; Jordan:2013iaa
and shown to also lead to a semiclassical droplet phase with similar
properties to that of standard CDT. On the other hand, even if such
generalized CDT model would turn out to be indeed in the same universality
class of usual CDT, this would not tell us what is their continuum
interpretation: it might well be that the FPD-invariant universality class is
robust under small departures from the foliated structure at the discrete
level. Moreover, as argued in the introduction, the very fact that observables
that are local in time are available and are being studied in CDT is by itself
a signal of a likely departure from a full-diffeomorphism invariant theory.

  * •

Analogies in the phase diagram: This was discussed for example in
Ambjorn:2010hu , and it is based on the observation that a Lifshitz point is
typically a tricritical point at the intersection of feromagnetic,
paramagnetic and modulated phases, and that such phases, with their analog
solutions in HL gravity, have a resemblance to the original three phases
observed in four-dimensional CDT. The picture was slightly disturbed by the
discovery of a new “bifurcation” phase in CDT Ambjorn:2014mra ;
Ambjorn:2016mnn , but ultimately in our opinion its most concrete realization
is in the kind of analysis we reviewed here: as reviewed in Sec. 4.2, the BIB
model based on HL gravity leads to the phase diagram of Fig. 7, which at
$b_{1}>0$ matches the phase diagram of CDT. A similar result holds also in
four dimensions Bogacz:2012sa : in this case, no higher-order terms in the
curvature are used, but nevertheless the fact that the coefficients of the
kinetic and potential terms are tuned independently in order to reach the
different phases suggests again a relation to HL gravity, as the ratio between
the two would instead be fixed in the Einstein-Hilbert action. It would be
interesting to add subleading terms in the potential of the BIB model for the
four-dimensional case (see Ambjorn:2012pp  for some results on such terms in
CDT) to see if the new bifurcation phase of CDT can also be reproduced,
although it has also been suggested that this might not be a conventional
Landau-type phase transition Ambjorn:2020rcn .

  * •

Short-scale spectral dimension: The fact that in the droplet phase of CDT the
spectral dimension matches the classical value at (not too) large scales, and
decreases at short scales Ambjorn:2005db  was shown to be reproduced by HL
gravity in Horava:2009if . However, several other approaches to quantum
gravity claimed a similar result (e.g. in asymptotic safety Lauscher:2005qz ),
thus weakening this point of contact. On the other hand, in Benedetti:2009ge
it was pointed out that in three dimensions asymptotic safety and HL gravity
give different predictions, and numerical evidence was produced in CDT
favoring the HL scenario over the asymptotic safety one.

  * •

Large-scale geometry: In Benedetti:2009ge  it was also observed that the
behavior of the spectral dimension at large scales, where it decays due to the
compactness of spacetime, is better fitted by a stretched sphere geometry,
rather than a round sphere. And the former is a solution to HL gravity.

  * •

Absence of conformal factor problem: As we discussed in Sec. 4, the effective
actions controlling the dynamics of the spatial volumes in CDT in $d=3$ and 4,
have a kinetic term with the good sign, that is, there is no conformal
sickness. While perhaps this could be interpreted as being due to
nonperturbative contributions from the functional measure, as suggested in
Dasgupta:2001ue , it might also have the much simpler explanation that in HL
gravity the conformal factor problem can be avoided by an appropriate choice
of the coupling $\lambda$ in (29). More investigations are probably needed in
order to decide between the two scenarios, but one should keep in mind that if
nonperturbative contributions could change the sign of the kinetic term for
the conformal mode with respect to usual Euclidean quantum gravity
Gibbons:1978ac , then either they would also spoil the good sign of the
graviton term in the latter, or they would be hard to reconcile with general
covariance of the theory, as the relative sign between conformal and graviton
terms is fixed by full-diffeomorphism invariance. Unfortunately no results are
available in CDT about a possible graviton effective action, but the
possibility that only the conformal factor’s sign is changed with respect to
usual Euclidean quantum gravity is corroborated by the results from three-
dimensional CDT with toroidal slices Budd:2011zm , according to which both the
conformal factor and the moduli have the good sign.

  * •

Potential versus higher-derivatives: Results from Ambjorn:2011ph  indicate
that in four-dimensional CDT higher-order time derivatives in the effective
action are subleading with respect to potential terms, that in a full action
would correspond to higher-order spatial derivatives. This is suggestive again
of an anisotropy.

  * •

Exact relation in two dimensions: Lastly, two-dimensional CDT is exactly
solvable and in the continuum limit it reproduces HL gravity, as reviewed in
Sec. 2.3, and more in detail in Sato:2022ory . As there is currently no hint
of a possible peculiarity of two dimensions when it comes to the CDT
foliation,252525There is of course a difference in the HL action leading to a
renormalizable theory, but this is irrelevant for an effective field theory
argument. this makes a strong plausibility argument for a relation between CDT
and HL gravity also in higher dimensions.

It should be stressed that clearly CDT in its usual definition has less
couplings than renormalizable HL gravity models in three or four dimensions.
Therefore, such renormalizable models are probably not directly relevant for
comparisons to CDT, unless one extends the latter with appropriate new terms
in the action, as done in Anderson:2011bj . However, as we argued in the
introduction and in Sec. 2.3, the fact that one starts in CDT with a
discretization of the Einstein-Hilbert action262626And moreover we remark
again that in four dimensions the CDT action has one more coupling than the
Einstein-Hilbert action. does not imply that its symmetries will be preserved.
The crucial point, on which we focused in this review by indirect means, is to
assess in which theory space a renormalization group flow of CDT will lead us,
or in other words what is the effective field theory of CDT. Once that is
established, only in a second step we will be able to worry about how to take
the continuum limit with propagating degrees of freedom corresponding to our
desired target theory. It is at this step that the fact that CDT has only few
tunable parameters might turn out to be a limitation. At present, the two
competing scenarios that can serve as justification for a program like CDT are
asymptotic safety and HL gravity. While the latter is based on a Gaussian
fixed point, and so it is straightforward to count the number of its relevant
couplings,272727It is instead not so straightforward to study its
renormalization group flow Benedetti:2013pya ; Barvinsky:2017kob ;
Barvinsky:2019rwn ; Barvinsky:2021ubv . the former is based on a conjectural
non-Gaussian fixed point, and we do not even know for sure if the number of
relevant couplings is finite (except in special simple approximations
Benedetti:2013jk ; Mitchell:2021qjr ). However, all evidence so far suggests
that if such a nontrivial fixed point exists, then it has at least three
relevant perturbations, which include higher-order curvature terms
Codello:2008vh ; Benedetti:2009rx ; Falls:2014tra . Therefore, continuum
scenarios and their results suggest that at some point one might need to
introduce more couplings in CDT.

Lastly, we stress again that if indeed the effective field theory of CDT turns
out to be a FPD-invariant one, this does not exclude the existence of a
subspace of its theory space which is invariant under full diffeomorphisms. In
other words, one might hope to embed asymptotically safe Euclidean gravity in
the HL theory space. Reaching such subspace would probably require again a
very delicate fine tuning of several breaking terms, yet perhaps in a less
dramatic way than a full breaking of diffeomorphism invariance reuires, as in
the functional renormalization group approach to asymptotic safety.

###### Acknowledgements.

I would like to thank Joe Henson and James Ryan for the precious collaboration
on the work reviewed here, and Jan Ambjorn and Renate Loll for the invitation
to contribute a chapter to the CDT section of the Handbook of Quantum Gravity.

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