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We investigate the geometry of a quantum universe with the topology of the four-torus. The study of non-contractible geodesic loops reveals that a typical quantum geometry consists of a small semi-classical toroidal bulk part, dressed with many outgrowths, which contain most of the four-volume and which have almost spherical topologies, but nevertheless are quite fractal., Comment: 16 pages, 16 figures

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., 4 pages, 2 figures

Using Monte-Carlo computer simulations, we study the impact of matter fields on the geometry of a typical quantum universe in the CDT model of lattice quantum gravity. The quantum universe has the size of a few Planck lengths and the spatial topology of a three-torus. The matter fields are multi-component scalar fields taking values in a torus with circumference $\delta$ in each spatial direction, which acts as a new parameter in the CDT model. Changing $\delta$, we observe a phase transition caused by the scalar field. This discovery may have important consequences for quantum universes with non-trivial topology, since the phase transition can change the topology to a simply connected one., Comment: 5 pages, 5 figures

The paper presents derivation of the impedance matrix based on the rigorous solution of the wave equation obtained by the Wiener-Hopf technique for a semi-infinite unflanged cylindrical duct. The impedance matrix allows, in turn, calculate the acoustic impedance along the duct and, as a special case, the radiation impedance. The analysis is carried out for a multimode incident wave accounting for modes coupling on the duct outlet not only qualitatively but also quantitatively for a selected source operating inside. The quantitative evaluation of the acoustic impedance requires setting of modes amplitudes which has been obtained applying the mode decomposition method to the far-field pressure radiation measurements and theoretical formulae for single mode directivity characteristics for an unflanged duct. Calculation of the acoustic impedance for a non-uniform distribution of the sound pressure and the sound velocity on a duct cross section requires determination of the acoustic power transmitted along/radiated from a duct. In the paper, the impedance matrix, the power, and the acoustic impedance were derived as functions of Helmholtz number and distance from the outlet.

We investigate the transition between the phases $B$ and $C_b$ observed in four-dimensional Causal Dynamical Triangulations (CDT). We find that the critical properties of CDT with toroidal spatial topology are the same as earlier observed in spherical spatial topology where the $B-C_b$ transition was found to be higher-order. This may have important consequences for the existence of the continuum limit of CDT, describing the perspective UV limit of quantum gravity, which potentially can be investigated in the toroidal model., Comment: 17 pages, 10 figures

Causal Dynamical Triangulations (CDT), a candidate theory of nonperturbative quantum gravity in 4D, turns out to have a rich phase structure. We investigate the recently discovered bifurcation phase $C_b$ and relate some of its characteristics to the presence of singular vertices of very high order. The transition lines separating this phase from the "time-collapsed" $B$-phase and the de Sitter phase $C_{dS}$ are of great interest when searching for physical scaling limits. The work presented here sheds light on the mechanisms behind these transitions. First, we study how the $B$-$C_b$ transition signal depends on the volume-fixing implemented in the simulations, and find results compatible with the previously determined second-order character of the transition. The transition persists in a transfer matrix formulation, where the system's time extension is taken to be minimal. Second, we relate the new $C_b$-$C_{dS}$ transition to the appearance of singular vertices, which leads to a direct physical interpretation in terms of a breaking of the homogeneity and isotropy observed in the de Sitter phase when crossing from $C_{dS}$ to the bifurcation phase $C_b$., Comment: 32 pages, 17 figures

The paper presents a method to generate a single user-selected waveguide cut-on mode by means of a mode synthesiser consisting of a set of point sources, each of them modelled by a thin pipe outlet and driven by a separate loudspeaker. Analysis of sound propagation in duct-like cylindrical facilities frequently needs considering the presence of higher cut-on modes together with mode reflection/coupling phenomena occurring on the duct outlet, impedance variations, etc. With increasing reduced frequency (Helmholtz number), description of the field becomes more and more complex. To avoid this complexity, at least in the case of tests carried out in laboratory conditions, the simplest approach seems to be able to generate a single cut-on mode by means of a properly constructed system, because to analyse the multi-mode wave, the contribution to the overall field from individual higher order modes must be established in advance. Based on in-duct field properties, the authors designed a mode synthesiser composed of N ⩽ 13 point sources which was constructed and integrated into the measurement duct. Two methods were used to approach the problem of generating a single higher mode. In the first one, the amplitudes of the individual point sources were calculated by means of the in-duct Green’s function. In the second method, amplitudes of individual sources were calibrated based on sound pressure measurement data taken on the duct cross-section with only one source operating at a time. These measurements, fully automated at a given cross-section, were carried out for some reduced frequencies allowing for propagation of at least four cut-on modes. The advantages and disadvantages of the two methods and the causes of discrepancies between theoretical predictions and the results obtained were analysed.

