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![(a) The known bounds [25, 28, 29] on the Hausdorff dimension dγ together with the predictions by Watabiki and Ding & Gwynne. (b) A zoomed-in version shifted by Watabiki’s prediction together with numerical estimates: [17] in brown, [19] in purple, [21] in orange, [22] in black, [23] in green.](publication/336668990/figure/fig2/AS:11431281079664675@1660815318657/a-The-known-bounds-25-28-29-on-the-Hausdorff-dimension-dg-together-with-the.jpg)
(a) The known bounds [25, 28, 29] on the Hausdorff dimension dγ together with the predictions by Watabiki and Ding & Gwynne. (b) A zoomed-in version shifted by Watabiki’s prediction together with numerical estimates: [17] in brown, [19] in purple, [21] in orange, [22] in black, [23] in green.
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![(a) The known bounds [25, 28, 29] on the Hausdorff dimension dγ...](publication/336668990/figure/fig2/AS:11431281079664675@1660815318657/a-The-known-bounds-25-28-29-on-the-Hausdorff-dimension-dg-together-with-the_Q320.jpg)
Two-dimensional quantum gravity, defined either via scaling limits of random discrete surfaces or via Liouville quantum gravity, is known to possess a geometry that is genuinely fractal with a Hausdorff dimension equal to 4. Coupling gravity to a statistical system at criticality changes the fractal properties of the geometry in a way that depends...
Citations
... Broadly speaking, it is extracted by comparing volumes with their characteristic linear size, measured in terms of a geodesic distance. There have been many studies of the Hausdorff dimension in the context of DT and CDT quantum gravity, including extensive investigations in two-dimensional Euclidean DT models with and without matter (see, for example, [76,77,96,128] and references therein). Following [77], we will investigate a local and a global ("cosmological") variant of this observable on the spatial slices. ...
... To perform a smooth rescaling, we first construct continuous functions that interpolate between these discrete values, which by slight abuse of notation we continue to call n V2 (r). Following the methodology of [128], we then rescale, for each system volume V 2 separately, the corresponding distribution n V2 (r) such that it maximally overlaps with the normalized distribution n Vmax (r) for the largest slice volume V max in the simulation, which we are using as a reference distribution. We denote these rescaled distance profiles byñ V2 (r), wherer is a rescaled length variable. ...
... where the two fit parameters are a rescaling dimension d and a phenomenological shift a [76,128] that corrects for discretization effects at small r, similar to the prescription (5.13) we used for the local Hausdorff dimension. Note thatr Vmax is equal to the original discrete length parameter r, so we can use them interchangeably, and that n Vmax = n Vmax . ...
This thesis investigates low-dimensional models of nonperturbative quantum gravity, with a special focus on Causal Dynamical Triangulations (CDT). We define the so-called curvature profile, a new quantum gravitational observable based on the quantum Ricci curvature. We subsequently study its coarse-graining capabilities on a class of regular, two-dimensional polygons with isolated curvature singularities, and we determine the curvature profile of (1+1)-dimensional CDT with toroidal topology. Next, we focus on CDT in 2+1 dimensions, intvestigating the behavior of the two-dimensional spatial slice geometries. We then turn our attention to matrix models, exploring a differential reformulation of the integrals over one- and two-matrix ensembles. Finally, we provide a hands-on introduction to computer simulations of CDT quantum gravity.
... Broadly speaking, it is extracted by comparing volumes with their characteristic linear size, measured in terms of a geodesic distance. There have been many studies of the Hausdorff dimension in the context of DT and CDT quantum gravity, including extensive investigations in two-dimensional Euclidean DT models with and without matter (see, for example, [20,[62][63][64] and references therein). Following [62], we will investigate a local and a global ("cosmological") variant of this observable on the spatial slices. ...
... To perform a smooth rescaling, we first construct continuous functions that interpolate between these discrete values, which by slight abuse of notation we continue to call n V 2 ðrÞ. Following the methodology of [64], we then rescale, for each system volume V 2 separately, the corresponding distribution n V 2 ðrÞ such that it maximally overlaps with the normalized distribution n V max ðrÞ for the largest slice volume V max in the simulation, which we are using as a reference distribution. We denote these rescaled distance profiles byñ V 2 ðrÞ, wherer is a rescaled length variable. ...
