Causal Dynamical Triangulation
Causal Dynamical Triangulations (CDT) provides an explicit definition of the path integral of quantum gravity as a sum over a
class of geometries one constructs from elementary building blocks. These building blocks, four-dimensional regular simplices,
come with a side length $a$ that acts as a ultraviolet cut-off of the quantum field theory. The geometries can all be rotated from geometries with
Lorentzian signature to geometries with Euclidean geometries and the corresponding path integral can be studied by Monte Carlo methods.
Although no background geometry is put in by hand, a four-dimensional de Sitter like world with quantum fluctuations emerges, the diameter
of the Euclidean de Sitter universe being around 20 Planck lengths for the largest universes that fits into the computer. The computer simulations
indicate the existence of a UV fixed point of the theory where the UV cut-off $a$ can be taken to zero, and hopefully then defining a continuum, non-trivial
theory of quantum gravity.
Contents |
Motivation for a lattice theory of quantum gravity
Classical General Relativity (GR) is defined as a metric theory by the Einstein-Hilbert (EH) action \begin{equation} S[g_{\mu\nu};G,\Lambda] = \frac{ 1}{16 \pi G}\int d^4 x \sqrt{-g(x)} \, \Big( R(x) - 2 \Lambda\Big), \tag{1} \end{equation} where $G$ denotes the gravitational constant, $\Lambda$ the cosmological constant and where $c = \hbar =1$ units are used. $g_{\mu\nu}$ is the four-metric of spacetime and $g$ the determinant of $g_{\mu\nu}$. The corresponding quantum theory is formally defined by the path integral \begin{equation} Z(G,\Lambda) = \int {\cal D} [g_{\mu\nu}] \; e^{iS[g_{\mu\mu};G,\Lambda]} \tag{2} \end{equation} where the integration is over four-geometries $ [g_{\mu\nu}]$. A priori it is not clear which geometries should be included in the path integral, but judging from the path integral for ordinary quantum field theories one would expect the integral to include at least all continuous four-geometries of a manifold with fixed topology. It is unclear how to include a summation over four-manifolds with different topologies since these cannot even be classified.
The quantum theory 'defined' by eq. (1) becomes even more formal because it is not a renormalizable theory when one expands $g_{\mu\nu}$ as a background metric $g_{\mu\nu}^{bg}$ plus quantum fluctuations around this background metric: \begin{equation} g_{\mu\nu} = g_{\mu\nu}^{bg} + \sqrt{G} \, h_{\mu\nu}. \tag{3} \end{equation} Thus $Z(G, \Lambda)$ cannot be defined by a perturbative expansion in $G$. One needs a non-perturbative definition. This is possible using so-called Causal Dynamical Triangulations (CDT). It is a lattice version of $Z(G,\Lambda)$ and it comes with a UV cut-off, the length $a$ of the lattice links. That makes the theory well defined. To take the continuum limit of this lattice theory one should study the limit $a \to 0$. It will be described below.
Definition of Causal Dynamical Triangulations
Like in attempts to quantize gravity by canonical methods, CDT supposes that spacetime is foliated into a family of spacelike surfaces $\Omega(t)$, labelled by their time coordinate $t$. It is further assumed that the topology of $\Omega(t)$ is not changing with $t$. The task is now to define the path integral over a suitable class of geometries associated with such a globally hyperbolic manifold. In CDT this is done in the following way: first $t$ is discretized to $t_i$. Next we consider the continuous geometries on $\Omega(t_i)$ one obtains by gluing together regular tetrahedra, flat in the interior and with equal sidelengths $a$, such that the topology matches that of $\Omega(t_i)$. Given one such triangulation of $\Omega(t_i)$ and another one of $\Omega(t_{i+1})$, the (four-dimensional) slab between the two spacelike surfaces is turned into a four-dimensional triangulations in the following way: each tetrahedron in $\Omega(t_i)$ is connected to a vertex in the triangulation of $\Omega(t_{i+1})$, thus forming a four-simplex. Each triangle in the triangulation of $\Omega(t_i)$ is connected to a link in the triangulation of $\Omega(t_{i+1})$, again forming a four-simplex. Similarly links in $\Omega(t_i)$ are connected to triangles in $\Omega(t_{i+1})$ and vertices in $\Omega(t_i)$ to tetrahedra in $\Omega(t_{i+1})$. The various four-simplices are denoted (4,1)-, (3,2)-, (2,3)- and (1,4)-simplices, respectively. The construction is illustrated in Fig. 1.