The paper presents a new approach to the theory of acoustic multi-port networks assuming propagation of multimode wave through the system and applying the scattering matrix formalism. A procedure is proposed which can be applied to improve accuracy of results obtained when analysing acoustic mufflers of complex geometry or other systems composed of several sub-systems joined by duct-like elements. In what follows, the multi-port order is defined by the number of ducts and a selected cross-section of a duct transmitting multimode wave is called the multimode port. The main feature of the presented method is that it allows to analyse each sub-system as a separate multi-port. The scattering matrix representing two selected consecutive sub-systems having a common multimode port could be transformed according to the derived formulae which results in reducing the number of multimode ports by two. The source term can be present in each sub-system. By executing this procedure step by step, the scattering matrix of the whole system can be determined. To ensure good accuracy of final results, the scattering matrix of a discontinuity constituting a degenerated multi-port is calculated with the near field effects taken into account adjusting the number of cut-off modes included in the analysis to the specified continuity conditions at junctions. The next step consists in reduction of the dimension of the “near-field scattering matrix” to the matrix accounting only for cut-on modes propagating in the adjoined ducts. The main advantage and the novelty of the presented procedure is that it can be executed step by step and thus applied to mufflers of complex geometry never leading to the scattering matrix of troublesomely large dimensions and at the same time, ensuring significant accuracy of the obtained results. The proposed method was validated in two ways: by comparing its results with those published in the literature and applying it to some other example mufflers of complex geometry.

CDT is an attempt to formulate a non-perturbative lattice theory of quantum gravity. We describe the phase diagram and analyse the phase transition between phase B and phase C (which is the analogue of the de Sitter phase observed for the spherical spatial topology). This transition is accessible to ordinary Monte Carlo simulations when the topology of space is toroidal. We find that the transition is most likely first order, but with unusual properties. The end points of the transition line are candidates for second order phase transition points where an UV continuum limit might exist., Comment: 18 pages, 7 figures

Causal Dynamical Triangulations is a non-perturbative quantum gravity model, defined with a lattice cut-off. The model can be viewed as defined with a proper time but with no reference to any three-dimensional spatial background geometry. It has four phases, depending on the parameters (the coupling constants) of the model. The particularly interesting behavior is observed in the so-called de Sitter phase, where the spatial three-volume distribution as a function of proper time has a semi-classical behavior which can be obtained from an effective mini-superspace action. In the case of the three-sphere spatial topology, it has been difficult to extend the effective semi-classical description in terms of proper time and spatial three-volume to include genuine spatial coordinates, partially because of the background independence inherent in the model. However, if the spatial topology is that of a three-torus, it is possible to define a number of new observables that might serve as spatial coordinates as well as new observables related to the winding numbers of the three-dimensional torus. The present paper outlines how to define the observables, and how they can be used in numerical simulations of the model., 26 pages, 15 figures

Four-dimensional CDT (causal dynamical triangulations) is a lattice theory of geometries which one might use in an attempt to define quantum gravity non-perturbatively, following the standard procedures of lattice field theory. Being a theory of geometries, the phase transitions which in usual lattice field theories are used to define the continuum limit of the lattice theory will in the CDT case be transitions between different types of geometries. This picture is interwoven with the topology of space which is kept fixed in the lattice theory, the reason being that "classical" geometries around which the quantum fluctuations take place depend crucially on the imposed topology. Thus it is possible that the topology of space can influence the phase transitions and the corresponding critical phenomena used to extract continuum physics. In this article we perform a systematic comparison between a CDT phase transition where space has spherical topology and the "same" transition where space has toroidal topology. The "classical" geometries around which the systems fluctuate are very different it the two cases, but we find that the order of phase transition is not affected by this., Comment: 41 pages, many figures