... where the two fit parameters are a rescaling dimension d and a phenomenological shift a [63,64] that corrects for discretization effects at small r, similar to the prescription (13) we used for the local Hausdorff dimension. Note thatr V max is equal to the original discrete length parameter r, so we can use them interchangeably, and thatñ V max ¼ n V max . ...
Three-dimensional Lorentzian quantum gravity, expressed as the continuum limit of a nonperturbative sum over spacetimes, is tantalizingly close to being amenable to analytical methods, and some of its properties have been described in terms of effective matrix and other models. To gain a more detailed understanding of three-dimensional quantum gravity, we perform a numerical investigation of the nature of spatial hypersurfaces in three-dimensional causal dynamical triangulations (CDT). We measure and analyze several quantum observables, the entropy exponent, the local and global Hausdorff dimensions, and the quantum Ricci curvature of the spatial slices, and try to match them with known continuum properties of systems of two-dimensional quantum geometry. Above the first-order phase transition of CDT quantum gravity, we find strong evidence that the spatial dynamics lies in the same universality class as two-dimensional Euclidean (Liouville) quantum gravity. Below the transition, the behavior of the spatial slices does not match that of any known quantum gravity model. This may indicate the existence of a new type of two-dimensional quantum system, induced by the more complex nature of the embedding three-dimensional quantum geometry.
... Therefore, our fit for the Ising model, in line with further supports the thesis. Our measurements are consistent with the conjectural relation between the Hausdorff dimension and central charge proposed by Watabiki in [11], yielding d W ≈ 4.2122, and it is consistent with the bounds rigorously derived by Gwynne 4.1892 < d W < 4.2156 [12], and with the state of the art numerical results from Budd, Ambjørn and Barkley [18,19]. As a practical note, we point out that computational complexity for the MCMC simulation of free fermions as we detailed scales as N 3 , with N the size of the triangulation. ...
... where we define S K to be the contribution to the action that only depends on the contorsion, and D ij is an analogous term to the Dirac-Wilson operator in (19), this time also accounting for torsion effects. The path integral (32) can then be computed as a series in powers of g and simulated with MCMC in a similar way as with free fermions. ...
Geometric properties of lattice quantum gravity in two dimensions are studied numerically via Monte Carlo on Euclidean Dynamical Triangulations. A new computational method is proposed to simulate gravity coupled with fermions, which allows the study of interacting theories on a lattice, such as non-Riemannian gravity models. This was tested on Majorana spinors, where we obtained a Hausdorff dimension dW = 4.22 +/- 0.03, consistent with the bounds from the literature 4.19 < dH < 4.21.
... Broadly speaking, it is extracted by comparing volumes with their characteristic linear size, measured in terms of a geodesic distance. There have been many studies of the Hausdorff dimension in the context of DT and CDT quantum gravity, including extensive investigations in two-dimensional Euclidean DT models with and without matter (see, for example, [20,[63][64][65] and references therein). Following [63], we will investigate a local and a global ("cosmological") variant of this observable on the spatial slices. ...
... Following the methodology of [65], we then rescale, for each system volume V 2 separately, the corresponding distribution n V 2 (r) such that it maximally overlaps with the normalized distribution n Vmax (r) for the largest slice volume V max in the simulation, which we are using as a reference distribution. We denote these rescaled distance profiles byñ V 2 (r), wherer is a rescaled length variable. ...
... where the two fit parameters are a rescaling dimension d and a phenomenological shift a [64,65] that corrects for discretization effects at small r, similar to the prescription (13) we used for the local Hausdorff dimension. Note thatr Vmax is equal to the original discrete length parameter r, so we can use them interchangeably, and thatñ Vmax = n Vmax . ...
