The Experts below are selected from a list of 303 Experts worldwide ranked by ideXlab platform
John F. Wheater - One of the best experts on this subject based on the ideXlab platform.
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multigraph models for causal quantum gravity and scale dependent Spectral Dimension
Journal of Physics A, 2012Co-Authors: Georgios Giasemidis, John F. Wheater, Stefan ZohrenAbstract:We study random walks on ensembles of a specific class of random multigraphs which provide an ‘effective graph ensemble’ for the causal dynamical triangulation (CDT) model of quantum gravity. In particular, we investigate the Spectral Dimension of the multigraph ensemble for recurrent as well as transient walks. We investigate the circumstances in which the Spectral Dimension and Hausdorff Dimension are equal and show that this occurs when ρ, the exponent for anomalous behaviour of the resistance to infinity, is zero. The concept of scale dependent Spectral Dimension in these models is introduced. We apply this notion to a multigraph ensemble with a measure induced by a size biased critical Galton–Watson process which has a scale dependent Spectral Dimension of two at large scales and one at small scales. We conclude by discussing a specific model related to four Dimensional CDT which has a Spectral Dimension of four at large scales and two at small scales.
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Spectral Dimension flow on continuum random multigraph
2012Co-Authors: Georgios Giasemidis, John F. Wheater, Stefan ZohrenAbstract:We review a recently introduced effective graph approximation of causal dynamical triangulations (CDT), the multigraph ensemble. We argue that it is well suited for analytical computations and that it captures the physical degrees of freedom which are important for the reduction of the Spectral Dimension as observed in numerical simulations of CDT. In addition multigraph models allow us to study the relationship between the Spectral Dimension and the Hausdorff Dimension, thus establishing a link to other approaches to quantum gravity.
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continuum random combs and scale dependent Spectral Dimension
Journal of Physics A, 2011Co-Authors: Max R. Atkin, Georgios Giasemidis, John F. WheaterAbstract:Numerical computations have suggested that in causal dynamical triangulation models of quantum gravity (CDT) the effective Dimension of spacetime in the ultraviolet (UV) is lower than in the infrared (IR). In this paper we develop a simple model based on the previous work on random combs, which share some of the properties of CDT, in which this effect can be shown to occur analytically. We construct a definition for short- and long-distance Spectral Dimensions and show that the random comb models exhibit scale-dependent Spectral Dimension defined in this way. We also observe that a hierarchy of apparent Spectral Dimensions may be obtained in the cross-over region between UV and IR regimes for suitable choices of the continuum variables. Our main result is valid for a wide class of tooth length distributions thereby extending previous work on random combs by Durhuus et al.
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Continuum Random Combs and Scale Dependent Spectral Dimension
Journal of Physics A: Mathematical and Theoretical, 2011Co-Authors: Max R. Atkin, Georgios Giasemidis, John F. WheaterAbstract:Numerical computations have suggested that in causal dynamical triangulation models of quantum gravity the effective Dimension of spacetime in the UV is lower than in the IR. In this paper we develop a simple model based on previous work on random combs, which share some of the properties of CDT, in which this effect can be shown to occur analytically. We construct a definition for short and long distance Spectral Dimensions and show that the random comb models exhibit scale dependent Spectral Dimension defined in this way. We also observe that a hierarchy of apparent Spectral Dimensions may be obtained in the cross-over region between UV and IR regimes for suitable choices of the continuum variables. Our main result is valid for a wide class of tooth length distributions thereby extending previous work on random combs by Durhuus et al.
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On the Spectral Dimension of Causal Triangulations
Journal of Statistical Physics, 2010Co-Authors: Bergfinnur Durhuus, Thordur Jonsson, John F. WheaterAbstract:We introduce an ensemble of infinite causal triangulations, called the uniform infinite causal triangulation, and show that it is equivalent to an ensemble of infinite trees, the uniform infinite planar tree. It is proved that in both cases the Hausdorff Dimension almost surely equals 2. The infinite causal triangulations are shown to be almost surely recurrent or, equivalently, their Spectral Dimension is almost surely less than or equal to 2. We also establish that for certain reduced versions of the infinite causal triangulations the Spectral Dimension equals 2 both for the ensemble average and almost surely. The triangulation ensemble we consider is equivalent to the causal dynamical triangulation model of two-Dimensional quantum gravity and therefore our results apply to that model.