The paper presents the procedure of experimental derivation of an acoustic two-port scattering matrix for a multimode wave propagating, for example, in a muffler of a complicated design, applying the single-mode generator. The self-designed single-mode generator is constructed in the form of a matrix of acoustic monopoles and is capable to generate a selected single mode. The scattering matrix formalism is a new approach to analyse the acoustical systems containing one or more duct-like elements (joints) connected by some undefined elements such as, for example, acoustic mufflers of different types and so the formalism is frequently applied to analyse the phenomena of noise reduction in such systems. Experimental derivation of the scattering matrix for a wave composed of $N$ modes demands, in general, excitation of $N$ different sound fields at each side of the two-port first. Finally the matrix composed of the modal pressures measurements obtained for each of the exited fields must be inversed to obtain the scattering matrix. Generating single $N$ modes the matrix of the incoming pressure data is, theoretically, diagonal which makes derivation of the scattering matrix much easier. Generation of independent sound fields in the shape of single modes ensures the matrix to be well conditioned.

The article analyses discrepancies in the results of the acoustic pressure measurement carried out for measuring points located on the duct axis at a chosen distance from the sound sources. The measurement set-up consists of a self-designed single mode synthesizer located inside anechoic termination and a measuring section with the microphone moving along the waveguide radius. Several measurement series were carried out for each of the generated consecutive single modes, i.e. (0,0), (1,1), (2,1), (0,1) and a frequency wideband signal for an axisymmetrical and non - axisymmetrical position of a single sound source. For each case, calculations of the mean value and the type A standard uncertainty were executed by means of the classical method but also by the bootstrap estimate of the mean value and the bootstrap estimation of the uncertainty. The results obtained applying each of the listed methods were discussed and compared with each other. The non-zero result of the standard deviation indicates some eccentricity in the measurement set - up and so it can serve as a hint to improve axiality of the system.

We investigate the impact of topology on the phase structure of four-dimensional Causal Dynamical Triangulations (CDT). Using numerical Monte Carlo simulations we study CDT with toroidal spatial topology. We confirm existence of all four distinct phases of quantum geometry earlier observed in CDT with spherical spatial topology. We plot the toroidal CDT phase diagram and find that it looks very similar to the case of the spherical spatial topology., 24 pages, 15 figures

It is convenient to have a device and a method of generating single cut-on modes in cylindrical hard-walled waveguides or at least in laboratory models of such systems. This allows to examine, among other things, properties of various active and/or passive elements inserted in a cylindrical duct by testing them in conditions when the incident (input) wave comprises only one cut-on mode and determining the reflection and transmission coefficients for single selected incident modes. As it has been already demonstrated by the present authors, it is possible to generate single cut-on modes in a circular duct using a small (although increasing with mode order) number of acoustic monopoles arranged properly on a duct cross-section and driven with appropriate acoustic volume amplitudes and phases. Laboratory models of such sources are proposed in this paper and results of tests verifying their directional properties are presented. The other technical issue relating to practical utilization of the proposed method is the possible error introduced by the apparatus used for scanning the acoustic field inside the duct model. It is shown that insertion of the measuring probe changes the total energy radiated into the free space only by a fraction of a decibel.

We show that for 1+1 dimensional Causal Dynamical Triangulations (CDT) coupled to 4 massive scalar fields one can construct an effective transfer matrix if the masses squared is larger than or equal to 0.05. The properties of this transfer matrix can explain why CDT coupled to matter can behave completely different from "pure" CDT. We identify the important critical exponent in the effective action, which may determine the universality class of the model., Comment: 14 pages,lot of figures

We study matter with central charge $c >1$ coupled to two-dimensional (2d) quantum gravity, here represented as causal dynamical triangulations (CDT). 2d CDT is known to provide a regularization of (Euclidean) 2d Ho��ava-Lifshitz quantum gravity. The matter fields are massive Gaussian fields, where the mass is used to monitor the central charge $c$. Decreasing the mass we observe a higher order phase transition between an effective $c=0$ theory and a theory where $c>1$. In this sense the situation is somewhat similar to that observed for "standard" dynamical triangulations (DT) which provide a regularization of 2d quantum Liouville gravity. However, the geometric phase observed for $c >1$ in CDT is very different from the corresponding phase observed for DT., 16 pages, 10 figures