Three-dimensional Lorentzian quantum gravity, expressed as the continuum limit of a nonperturbative sum over spacetimes, is tantalizingly close to being amenable to analytical methods, and some of its properties have been described in terms of effective matrix and other models. To gain a more detailed understanding of three-dimensional quantum gravity, we perform a numerical investigation of the nature of spatial hypersurfaces in three-dimensional Causal Dynamical Triangulations (CDT). We measure and analyze several quantum observables, the entropy exponent, the local and global Hausdorff dimensions, and the quantum Ricci curvature of the spatial slices, and try to match them with known continuum properties of systems of two-dimensional quantum geometry. Above the first-order phase transition of CDT quantum gravity, we find strong evidence that the spatial dynamics lies in the same universality class as two-dimensional Euclidean (Liouville) quantum gravity. Below the transition, the behaviour of the spatial slices does not match that of any known quantum gravity model. This may indicate the existence of a new type of two-dimensional quantum system, induced by the more complex nature of the embedding three-dimensional quantum geometry.
... The value d γ is precisely the Hausdorff dimension of this metric [44], which informally is saying that the µ φ -quantum area of a geodesic ball of radius r around any point is of order r dγ when r → 0. The exact value of d γ is only known for γ = 8/3, corresponding to the pure gravity universality class, where d √ 8/3 = 4. For γ = 8/3, rigorous bounds are known [45,46] as well as numerical estimates [30]. Moreover, as γ → 0 the dimension d γ approaches 2 (see [47] for bounds on the convergence rate) in accordance with the constant curvature solution g ab to the classical Liouville action at γ = 0. ...
... 1. This expression takes into account a leading-order correction and has proven to work well for Hausdorff dimension estimations in a similar setting [30]. The fitting procedure was tested by varying the range of volumes included, as well as the values of δ and d H while keeping a ≈ 1 and |b| 1. ...
... A distinction between the regions γ < 1 and γ ≥ 1 is made, since the latter is the domain analysed in [30]. The reason for choosing the four values γ = 1, 4/3, √ 2, 8/3 is that they are the values associated to the universality classes of Schnyder-wood-decorated triangulations, bipolaroriented triangulations, spanning-tree-decorated quadrangulations and uniform quadrangulations, respectively. ...
The search for scale-invariant random geometries is central to the Asymptotic Safety hypothesis for the Euclidean path integral in quantum gravity. In an attempt to uncover new universality classes of scale-invariant random geometries that go beyond surface topology, we explore a generalization of the mating of trees approach introduced by Duplantier, Miller and Sheffield. The latter provides an encoding of Liouville Quantum Gravity on the 2-sphere decorated by a certain random space-filling curve in terms of a two-dimensional correlated Brownian motion, that can be viewed as describing a pair of random trees. The random geometry of Liouville Quantum Gravity can be conveniently studied and simulated numerically by discretizing the mating of trees using the Mated-CRT maps of Gwynne, Miller and Sheffield. Considering higher-dimensional correlated Brownian motions, one is naturally led to a sequence of non-planar random graphs generalizing the Mated-CRT maps that may belong to new universality classes of scale-invariant random geometries. We develop a numerical method to efficiently simulate these random graphs and explore their possible scaling limits through distance measurements, allowing us in particular to estimate Hausdorff dimensions in the two- and three-dimensional setting. In the two-dimensional case these estimates accurately reproduce previous known analytic and numerical results, while in the three-dimensional case they provide a first window on a potential three-parameter family of new scale-invariant random geometries.
... It was later shown in [136,137] that its spectral dimension was two for the whole range γ ∈ (0, 2). The dependence of the Hausdorff dimension with the central charge is on the other hand less clear, with the propositions of Watabiki [138], and Ding and Goswami [139] closest to numeral simulations [140]. Other topologies than the sphere were also worked out, such as the disk [141] or the half-plane [142]. ...