The Figure show a $(3,2)$ four-simplex and a $(4,1)$ four-simplex. Finally all these four-simplices should be glued together in such a way that they have the topology of $[0,1] \times \Omega$. The links in the triangulation connecting $\Omega(t_i)$ and $\Omega(t_{i+1})$ are all assumed to be time-like with a proper length squared $\Delta s^2 = - \alpha \, a^2$. The total set of geometries entering in the CDT path integral will now be the set of continuous geometries obtained by summing over all abstract three-dimensional triangulations of each $\Omega(t_i)$ and connecting neighboring triangulated $\Omega(t_i)$s as described above in all possible ways compatible with the topology. The procedure will be well defined if one assumes that the $\Omega(t_i)$s as well of the range of $t$ are compact, since then only a finite number of four-simplices will enter the construction. In the following it is assumed for simplicity that the topology of $\Omega$ is that of $S^3$.
If one starts with a Lorentzian geometry constructed as above and with $\alpha =1$, one can associated an Euclidean geometry, obtained by rotating $\alpha$ from 1 to -1 in the lower complex half-plane. The corresponding Euclidean geometry is defined by the same abstract triangulation $T$ as its Lorentzian companion, only are all four-simplices now identical, with links of length $a$. The EH action (1) is defined for smooth geometries. However, Regge showed that there is a natural generalization of the EH action to so-called piecewiese flat geometries [Regge, 1961], of which the geometries discussed here constitute a subclass. Let $S_L[T,\alpha]$ denote the Regge action calculated for the Lorentzian geometry constructed above, and $S_E[T]$ denote the Regge action for the Euclidean geometry mentioned. Then, performing the rotation of $\alpha$ in the lower complex half-plane one obtains \begin{equation} S_L[T,\alpha =1] \to \lim_{\epsilon \to 0} S_L[T, \alpha = -1-i\epsilon] = i S_E[T]. \tag{4} \end{equation} In addition $S_E[T]$, constructed from identical four-simplices, becomes exceedingly simple. For a given triangulation $T$ one finds \begin{equation} S_E[T] = -\hat{k}_0 (N_0(T)-\chi(T)) + \hat{k}_4 N_4(T) \tag{5} \end{equation} where $N_0(T)$ and $N_4(T)$ denote the number of vertices and four-simplices in the triangulation $T$ and $\chi(T)$ is the Euler characteristic of $T$. The $N_i(T)$ will large be compared to $\chi(T)$ for the triangulations of interest, and the $\chi(T)$ term will be ignored in the following. Finally, the relations between the dimensionless lattice coupling constants $\hat{k}_0$ and $\hat{k}_4$ and the coupling constants $G$ and $\Lambda$ in the EH action (1) are \begin{equation} \hat{k}_0 = c_0 \frac{a^2}{G}, \qquad \hat{k}_4 = c_4 \frac{a^2}{G} + c_4' \frac{a^4 \Lambda}{G}, \tag{6} \end{equation} where $c_0$, $c_4$ and $c'_4$ are positive constants of order 1. Details of the above construction in spacetime dimensions 2, 3 and 4 can be found in [Ambjorn et al.,2012]
The CDT path integral
The CDT path integral is now defined as follows: \begin{equation} Z(G,\Lambda) = \int {\cal D} [g_{\mu\nu}] \; e^{iS[g_{\mu\mu};G,\Lambda]} \to Z^{L}_a = \sum_{T} \frac{1}{C_T} e^{i S_L[T;\alpha =1]}, \tag{7} \end{equation} where the summation is over the triangulations discussed above with link length $a$, and where $C_T$ is a symmetry factor, the order of the automorphism group of $T$, viewed as a graph.