A method was developed for the computerized volumetric assessment of the intracranial cerebrospinal fluid (CSF) distribution. The study involved 62 patients differentiated into two groups: with CSF resorption disorders (normal pressure hydrocephalus - 30 patients) and without CSF resorption disorders (various types of brain atrophy - 32 patients). The goal of the study was to ascertain whether the assessment, depending on the linear discriminant analysis of volumetric brain features, could be an effective tool differentiating the two groups. Volumetric measurements were performed using VisNow software. For each patient, five features were determined and subjected to discriminant analysis: CSF volume in the subarachnoid space and basal cisterns (SV), CSF volume in the intracranial ventricular system (VV), brain volume (BV), total intracranial CSF volume (FV), and total intracranial volume (TV). Discriminant analysis enables the achievement of a high percentage of correct classification of patients to the appropriate group determined on the result of a lumbar infusion test. The discriminator, based on three features: BV, SV, and VV, showed a complete separation of the groups; irrespective of age. The squared Mahalanobis distance was 70.8. The results confirmed the applicability of the volumetric method. Discriminant analysis seems a useful tool leading to the acquisition of a computer-aided method for the differential diagnosis of CSF resorption disorders.

Modi cation of the Measurement Setup for Examination of Acoustic Field Distribution Inside Cylindrical Ducts for Axisymmetric Excitation Š. Gorazda,∗, J. Jurkiewicz, S. Raab, A. Snakowska AGH University of Science and Technology a Faculty of Mechanical Engineering and Robotics b Faculty of Electrical Engineering, Automatics, Computer Science and Biomedical Engineering al. A. Mickiewicza 30, 30-059 Krakow, Poland

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., 15 pages, 11 figures. Conforms with version accepted for publication in Phys. Rev. D

3+1 dimensional Causal Dynamical Triangulations (CDT) describe a quantum theory of fluctuating geometries without the introduction of a background geometry. If the topology of space is constrained to be that of a three-dimensional torus we show that the system will fluctuate around a dynamically formed background geometry which can be understood from a simple minisuperspace action which contains both a classical part and a quantum part. We determine this action by integrating out degrees of freedom in the full model, as well as by transfer matrix methods., Comment: 28 pages, 15 figures

The first teacher was Jesus Christ. He prepared the apostles to preach the Gospel. Then prepare for the priesthood, took various forms. In the first centuries of Christianity, it was important ascetical and spiritual preparation for the priesthood. Appropriate knowledge and holy life – it was a program to prepare for the priesthood. Created a community where young people prepare for the priesthood.

The paper presents a method of theoretical derivation and numerical calculation of the open-end correction coefficient for an arbitrary cut-on mode propagating in acoustic waveguide. Actually, the so-called open-end correction coefficient of acoustic tube, frequently discussed in literature, refers to specific conditions, when the wave heading the outlet is the plane wave. It follows from the fact that the plane wave is a commonly applied approximation when considering phenomena in duct-like devices or systems (tubes, musical instruments, heating or ventilation systems). The aim of the paper is to extend the concept of the open-end correction on the so-called higher Bessel modes, that under some conditions can also propagate in a duct. Theoretical results, forming the basis for numerical calculations, were obtained by considering diffraction at the duct end and applying the Wiener–Hopf factorization method. As a result, the formula for the acoustic field inside the duct was derived. For each Bessel mode present in the incident wave the reflected wave is composed of all cut-on modes of the same circumferential order. Each mode present in the reflected wave is characterized by the complex reflection/coupling coefficient, argument of which describes phase change at the duct end and therefore the open-end correction coefficient can be attributed to each coupled pair of modes.

In the paper, the specific acoustic impedance of outlet of an unbaffled duct for incident wave composed ofoneormanycircumferentialcut-onductmodeshasbeenderivedandpresentedongraphs. Whenconsideringthemultimodeexcitationtheequalenergypermodeprincipleandrandomphaseshavebeenassumed. Experimentallyverified assumption on equal energy per mode allows for quantitative analysis of the sound field and thusprovides more physical insight into the in-duct and out-duct phenomena. The assumption of random phase wasimplemented by means of the Monte Carlo method and so the expected value, the standard deviation and thepercentilecurvesoftheimpedancewerepresented. Numericaldataobtainedaccordingtothederivedtheoreticalformulaeforsingleandmultimodeincidentwavemanifeststrongmodalcharacteranddependenceonthemodesphases. The assumption of multimode incident wave has brought the theoretical model closer to what is beingobserved in practice, especially when ducts of large radius, such as heating and air conditioning systems orturbojetengines,areconsidered.PACS:43.20.Mv,43.20.Rz

We apply the concept of free random variables to doubly correlated (Gaussian) Wishart random matrix models, appearing, for example, in a multivariate analysis of financial time series, and displaying both inter-asset cross-covariances and temporal auto-covariances. We give a comprehensive introduction to the rich financial reality behind such models. We explain in an elementary way the main techniques of free random variables calculus, with a view to promoting them in the quantitative finance community. We apply our findings to tackle several financially relevant problems, such as a universe of assets displaying exponentially decaying temporal covariances, or the exponentially weighted moving average, both with an arbitrary structure of cross-covariances.