This thesis focuses on renormalization of quantum field theories. Its first part considers three tensor models in three dimensions, a Fermionic quartic with tensors of rank-3 and two Bosonic sextic, of ranks 3 and 5. We rely upon the large-$N$ melonic expansion of tensor models. For the first model, invariant under $U(N)^3$, we obtain the RG flow of the two melonic couplings and the vacuum phase diagram, from a reformulation with a diagonalizable matrix intermediate field. The discrete chiral symmetry breaks spontaneously and we compare with the three-dimensional Gross-Neveu model. Beyond the massless $U(N)^3$ symmetric phase, we also observe a massive phase of same symmetry and another where the symmetry breaks into $U(N^2)\times U(N/2)\times U(N/2)$. A matrix model invariant under $U(N)\times U(N^2)$, with close properties, is also studied. For the other models, with symmetry groups $U(N)^3$ and $O(N)^5$, a non-melonic coupling (the "wheel") with an optimal scaling in $N$ drives us to a generalized melonic expansion. The kinetic terms are taken of short- and long-range, and we analyze perturbatively, at large-$N$, the RG flows of the sextic couplings up to four loops. Only the rank-3 model displays non-trivial fixed points (two real Wilson-Fisher-like in the short-range case and a line of fixed points in the other). We finally obtain the real conformal dimensions of the primary bilinear operators. In the second part, we establish the first results of perturbative multi-scale renormalization for a quartic scalar field on critical Galton-Watson trees, with a long-range kinetic term. At criticality, an emergent infinite spine provides a space of effective dimension $4/3$ on which to compute averaged correlation fonctions. This approach formalizes the notion of a QFT on a random geometry. We use known probabilistic bounds on the heat-kernel on a random graph reviewed in detail.
... It was later shown in [136,137] that its spectral dimension was two for the whole range γ ∈ (0, 2). The dependence of the Hausdorff dimension with the central charge is on the other hand less clear, with the propositions of Watabiki [138], and Ding and Goswami [139] closest to numeral simulations [140]. Other topologies than the sphere were also worked out, such as the disk [141] or the half-plane [142]. ...
This thesis divides into two parts, focusing on the renormalization of quantum field theories. The first part considers three tensor models in three dimensions, a fermionic quartic with tensors of rank-3 and two bosonic sextic, of ranks 3 and 5. We rely upon the large-N melonic expansion of tensor models. For the first model, invariant under U(N)³, we compute the renormalization group flow of the two melonic couplings and establish the vacuum phase diagram, from a reformulation with a diagonalizable matrix intermediate field. Noting a spontaneous symmetry breaking of the discrete chiral symmetry, the comparison with the three-dimensional Gross-Neveu model is made. Beyond the massless U(N)³ symmetric phase, we also observe a massive phase of same symmetry and another where the symmetry breaks into U(N²) x U(N/2) x U(N/2). A matrix model invariant under U(N) x U(N²), sharing the same properties, is also studied.For the two other tensor models, with symmetry groups U(N)³ and O(N)⁵, a non-melonic coupling (the ``wheel") with an optimal scaling in N drives us to a generalized melonic expansion. The kinetic terms are taken of short and long range, and we analyze perturbatively, at large-N, the renormalization group flows of the sextic couplings up to four loops. While the rank-5 model doesn't present any non-trivial fixed point, that of rank 3 displays two real non-trivial Wilson-Fisher fixed points in the short-range case and a line of fixed points in the other. We finally obtain the real conformal dimensions of the primary operators bilinear in the fundamental field.In the second part, we establish the first results of constructive multi-scale renormalization for a quartic scalar field on critical Galton-Watson trees, with a long-range kinetic term. At the critical point, an emergent infinite spine provides a space of effective dimension 4/3 on which to compute averaged correlation fonctions. This approach formalizes the notion of a quantum field theory on a random geometry. We use known probabilistic bounds on the heat-kernel on a random graph. At the end, we sketch the extension of the formalism to fermions and to a compactified spine.
... The formula (1.2) is consistent with all known bounds for d γ . Moreover, recent numerical simulations by Barkley and Budd [BB19] fit much more closely with (1.2) than with (1.1). However, there is currently no theoretical justification, even at a heuristic level, for (1.2). ...
Let \(\gamma \in (0,2)\), let h be the planar Gaussian free field, and consider the \(\gamma \)-Liouville quantum gravity (LQG) metric associated with h. We show that the essential supremum of the Hausdorff dimension of the boundary of a \(\gamma \)-LQG metric ball with respect to the Euclidean (resp. \(\gamma \)-LQG) metric is \(2 - \frac{\gamma }{d_\gamma }\left( \frac{2}{\gamma } + \frac{\gamma }{2} \right) + \frac{\gamma ^2}{2d_\gamma ^2}\) (resp. \(d_\gamma -1\)), where \(d_\gamma \) is the Hausdorff dimension of the whole plane with respect to the \(\gamma \)-LQG metric. For \(\gamma = \sqrt{8/3}\), in which case \(d_{\sqrt{8/3}}=4\), we get that the essential supremum of Euclidean (resp. \(\sqrt{8/3}\)-LQG) dimension of a \(\sqrt{8/3}\)-LQG ball boundary is 5/4 (resp. 3). We also compute the essential suprema of the Euclidean and \(\gamma \)-LQG Hausdorff dimensions of the intersection of a \(\gamma \)-LQG ball boundary with the set of metric \(\alpha \)-thick points of the field h for each \(\alpha \in \mathbb R\). Our results show that the set of \(\gamma /d_\gamma \)-thick points on the ball boundary has full Euclidean dimension and the set of \(\gamma \)-thick points on the ball boundary has full \(\gamma \)-LQG dimension.