As mentioned it is difficult to perform any calculations related to the lhs of eq. (7), but it is almost equally difficult using the rhs. However, if one performs the analytic continuation to Euclidean triangulations under which the action $S_L[T,\alpha=1]$ transforms as in eq. (4), it follows that $Z_a^L$ will transform into an Euclidean path integral: \begin{equation} Z_a^L \to Z_a^E(\hat{k}_0,\hat{k}_4) = \sum_{T} \frac{1}{C_T} e^{-S_E[T;\hat{k}_0,\hat{k}_4]}. \tag{8} \end{equation} The advantage of the partition function $Z_a^E(\hat{k}_0,\hat{k}_4)$ is that the weight of each $T$ enters with a positive Boltzmann weight and one can study the corresponding statistical system using Monte Carlo simulations. The results obtained by such Monte Carlo simulations will be discussed below, but first it is natural to generalize the Euclidean action (5) slightly: In the Lorentzian Regge action associated with a triangulation $T$, the (2,3)-- and (3,2)--simplices appear with a different weight than the (1,4)-- and the (4,1)--simplices, caused by their different spacetime structure. It is not unnatural to allow for such a different weight also in the Euclidean triangulations, in particular from a Wilsonian point of view where one should allow many different terms in the action. Thus the Euclidean action is generalized to \begin{eqnarray} S_E[T;k_0k_{23},k_{14}] &=& -k_0 N_0(T) + k_{23} N_4^{(23)}(T) + k_{14} N_4^{(14)}(T) \tag{9} \\ S_E[T;k_0,\Delta,k_4] &=& (k_0 + 6\Delta) N_0(T) +k_4 N_4(T) + \Delta N_4^{(14)}. \tag{10} \end{eqnarray} In eq. (9) $N_4^{(23)}(T)$ is the number of (2,3)-- and (3,2)--simplices and $N_4^{(14)}$ the number of (1,4)-- and (4,1)--simplices in $T$, respectively. Using the Dehn-Sommerville relations it can be shown to be the most general expression that is linear in the number of (sub)simplices in $T$. Eq. (10) is just a rewriting of eq. (9), a parametrization that has been used in the Monte Carlo simulations. For $\Delta =0$ it reduces to eq. (5). Thus our final expression for the CDT path integral is \begin{equation} Z_E[k_0,\Delta,k_4] = \sum_{T} \frac{1}{C_T} e^{-S_E[T;k_0,\Delta,{k}_4]} = \sum_{N_4} e^{-k_4 N_4} Z_E(k_0,\Delta,N_4) \tag{11} \end{equation} Here $Z_E(k_0,\Delta,N_4)$ is the partition function for a fixed number of four-simplices. It grows exponentially with $N_4$, \begin{equation} Z_E( (k_0,\Delta,N_4) = \!\!\sum_{T~{\rm where} \atop N_4(T) =N_4} \frac{1}{C_{T}} e^{(k_0+6\Delta)N_0(T) - \Delta N^{(14)}_4 (T)} = e^{k_4^c(k_0,\Delta) N_4} F(k_0,\Delta,N_4) \tag{12} \end{equation} where $F(k_0,\Delta,N_4)$ is subleading as a function of $N_4$. It follows that for given values of $k_0,\Delta$ the sum (11) is convergent for $k_4 > k_4^c(k_0,\Delta)$. In this region of the coupling constant space the CDT path integral is thus well defined. In eq. (11) the reference to the lattice spacing $a$ is dropped since one for the generalized action (10) no longer has the relation (6). $k_0,\Delta,k_4$ are then viewed as bare dimensionless lattice coupling constants. The lattice cut-off $a$ and the continuum coupling constants will reappear when one studies how observables behave in the limit of the coupling constant space $(k_0,\Delta,k_4)$ where $\langle N_4 \rangle \to \infty$.