The aim of this paper consists in presentation of spatial distribution of energy radiated from the outlet of a cylindrical duct in case of multimodal excitation. As the mathematical model, the semi-infinite unflanged hard duct was applied, so the results account for the diffraction phenomena occurring at the duct opening. They are based on solution of the wave equation with Neumann boundary conditions obtained by means of the Wiener-Hopf method. The multimode excitation was considered assuming the incident wave composed of all propagating modes representing solutions of the wave equation at a given diffraction parameter (Helmholtz number) and equal energy per mode (EEpM) allowing to derive mode amplitudes. The paper presents mathematical formulae and numerical results for the power radiated outside, the power reflection coefficients and the power-gain function for different values of the diffraction parameter ka. The presented results can be applied for modeling of passive and/or active noise reducing systems.

We investigate the impact of spatial topology in 3+1 dimensional causal dynamical triangulations (CDT) by performing numerical simulations with toroidal spatial topology instead of the previously used spherical topology. In the case of spherical spatial topology we observed in the so-called phase C an average spatial volume distribution n(t) which after a suitable time redefinition could be identified as the spatial volume distribution of the four-sphere. Imposing toroidal spatial topology we find that the average spatial volume distribution n(t) is constant. By measuring the covariance matrix of spatial volume fluctuations we determine the form of the effective action. The difference compared to the spherical case is that the effective potential has changed such that it allows a constant average n(t). This is what we observe and this is what one would expect from a minisuperspace GR action where only the scale factor is kept as dynamical variable. Although no background geometry is put in by hand, the full quantum theory of CDT is also with toroidal spatial toplogy able to identify a classical background geometry around which there are well defined quantum fluctuations., Comment: 22 pages, 10 figures

We search for a continuum limit in the causal dynamical triangulation (CDT) approach to quantum gravity by determining the change in lattice spacing using two independent methods. The two methods yield similar results that may indicate how to tune the relevant couplings in the theory in order to take a continuum limit., Comment: 19 pages, 8 figures. Title change and journal reference added

This article considers non-Gaussian random matrices consisting of random variables with heavy-tailed probability distributions. In probability theory heavy tails of distributions describe rare but violent events which usually have a dominant influence on the statistics. Furthermore, they completely change the universal properties of eigenvalues and eigenvectors of random matrices. This article focuses on the universal macroscopic properties of Wigner matrices belonging to the Lévy basin of attraction, matrices representing stable free random variables, and a class of heavy-tailed matrices obtained by parametric deformations of standard ensembles. It first examines the properties of heavy-tailed symmetric matrices known as Wigner–Lévy matrices before discussing free random variables and free Lévy matrices as well as heavy-tailed deformations. In particular, it describes random matrix ensembles obtained from standard ensembles by a reweighting of the probability measure. It also analyses several matrix models belonging to heavy-tailed random matrices and presents methods for integrating them.

Imaging studies make it possible not only to visualize the general structure of the brain but also to take precise measurements of brain tissue volume and the size of individual lobes and their structure. The aim of this study was to determine the ratio of cerebrospinal fluid (CSF) volume contained in the intracranial ventricular system to the brain tissue volume and the ratio of CSF volume in the subarachnoid space and basal cisterns to the brain tissue volume.The evaluation of volumetric measurements of computed tomographic (CT) images was undertaken on 23 male and female patients (average age 56.9 ± 6.1) diagnosed with normal pressure hydrocephalus (NPH) and 27 male and female patients (average age 70.6 ± 5.2) diagnosed with brain atrophy (BA). In the CT imaging studies, the total brain tissue volume and CSF volume collected in the intracranial fluid cavities were mapped to a colour scale. The VisNow software was used for volumetric evaluation. The groups were compared by means of the non-parametric Kolmogorov-Smirnov test (K-S) for independent samples. Paired data were compared by means of the nonparametric Wilcoxon test.The volumes of brain (brain volume - BV) and cerebrospinal fluid (fluid volume - FV) differ greatly from each other in both groups BA and NPH. The SBR (subarachnoid space and basal cisterns-to-brain ratio) and VBR (ventricle-to-brain ratio) indicators differ significantly and very much within the NPH group as well as within the BA group. In the NPH group a clearly higher value of VBR can be observed in comparison with the BA group. There was a higher value of SBR in the BA than the NPH group.The simultaneous use of two indicators, VBR and SBR, on a study group of 50 patients enabled the total separation of NPH and BA groups. This differentiation can have real diagnostic value. Thus the volumetric assessment of the volume of CSF and brain tissue based on CT of the head can become an important part of the differential diagnosis of hydrocephalus and brain atrophy.

