Evidence for Asymptotic Safety from Dimensional Reduction in Causal Dynamical Triangulations

We calculate the spectral dimension for a nonperturbative lattice approach to quantum gravity, known as causal dynamical triangulations (CDT), showing that the dimension of spacetime smoothly decreases from approximately 4 on large distance scales to approximately 3/2 on small distance scales. This novel result may provide a possible resolution to a long-standing argument against the asymptotic safety scenario. A method for determining the relative lattice spacing within the physical phase of the CDT parameter space is also outlined, which might prove useful when studying renormalization group flow in models of lattice quantum gravity.Comment: 21 pages, 8 figures, 4 tables. Typos corrected, 3 tables added. Conclusions unchanged. Conforms with version published in JHE

New higher-order transition in causal dynamical triangulations

We reinvestigate the recently discovered bifurcation phase transition in Causal Dynamical Triangulations (CDT) and provide further evidence that it is a higher order transition. We also investigate the impact of introducing matter in the form of massless scalar fields to CDT. We discuss the impact of scalar fields on the measured spatial volumes and fluctuation profiles in addition to analysing how the scalar fields influence the position of the bifurcation transition.Comment: 15 pages, 11 figures. Conforms with version accepted for publication in Phys. Rev.

Signal from noise retrieval from one and two-point Green's function - comparison

We compare two methods of eigen-inference from large sets of data, based on the analysis of one-point and two-point Green's functions, respectively. Our analysis points at the superiority of eigen-inference based on one-point Green's function. First, the applied by us method based on Pad?e approximants is orders of magnitude faster comparing to the eigen-inference based on uctuations (two-point Green's functions). Second, we have identified the source of potential instability of the two-point Green's function method, as arising from the spurious zero and negative modes of the estimator for a variance operator of the certain multidimensional Gaussian distribution, inherent for the two-point Green's function eigen-inference method. Third, we have presented the cases of eigen-inference based on negative spectral moments, for strictly positive spectra. Finally, we have compared the cases of eigen-inference of real-valued and complex-valued correlated Wishart distributions, reinforcing our conclusions on an advantage of the one-point Green's function method.Comment: 14 pages, 8 figures, 3 table

The moduli space of isometry classes of globally hyperbolic spacetimes

This is the last article in a series of three initiated by the second author. We elaborate on the concepts and theorems constructed in the previous articles. In particular, we prove that the GH and the GGH uniformities previously introduced on the moduli space of isometry classes of globally hyperbolic spacetimes are different, but the Cauchy sequences which give rise to well-defined limit spaces coincide. We then examine properties of the strong metric introduced earlier on each spacetime, and answer some questions concerning causality of limit spaces. Progress is made towards a general definition of causality, and it is proven that the GGH limit of a Cauchy sequence of $\mathcal{C}^{\pm}_{\alpha}$, path metric Lorentz spaces is again a $\mathcal{C}^{\pm}_{\alpha}$, path metric Lorentz space. Finally, we give a necessary and sufficient condition, similar to the one of Gromov for the Riemannian case, for a class of Lorentz spaces to be precompact.Comment: 29 pages, 9 figures, submitted to Class. Quant. Gra

Euler characteristic of coherent sheaves on simplicial torics via the Stanley-Reisner ring

We combine work of Cox on the total coordinate ring of a toric variety and results of Eisenbud-Mustata-Stillman and Mustata on cohomology of toric and monomial ideals to obtain a formula for computing the Euler characteristic of a Weil divisor D on a complete simplicial toric variety in terms of graded pieces of the Cox ring and Stanley-Reisner ring. The main point is to use Alexander duality to pass from the toric irrelevant ideal, which appears in the computation of the Euler characteristic of D, to the Stanley-Reisner ideal of the fan, which is used in defining the Chow ring. The formula also follows from work of Maclagan-Smith.Comment: 9 pages 1 figur

Lattice Gauge Theory -- Present Status

Lattice gauge theory is our primary tool for the study of non-perturbative phenomena in hadronic physics. In addition to giving quantitative information on confinement, the approach is yielding first principles calculations of hadronic spectra and matrix elements. After years of confusion, there has been significant recent progress in understanding issues of chiral symmetry on the lattice. (Talk presented at HADRON 93, Como, Italy, June 1993.)Comment: 11 pages, BNL-4946

Random walks on combs

We develop techniques to obtain rigorous bounds on the behaviour of random walks on combs. Using these bounds we calculate exactly the spectral dimension of random combs with infinite teeth at random positions or teeth with random but finite length. We also calculate exactly the spectral dimension of some fixed non-translationally invariant combs. We relate the spectral dimension to the critical exponent of the mass of the two-point function for random walks on random combs, and compute mean displacements as a function of walk duration. We prove that the mean first passage time is generally infinite for combs with anomalous spectral dimension.Comment: 42 pages, 4 figure

First-order phase transition in the tethered surface model on a sphere

We show that the tethered surface model of Helfrich and Polyakov-Kleinert undergoes a first-order phase transition separating the smooth phase from the crumpled one. The model is investigated by the canonical Monte Carlo simulations on spherical and fixed connectivity surfaces of size up to N=15212. The first-order transition is observed when N>7000, which is larger than those in previous numerical studies, and a continuous transition can also be observed on small-sized surfaces. Our results are, therefore, consistent with those obtained in previous studies on the phase structure of the model.Comment: 6 pages with 7 figure

A non-perturbative Lorentzian path integral for gravity

A well-defined regularized path integral for Lorentzian quantum gravity in three and four dimensions is constructed, given in terms of a sum over dynamically triangulated causal space-times. Each Lorentzian geometry and its associated action have a unique Wick rotation to the Euclidean sector. All space-time histories possess a distinguished notion of a discrete proper time. For finite lattice volume, the associated transfer matrix is self-adjoint and bounded. The reflection positivity of the model ensures the existence of a well-defined Hamiltonian. The degenerate geometric phases found previously in dynamically triangulated Euclidean gravity are not present. The phase structure of the new Lorentzian quantum gravity model can be readily investigated by both analytic and numerical methods.Comment: 11 pages, LaTeX, improved discussion of reflection positivity, conclusions unchanged, references update

Cosmic voids and filaments from quantum gravity

Using computer simulations we study the geometry of a typical quantum universe, i.e. the geometry one might expect before a possible period of inflation. We display it using coordinates defined by means of four classical scalar fields satisfying the Laplace equation with non-trivial boundary conditions. The field configurations reveal cosmic web structures surprisingly similar to the ones observed in the present-day Universe. Inflation might make these structures relevant for our Universe.Comment: 4 pages, 2 figure