Some Monte Carlo Simulation results
From the action (10) it is clear that the average number of $N_4$ in the ensemble of triangulation entering (11) will be monitored by $k_4$ and will be largest when $k_4 \to k_4^c(k_0,\Delta)$ from above. We are interested in large values of $N_4$ and from a computer simulation point of view it is far more convenient to fix $N_4$ and perform the simulation for different values of $N_4$, rather than trying to fine tune $k_4$ to $k_4^c(k_0,\Delta)$ (which one does not know analytically). This means that we are using $Z_E(k_0,\Delta,N_4)$ as the (canonical) partition function, but nothing is lost if we know it for all $N_4$ since it is connected to the (grand canonical) partition function $Z_E(k_0,\Delta,k_4)$ by a discrete Laplace transformation as eq. (11) shows.
Depending on the choice of coupling constants $k_0,\Delta$ one observes classes of vastly different geometries and phase transitions between these classes of geometries. The phase diagram is shown in Fig. 2. In phase $A$ different time slices seem not to couple. In phase $B$ the time extension of the universe is only one time-slice. In phase $C_b$ the time extension of universe is larger, but it does not scale when $N_4$ is increased. Only phase $C_{\rm dS}$ seems to represent a four-dimensional universe. Thus only the so-called de Sitter phase $C_{\rm dS}$ will be of interest here. The others are viewed as lattice artifacts.
An observable to measure in the simulations is average spatial lattice volumes at times $t_i$, denoted $\langle N_3(i) \rangle_{N_4}$, where $N_3(i)$ is the number of tetrahedra in the spatial slice at time $t_i$. Another observable is the connected correlator between $N_3(i_1)$ and $N_3(i_2)$, defined by \begin{equation} C(i_1,i_2;N_4) = \langle \big(N_3(i_1) - \langle N_3(i_1) \rangle\big) \big(N_3(i_2) - \langle N_3(i_2) \rangle\big) \rangle_{N_4} \tag{13} \end{equation} There are subtleties related to the definition of the averages (see [Ambjorn et al.,2008]), but the results for sufficiently large $N_4$ are: \begin{equation} \langle N_3(i) \rangle_{N_4} \propto N_4 \; \frac{1}{ \omega N_4^{1/4}} \cos^3\Big(\frac{i}{\omega N_4^{1/4}} \Big), \tag{14} \end{equation} \begin{equation} C(i_1,i_2,N_4) \propto \Gamma \, N_4 \; H\Big(\frac{i_1}{\omega N^{1/4}} ,\frac{i_2}{\omega N^{1/4}}\Big), \quad H(0,0) =1 \tag{15} \end{equation} Only the constants $\omega$ and $\Gamma$ depend on the coupling constants $k_0,\Delta$ as long as they belong to phase $C_{\rm dS}$.
Fig. 3 shows the saw-tooth picture of a computer measurement of $N_3(i)$ for $N_4 = 362000$ (the blue curve). The average over many configurations, $\langle N_3(i) \rangle_{N_4}$, is the red curve. On the Figure one observes a 'blob'. It is indistinguishable from the analytic formula (14) for a suitable value of $\omega$. The height scales as $N_4^{3/4}$ while the width scales as $N_4^{1/4}$. Outside the blob the value of $\langle N_3(i)\rangle$ is close to the cut-off scale: the computer program does not allow $N_3(i) =0$, but instead it stays close to the minimal non-zero value required to form a triagulation of $S^3$, namely $N_3(i) = 5$. This region is denoted the 'stalk'. From the distribution of measured $N_3(i)$ one can also determine the fluctuations of $N_3(i)$ and they scale as $N_4^{1/2}$ in the blob region. This implies that in the blob the relative size of the fluctuations of $N_3(i)$ scales to zero as $N_4^{-1/4}$, as will be discussed below.