A variety of diagnostic methods in normal pressure hydrocephalus and brain atrophy, it testifies to the difficulties in contrast to both teams. Develop a safe diagnostic tool having comparable statistical accuracy of the infusion test results would be of great clinical significance. The chances of implementing such a procedure in clinical practice, it may be a combination of these syndromes in the functional assessment (Posturography) the assessment of morphological CNS (volumetric).To evaluate and summary of the results of two non-invasive diagnostic methods morphological (posturography) and functional (volumetric) with results that were obtained from tests infusion.The evaluation of volumetric and posturographic research was conducted in 39 patients including 27 patients with brain atrophy (BA) and in 12 patients with hydrocephalus (NPH). Volumetric measurements were made using proprietary software VisNow. Posturographic measurements performed using manual posturograph Pro-Med.Both the volume of CSF obtained by volumetric assessment, as well as the values of the analyzed parameters posturographic, highly correlate with the results of resistance resorption (R). The BA radius values sway with eyes open and closed R_OZ R_OO were consistent with test results infusion of 100% in the NPH radius values showed 75% compliance with the test results infusion. Volumetric parameters: the volume of CSF contained in the subarachnoid space and basal cisterns (SV) and the volume of CSF system intracranial ventricular (VV) were found in both patients NPH and BA 100% compatible resorption resistance (R).Simultaneous use of two non-standard methods of non-invasive (posturographic and volumetric) in the diagnostic process is a proposal to recognize normal pressure hydrocephalus and brain atrophy non-invasive methods, while maintaining very high statistical significance.

We compare two methods of eigeninference from large sets of data. Our analysis points at the superiority of our eigeninference method based on one-point Green's functions and Pade approximants over a method based on fluctuations and two-point Green's functions. The first method is orders of magnitude faster than the second one; moreover, we found a source of potential instability of the second method and identified it as arising from the spurious zero and negative modes of the estimator for the variance operator of a certain multidimensional Gaussian distribution, inherent for that method. We also present eigeninference based on spectral moments of negative orders, for strictly positive spectra. Finally, we compare the cases of eigeninference of real-valued and complex-valued correlated Wishart distributions, reinforcing our conclusions on the advantage of the one-point Green's function method.

We study the effective transfer matrix within the semiclassical and bifurcation phases of CDT quantum gravity. We find that for sufficiently large lattice volumes the kinetic term of the effective transfer matrix has a different sign in each of the two phases. We argue that this sign change can be viewed as a Wick rotation of the metric. We discuss the likely microscopic mechanism responsible for the bifurcation phase transition, and propose an order parameter that can potentially be used to determine the precise location and order of the transition. Using the effective transfer matrix we approximately locate the position of the bifurcation transition in some region of coupling constant space, allowing us to present an updated version of the CDT phase diagram., 16 pages, 9 figures

By explicit construction, we show that one can in a simple way introduce and measure gravitational holonomies and Wilson loops in lattice formulations of nonperturbative quantum gravity based on (causal) dynamical triangulations. We use this setup to investigate a class of Wilson line observables associated with the world line of a point particle coupled to quantum gravity, and deduce from their expectation values that the underlying holonomies cover the group manifold of SO(4) uniformly.