Piero Nicolini - One of the best experts on this subject based on the ideXlab platform.
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Un-Spectral Dimension and quantum spacetime phases
Physics Letters B, 2011Co-Authors: Piero Nicolini, Euro SpallucciAbstract:In this Letter, we propose a new scenario emerging from the conjectured presence of a minimal length $\ell$ in the spacetime fabric, on the one side, and the existence of a new scale invariant, continuous mass spectrum, of un-particles on the other side. We introduce the concept of \textit{un-Spectral Dimension} $\mathbb{D}_U$ of a $d$-Dimensional, euclidean (quantum) spacetime, as the Spectral Dimension measured by an "un-particle" probe. We find a general expression for the un-Spectral Dimension $\mathbb{D}_U$ labelling different spacetime phases: a semi-classical phase, where ordinary Spectral Dimension gets contribution from the scaling Dimension $d_U$ of the un-particle probe ; a critical "Planckian phase", where four-Dimensional spacetime can be effectively considered two-Dimensional when $d_U=1$; a "Trans-Planckian phase", which is accessible to un-particle probes only, where spacetime as we currently understand it looses its physical meaning
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Un-Spectral Dimension and quantum spacetime phases
Physics Letters B, 2010Co-Authors: Piero Nicolini, Euro SpallucciAbstract:Abstract In this Letter, we propose a new scenario emerging from the conjectured presence of a minimal length l in the spacetime fabric, on the one side, and the existence of a new scale invariant, continuous mass spectrum, of un-particles on the other side. We introduce the concept of un-Spectral Dimension D U of a d -Dimensional, euclidean (quantum) spacetime, as the Spectral Dimension measured by an “un-particle” probe. We find a general expression for the un-Spectral Dimension D U labelling different spacetime phases: a semi-classical phase, where ordinary Spectral Dimension gets contribution from the scaling Dimension d U of the un-particle probe; a critical “Planckian phase”, where four-Dimensional spacetime can be effectively considered two-Dimensional when d U = 1 ; a “Trans-Planckian phase”, which is accessible to un-particle probes only, where spacetime as we currently understand it looses its physical meaning.
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Spectral Dimension of a quantum universe
Physical Review D, 2010Co-Authors: Leonardo Modesto, Piero NicoliniAbstract:In this paper, we calculate in a transparent way the Spectral Dimension of a quantum spacetime, considering a diffusion process propagating on a fluctuating manifold. To describe the erratic path of the diffusion, we implement a minimal length by averaging the graininess of the quantum manifold in the flat space case. As a result we obtain that, for large diffusion times, the quantum spacetime behaves like a smooth differential manifold of discrete Dimension. On the other hand, for smaller diffusion times, the spacetime looks like a fractal surface with a reduced effective Dimension. For the specific case in which the diffusion time has the size of the minimal length, the spacetime turns out to have a Spectral Dimension equal to 2, suggesting a possible renormalizable character of gravity in this regime. For smaller diffusion times, the Spectral Dimension approaches zero, making any physical interpretation less reliable in this extreme regime. We extend our result to the presence of a background field and curvature. We show that in this case the Spectral Dimension has a more complicated relation with the diffusion time, and conclusions about the renormalizable character of gravity become less straightforward with respect to what we found with the flat space analysis.
Euro Spallucci - One of the best experts on this subject based on the ideXlab platform.