In fact the blob region shows perfect finite size scaling: If we introduce scaling variables \begin{equation} s_i = \frac{i}{N_4^{1/4}}, \quad n_3(s_i) = \frac{N_3(i)}{N_4^{3/4}}, \tag{16} \end{equation} the curves for $\langle N_3(i) \rangle_{N_4}$ and for $ \sqrt{N_4} \,C(i_1,i_2,N_4) $ for different $N_4$ collapse to two curves \begin{equation} \langle n_3(s) \rangle \propto \frac{3}{4\omega } \, \cos^3 \Big( \frac{ s}{\omega}\Big), \qquad \sqrt{N_4}\; C(s_1,s_2) \propto \Gamma\; H\Big( \frac{s_1}{\omega},\frac{s_2}{\omega}\Big). \tag{17} \end{equation} Further, these results can be derived from the following effective action \begin{equation} S_{\rm eff}[k_0,\Delta] = \frac{1}{\Gamma} \sum_i \left( \frac{\big(N_3({i+1}) -N_3(i)\big)^2}{ N_3({i})\big)} + \delta \, N_3^{1/3}(i) \right), \tag{18} \end{equation} or, expressed in scaling variables (with $ds_i = 1/N_4^{1/4}$) \begin{equation} S_{\rm eff}[k_0,\Delta] = \frac{\sqrt{N_4}}{\Gamma} \int ds \; \left( \frac{(\dot{n}_3^2(s)}{ (n_3(s)} + \delta \, n_3^{1/3}(s) \right), \quad \int ds \, n_3(s) = 1, \tag{19} \end{equation} where the summation is replaced by an integration for $N_4$ sufficiently large. The solution to the 'classical' eom associated with $S_{\rm eff}$ is precisely $\langle n_3(s) \rangle$ given by eq. (17) provided $\delta$ and $\omega$ are related as follows \begin{equation} \frac{\delta}{\delta_0} = \left(\frac{\omega_0}{\omega}\right)^{8/3}, \qquad \delta_0 = 9 (2\pi^2)^{2/3}, \quad \omega_0 = \frac{3}{\sqrt{2}} \frac{1}{\delta_0^{3/8}}. \tag{20} \end{equation} Finally the fluctuation function $H$ is well described by Gaussian fluctuations around $\langle n_3(s) \rangle$.
If $\omega = \omega_0$, $\langle n_3(s) \rangle$ will describe a four-sphere with four-volume 1, $s$ being the geodesic distance from equator when using the line element \begin{equation} d\tau^2 = ds^2 + r^2(s) d \Omega_3, \quad n_3(s) = 2 \pi^2 r^3(s), \tag{21} \end{equation} $d\Omega_3$ being the line element of the unit three-sphere. $\omega$ is determined by the computer simulations, depends on the lattice coupling constants $k_0,\Delta$ and will in general not be equal to $\omega_0$. If $\omega < \omega_0$ $\langle n_3(s) \rangle$ will describe a four-sphere squeezed in the proper-time direction. For fixed values of $k_0,\Delta$ we can turn this squeezed four-sphere into a 'round' four-sphere by rescaling the link length $a_t$ in the time direction as \begin{equation} a_t = a \to a_t = \Big(\frac{\omega_0}{\omega}\Big)^{4/3} a. \tag{22} \end{equation} With this rescaling the effective action (19) will change to \begin{equation} S_{\rm eff}[k_0,\Delta] = \frac{\omega_0^2\sqrt{N_4}}{\omega^2\Gamma} \int ds \; \left( \frac{(\dot{n}_3^2(s)}{ (n_3(s)} + \delta_0 \, n_3^{1/3}(s) \right), \quad \int ds \, n_3(s) = 1, \tag{23} \end{equation} The Hartle-Hawking minisuperspace action is obtained by starting out with the EH action (1), rotating to Euclidean signature and restricting the degrees of freedom to the scale factor $r$ of the universe, as in eq. (21), and finally rotating the conformal mode such that the minisuperspace action is bounded from below. It is remarkable that this minisuperspace action can precisely be written as eq. (23) using suitably rescaled variables, provided one makes the identification \begin{equation} \frac{\omega_0^2\sqrt{N_4}}{\omega^2\Gamma} = \frac{\sqrt{V_4}}{24 \pi G} = \frac{\sqrt{6}}{24 \Lambda G}. \tag{24} \end{equation} Here $V_4$ is the volume of the four-sphere that is the solution to the Euclidean GR equations with cosmological constant $\Lambda$. What is equally remarkable is that the CDT action is derived from the complete path integral by integrating out all modes except the scale factor (here represented as the three-volume $V_3(t_i) = N_3(t_i) a^3$), while the Hartle-Hawking action is obtained by ignoring all degrees of freedom except the scale factor.