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., 21 pages, 8 figures, 4 tables. Typos corrected, 3 tables added. Conclusions unchanged. Conforms with version published in JHEP

We study the two-point correlation function in the model of branched polymers and its relation to the critical behaviour of the model. We show that the correlation function has a universal scaling form in the generic phase with the only scale given by the size of the polymer. We show that the origin of the singularity of the free energy at the critical point is different from that in the standard statistical models. The transition is related to the change of the dimensionality of the system., Comment: 10 Pages, Latex2e, uses elsart.cls, 1 figure included

We measure the effective action in all three phases of 4-dimensional Causal Dynamical Triangulations (CDT) using the transfer matrix method. The transfer matrix is parametrized by the total 3-volume of the CDT universe at a given (discrete) time. We present a simple effective model based on the transfer matrix measured in the de Sitter phase. It allows us to reconstruct the results of full CDT in this phase. We argue that the transfer matrix method is valid not only inside the de Sitter phase ('C') but also in the other two phases. A parametrization of the measured transfer matrix / effective action in the 'A' and 'B' phases is proposed and the relation to phase transitions is explained. We discover a potentially new 'bifurcation' phase separating the de Sitter phase ('C') and the 'collapsed' phase ('B')., Comment: 33 pages, 24 figures

Causal dynamical triangulations provide a nonperturbative regularization of a theory of quantum gravity. We describe how it connects to the asymptotic safety program and to the Hořava–Lifshitz gravity theory and present the most recent results from computer simulations.

A recently proposed renormalization group technique, based on the hierarchical structures present in theories with fluctuating geometry, is implemented in the model of branched polymers. The renormalization group equations can be solved analytically, and the flow in coupling constant space can be determined.

The Causal Dynamical Triangulation model of quantum gravity (CDT) is a proposition to evaluate the path integral over space-time geometries using a lattice regularization with a discrete proper time and geometries realized as simplicial manifolds. The model admits a Wick rotation to imaginary time for each space-time configuration. Using computer simulations we determined the phase structure of the model and discovered that it predicts a de Sitter phase with a four-dimensional spherical semi-classical background geometry. The model has a transfer matrix, relating spatial geometries at adjacent (discrete lattice) times. The transfer matrix uniquely determines the theory. We show that the measurements of the scale factor of the (CDT) universe are well described by an effective transfer matrix where the matrix elements are labelled only by the scale factor. Using computer simulations we determine the effective transfer matrix elements and show how they relate to an effective minisuperspace action at all scales., Comment: 6 pages, 6 figures, contribution to the MULTIVERSE conference, Szczecin, Poland, September 2012

We show how it is possible to formulate Euclidean two-dimensional quantum gravity as the scaling limit of an ordinary statistical system by means of dynamical triangulations, which can be viewed as a discretization in the space of equivalence classes of metrics. Scaling relations exist and the critical exponents have simple geometric interpretations. Hartle-Hawkings wave functionals as well as reparametrization invariant correlation functions which depend on the geodesic distance can be calculated. The discretized approach makes sense even in higher dimensional space-time. Although analytic solutions are still missing in the higher dimensional case, numerical studies reveal an interesting structure and allow the identification of a fixed point where we can hope to define a genuine non-perturbative theory of four-dimensional quantum gravity., Comment: Review, 44 pages, tar compressed uuencoded ps-file (after removing header, type csh filename.uu)

It is known that there are four-manifolds which are not algorithmically recognizable. This implies that there exist triangulations of these manifolds which are separated by large barriers from the point of view of the computer algorithm. We have not observed these barriers for triangulations of $S^4$., 9 p.+4 figures (uuencoded ps-files). Latex+epsf

We describe a method of Monte-Carlo simulations of simplicial quantum gravity coupled to matter fields. We concentrate mainly on the problem of implementing effectively the random, dynamical triangulation and building in a detailed-balance condition into the elementary transformations of the triangulation. We propose a method of auto-tuning the parameters needed to balance simulations of the canonical ensemble. This method allows us to prepare a whole set of jobs and therefore is very useful in systematic determining the phase diagram in two dimensional coupling space. It is of particular importance when the jobs are run on a parallel machine., Comment: 24 pages, PostScript

We measure by Monte Carlo simulations γstring for a model of random surfaces embedded in three dimensional Euclidean space-time. The action of the string is the usual Polyakov action plus an extrinsic curvature term. The system undergoes a phase transition at a finite valueβc of the extrinsic curvature coupling and at the transition point the numerically measured value ofγstring (λc) ≈ 0.27 ± 0.06. This is consistent with γ string (λ c ) = 1 4 , i.e. equal to the first of the non-trivial values of γstring between 0 and 1 2 .