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Un-Spectral Dimension and quantum spacetime phases
Physics Letters B, 2011Co-Authors: Piero Nicolini, Euro SpallucciAbstract:In this Letter, we propose a new scenario emerging from the conjectured presence of a minimal length $\ell$ in the spacetime fabric, on the one side, and the existence of a new scale invariant, continuous mass spectrum, of un-particles on the other side. We introduce the concept of \textit{un-Spectral Dimension} $\mathbb{D}_U$ of a $d$-Dimensional, euclidean (quantum) spacetime, as the Spectral Dimension measured by an "un-particle" probe. We find a general expression for the un-Spectral Dimension $\mathbb{D}_U$ labelling different spacetime phases: a semi-classical phase, where ordinary Spectral Dimension gets contribution from the scaling Dimension $d_U$ of the un-particle probe ; a critical "Planckian phase", where four-Dimensional spacetime can be effectively considered two-Dimensional when $d_U=1$; a "Trans-Planckian phase", which is accessible to un-particle probes only, where spacetime as we currently understand it looses its physical meaning
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Un-Spectral Dimension and quantum spacetime phases
Physics Letters B, 2010Co-Authors: Piero Nicolini, Euro SpallucciAbstract:Abstract In this Letter, we propose a new scenario emerging from the conjectured presence of a minimal length l in the spacetime fabric, on the one side, and the existence of a new scale invariant, continuous mass spectrum, of un-particles on the other side. We introduce the concept of un-Spectral Dimension D U of a d -Dimensional, euclidean (quantum) spacetime, as the Spectral Dimension measured by an “un-particle” probe. We find a general expression for the un-Spectral Dimension D U labelling different spacetime phases: a semi-classical phase, where ordinary Spectral Dimension gets contribution from the scaling Dimension d U of the un-particle probe; a critical “Planckian phase”, where four-Dimensional spacetime can be effectively considered two-Dimensional when d U = 1 ; a “Trans-Planckian phase”, which is accessible to un-particle probes only, where spacetime as we currently understand it looses its physical meaning.
Ginestra Bianconi - One of the best experts on this subject based on the ideXlab platform.
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Simplicial complexes: higher-order Spectral Dimension and dynamics
Journal of Physics: Complexity, 2020Co-Authors: Joaquín J Torres, Ginestra BianconiAbstract:Simplicial complexes constitute the underlying topology of interacting complex systems including among the others brain and social interaction networks. They are generalized network structures that allow to go beyond the framework of pairwise interactions and to capture the many-body interactions between two or more nodes strongly affecting dynamical processes. In fact, the simplicial complexes topology allows to assign a dynamical variable not only to the nodes of the interacting complex systems but also to links, triangles, and so on. Here we show evidence that the dynamics defined on simplices of different Dimensions can be significantly different even if we compare dynamics of simplices belonging to the same simplicial complex. By investigating the Spectral properties of the simplicial complex model called "Network Geometry with Flavor" we provide evidence that the up and down higher-order Laplacians can have a finite Spectral Dimension whose value increases as the order of the Laplacian increases. Finally we discuss the implications of this result for higher-order diffusion defined on simplicial complexes.
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Probing the Spectral Dimension of quantum network geometries
arXiv: Quantum Physics, 2020Co-Authors: Johannes Nokkala, Jyrki Piilo, Ginestra BianconiAbstract:We consider an environment for an open quantum system described by a "Quantum Network Geometry with Flavor" (QNGF) in which the nodes are coupled quantum oscillators. The geometrical nature of QNGF is reflected in the Spectral properties of the Laplacian matrix of the network which display a finite Spectral Dimension, determining also the frequencies of the normal modes of QNGFs. We show that an a priori unknown Spectral Dimension can be indirectly estimated by coupling an auxiliary open quantum system to the network and probing the normal mode frequencies in the low frequency regime. We find that the network parameters do not affect the estimate; in this sense it is a property of the network geometry, rather than the values of, e.g., oscillator bare frequencies or the constant coupling strength. Numerical evidence suggests that the estimate is also robust both to small changes in the high frequency cutoff and noisy or missing normal mode frequencies. We propose to couple the auxiliary system to a subset of network nodes with random coupling strengths to reveal and resolve a sufficiently large subset of normal mode frequencies.