The IR and and UV limits of CDT
In a conventional lattice field theory IR and UV fixed points are located on critical surfaces in the lattice coupling constant space where the correlation lengths of observables are infinite. If the bare lattice coupling constants are kept fixed, renormalized (continuum) coupling constants will flow to an IR fixed point when the correlation length goes to infinity. If the renormalized coupling constants are kept fixed, the bare lattice coupling constants will flow to a UV fixed point. Here the lattice couplings are $k_0$ and $\Delta$ and one can choose the dimensionless coupling constant $\Lambda G$ as a continuum renormalized coupling constant. Then the lhs of eq. (24) can be read as the lattice expression for $\Lambda G$, expressed in terms of $k_0$, $\Delta$ and $N_4$. Since one has finite size scaling one can view $N_4^{1/4}$ as a correlation length and the critical surface is thus $N_4 = \infty$.
According to eq. (24), keeping $k_0$ and $\Delta$ constant and taking $N_4 \to \infty$, $\Lambda G$ will then flow to an IR fixed point. Since $\Gamma$ and $\omega$ will go to constant values (that depend on $k_0,\Delta$), $\Lambda G \to 0$ for $N_4 \to \infty$. Using the continuum renormalization group one can study the flow of $\Lambda G$ and 0 is an IR Gaussian fixed point where $\Lambda \to 0$ and $G \to \ell_p^2$, the Planck length squared. The lattice picture of this is according to eqs. (22) and (24), \begin{equation} V_4 \propto \Big(\frac{\omega_0}{\omega}\Big)^{4/3}N_4\, a^4 \propto N_4 \ell_p^4 \qquad a \propto \Big(\frac{\omega_0}{\omega}\Big)^{2/3} \frac{ \sqrt{G}}{\sqrt{\Gamma}} \propto \ell_p. \tag{25} \end{equation} Any $(k_0,\Delta)$ in the interior of the $C_{\rm dS}$ region belongs to the critical surface related to the IR Gaussian fixed point of $\Lambda G$, and while the four-volume $V_4 \propto N_4 a^4$ goes to infinity, the lattice spacing $a$ does not scale to zero, but is of the order of the Planck length.
Eq. (25) allows an estimate of the size of the universes simulated on the computer. For typical values of $k_0,\Delta$ in phase $C_{\rm dS}$ the diameter will be around 20 Planck lengths for $N_4 \approx 400000$. It is surprising that global features of such small universes are well described by the action (23).
To locate a UV lattice fixed point one should follow a path $(k_0(N_4),\Delta(N_4))$ such that $\Lambda G$ on the rhs of eq. (24) stays constant for $N_4 \to \infty$, i.e. a path such that \begin{equation} \omega^2\big(k_0(N_4),\Delta(N_4)\big)\, \Gamma\big(k_0(N_4),\Delta(N_4)\big) \propto \sqrt{N_4} \quad {\rm for} \quad N_4 \to \infty. \tag{26} \end{equation} Numerical evidence suggests that this is only possible if the path leads to the $A$-$C_{\rm dS}$ phase transition line from to $C_{\rm dS}$ side. Quite surprising, one can find paths where $\omega^2 \Gamma$ behaves precisely like needed when one approaches the $A$-$C_{\rm dS}$ line (see [Ambjorn et al., 2024]): \begin{equation} \omega^2 \Gamma \propto N_4^\delta, \qquad \delta = 0.54 \pm 0.04. \tag{27} \end{equation} It is a hint that the $A$-$C_{\rm dS}$ phase transition line can be viewed as a UV critical line and that CDT can be used to define a continuum, non-trivial quantum theory of gravity. Eqs. (25) and (27) shows that the cut-off $a\to 0$ when one approaches this critical line. However, it is numerical quite demanding to improve the estimate (27), and the fact that $\omega(k_0,\Delta) \to 0$ at the $A$-$C_{\rm dS}$ phase transition line causes some additional problems for the interpretation, since the width of the blob in the time direction, $\omega(k_0,\Delta)N_4^{1/4}$, should go to infinity when $(k_0(N_4),\Delta(N_4))$ approaches a UV fixed point. On the other hand, clarifying these issues is only a question of additional Monte Carlo simulations.