... As one of its most intricate features, the composite operator formalism employed in this work could act as a connector between Asymptotic Safety [22,23] and more geometric approaches to quantum gravity based on causal dynamical triangulations [80,81] or random geometry. In d = 2 dimensions, a natural benchmark would involve a quantitative comparison of scaling properties associated with the geodesic length recently considered in Pagani and Reuter [19], Becker and Pagani [29,30], Becker et al. [31], and Houthoff et al. [32] and exact computations for random discrete surfaces in the absence of matter fields [21,82] as well as rigorous and numerical bounds arising from Liouville Gravity in the presence of matter [83,84]. On the renormalization group side this will involve taking limits akin to Nink and Reuter [85]. ...
The asymptotic safety program builds on a high-energy completion of gravity based on the Reuter fixed point, a non-trivial fixed point of the gravitational renormalization group flow. At this fixed point the canonical mass-dimension of coupling constants is balanced by anomalous dimensions induced by quantum fluctuations such that the theory enjoys quantum scale invariance in the ultraviolet. The crucial role played by the quantum fluctuations suggests that the geometry associated with the fixed point exhibits non-manifold like properties. In this work, we continue the characterization of this geometry employing the composite operator formalism based on the effective average action. Explicitly, we give a relation between the anomalous dimensions of geometric operators on a background d-sphere and the stability matrix encoding the linearized renormalization group flow in the vicinity of the fixed point. The eigenvalue spectrum of the stability matrix is analyzed in detail and we identify a “perturbative regime” where the spectral properties are governed by canonical power counting. Our results recover the feature that quantum gravity fluctuations turn the (classically marginal) R2-operator into a relevant one. Moreover, we find strong indications that higher-order curvature terms present in the two-point function play a crucial role in guaranteeing the predictive power of the Reuter fixed point.
... As one of its most intricate features, the composite operator formalism employed in this work could act as a connector between Asymptotic Safety [22,23] and more geometric approaches to quantum gravity based on causal dynamical triangulations [75,76] or random geometry. In d = 2 dimensions, a natural benchmark would involve a quantitative comparison of scaling properties associated with the geodesic length recently considered in [19,29,30,31,32] and exact computations for random discrete surfaces in the absence of matter fields [21,77] as well as rigorous and numerical bounds arising from Liouville Gravity in the presence of matter [78,79]. On the renormalization group side this will involve taking limits akin to [80]. ...
The asymptotic safety program builds on a high-energy completion of gravity based on the Reuter fixed point, a non-trivial fixed point of the gravitational renormalization group flow. At this fixed point the canonical mass-dimension of coupling constants is balanced by anomalous dimensions induced by quantum fluctuations such that the theory enjoys quantum scale invariance in the ultraviolet. The crucial role played by the quantum fluctuations suggests that the geometry associated with the fixed point exhibits non-manifold like properties. In this work, we continue the characterization of this geometry employing the composite operator formalism based on the effective average action. Explicitly, we give a relation between the anomalous dimensions of geometric operators on a background $d$-sphere and the stability matrix encoding the linearized renormalization group flow in the vicinity of the fixed point. The eigenvalue spectrum of the stability matrix is analyzed in detail and we identify a ``perturbative regime'' where the spectral properties are governed by canonical power counting. Our results recover the feature that quantum gravity fluctuations turn the (classically marginal) $R^2$-operator into a relevant one. Moreover, we find strong indications that higher-order curvature terms present in the two-point function play a crucial role in guaranteeing the predictive power of the Reuter fixed point.