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The Spectral Dimension of simplicial complexes: a renormalization group theory
Journal of Statistical Mechanics: Theory and Experiment, 2020Co-Authors: Ginestra Bianconi, Sergey N DorogovstevAbstract:Simplicial complexes are increasingly used to study complex system structure and dynamics including diffusion, synchronization and epidemic spreading. The Spectral Dimension of the graph Laplacian is known to determine the diffusion properties at long time scales. Using the renormalization group here we calculate the Spectral Dimension of the graph Laplacian of two classes of non-amenable $d$ Dimensional simplicial complexes: the Apollonian networks and the pseudo-fractal networks. We analyse the scaling of the Spectral Dimension with the topological Dimension $d$ for $d\to \infty$ and we point out that randomness such as the one present in Network Geometry with Flavor can diminish the value of the Spectral Dimension of these structures.
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Synchronization in network geometries with finite Spectral Dimension.
Physical review. E, 2019Co-Authors: Ana P Millán, Joaquín J Torres, Ginestra BianconiAbstract:Recently there is a surge of interest in network geometry and topology. Here we show that the Spectral Dimension plays a fundamental role in establishing a clear relation between the topological and geometrical properties of a network and its dynamics. Specifically we explore the role of the Spectral Dimension in determining the synchronization properties of the Kuramoto model. We show that the synchronized phase can only be thermodynamically stable for Spectral Dimensions above four and that phase entrainment of the oscillators can only be found for Spectral Dimensions greater than two. We numerically test our analytical predictions on the recently introduced model of network geometry called complex network manifolds, which displays a tunable Spectral Dimension.
Astrid Eichhorn - One of the best experts on this subject based on the ideXlab platform.
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Spectral Dimension on spatial hypersurfaces in causal set quantum gravity
Classical and Quantum Gravity, 2019Co-Authors: Astrid Eichhorn, Sumati Surya, Fleur VersteegenAbstract:An important probe of quantum geometry is its Spectral Dimension, defined via a spatial diffusion process. In this work we study the Spectral Dimension of a ``spatial hypersurface'' in a manifoldlike causal set using the induced spatial distance function. In previous work, the diffusion was taken on the full causal set, where the nearest neighbours are unbounded in number. The resulting super-diffusion leads to an increase in the Spectral Dimension at short diffusion times, in contrast to other approaches to quantum gravity. In the current work, by using a temporal localisation in the causal set, the number of nearest spatial neighbours is rendered finite. Using numerical simulations of causal sets obtained from $d=3$ Minkowski spacetime, we find that for a flat spatial hypersurface, the Spectral Dimension agrees with the Hausdorff Dimension at intermediate scales, but shows clear indications of Dimensional reduction at small scales, i.e., in the ultraviolet. The latter is a direct consequence of ``discrete asymptotic silence'' at small scales in causal sets.
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Spectral Dimension in causal set quantum gravity
Classical and Quantum Gravity, 2014Co-Authors: Astrid Eichhorn, Sebastian MizeraAbstract:We evaluate the Spectral Dimension in causal set quantum gravity by simulating random walks on causal sets. In contrast to other approaches to quantum gravity, we find an increasing Spectral Dimension at small scales. This observation can be connected to the non-locality of causal set theory that is deeply rooted in its fundamentally Lorentzian nature. Based on its large-scale behaviour, we conjecture that the Spectral Dimension can serve as a tool to distinguish causal sets that approximate manifolds from those that do not. As a new tool to probe quantum spacetime in different quantum gravity approaches, we introduce a novel Dimensional estimator, the causal Spectral Dimension, based on the meeting probability of two random walkers, which respect the causal structure of the quantum spacetime. We discuss a causal-set example, where the Spectral Dimension and the causal Spectral Dimension differ, due to the existence of a preferred foliation.