Fractal dimensions in CDT gravity
An ensemble of geometries. like the one generated by the CDT path integral, will be characterized by various fractal dimensions, like the Hausdorff dimension $D_H$ and the spectral dimension $D_S$.
The Hausdorff dimension is determined by the finite size scaling of the so-called two-point function, the correlator $C(r,N_4)$ between two point separated a geodesic distance (i.e. a shortest lattice distance) $r$ : \begin{equation} C(r,N_4) \propto x^{D_H} F(x), \qquad x = \frac{r}{N_4^{1/D_H}}. \tag{28} \end{equation} In fact we have already determined $D_H$ since $C(i_i,i_2;N_4)$ is just the two-point function integrated over two spatial hypersurfaces separated a geodesic distance $|i_1-i_2|$ (see [Ambjorn et al., 2020] for a detailed discussion). So $D_H =4$.
Consider a discrete diffusion process on a triangulation $T$. Denote the average return probability by $P_T(\sigma)$, where $\sigma$ is the (discrete) diffusion time. On the ensemble of CDT triangulations with $N_4$ four-simplices one defines the return probability as \begin{equation} P_{N_4}(\sigma) = \frac{1}{Z(k_0,\Delta,N_4)} \sum_{T~{\rm where} \atop N_4(T) = N_4} \frac{1}{C_T} \; e^{(k_0+6\Delta)N_0(T) - \Delta N_4^{(14)}(T)}\; P_T (\sigma), \tag{29} \end{equation} where the summation is over triangulations with $N_4$ four-simplices. One expects the return probability to have the form \begin{equation} P_{N_4}(\sigma) = \sigma^{-D_S/2} H\Big( \frac{\sigma}{N_4^{2/D_S}}\Big), \qquad H(0) > 0. \tag{30} \end{equation} Defining a $\sigma$ dependent $D_S(\sigma)$ by \begin{equation} D_S(\sigma) = -2 \frac{d \ln P_{N_4}(\sigma)}{d\sigma}, \tag{31} \end{equation} one expects $D_S(\sigma)$ to be almost constant for $\sigma < N_4^{2/d_S}$, and if this is the case in measurements of $P_{N_4}(\sigma)$, this is what one calls the spectral dimension of the ensemble of triangulations. However, one does not observe such a plateau of constant $D_S(\sigma)$ in the case of four-dimensional CDT.
Fig. 4 shows the spectral dimension $D_S$ as function of the diffusion time $\sigma$ measured for spacetime volume $N_4 = 181000$. The central curve shows the averaged of the measurements of $D_S(\sigma)$ together with a superimposed best fit $D_S(\sigma) = 4.02 - 119/(54+\sigma)$. The fit is so good that it is difficult the distinguish the two curves. The two outer curves represent error bars. Lattice discretisation effects prevents a reliable measurement of $D_S(\sigma)$ for $\sigma < 40$. Fig. 4 indicates that there could to be a genuine scale dependence of $D_S(\sigma)$ for small $\sigma$: $D_S(\sigma)$ increases gradually from around 2 for small $\sigma$ to around 4 for moderate $\sigma < N_4^{1/2}$. After this observation of a scale dependent $D_S(\sigma)$ was first observed in CDT [Ambjorn et al.,2005] it has also been obtained in a number of other approaches to quantum gravity.
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