Lorentzian and Euclidean Quantum Gravity - Analytical and Numerical Results

We review some recent attempts to extract information about the nature of quantum gravity, with and without matter, by quantum field theoretical methods. More specifically, we work within a covariant lattice approach where the individual space-time geometries are constructed from fundamental simplicial building blocks, and the path integral over geometries is approximated by summing over a class of piece-wise linear geometries. This method of ``dynamical triangulations'' is very powerful in 2d, where the regularized theory can be solved explicitly, and gives us more insights into the quantum nature of 2d space-time than continuum methods are presently able to provide. It also allows us to establish an explicit relation between the Lorentzian- and Euclidean-signature quantum theories. Analogous regularized gravitational models can be set up in higher dimensions. Some analytic tools exist to study their state sums, but, unlike in 2d, no complete analytic solutions have yet been constructed. However, a great advantage of our approach is the fact that it is well-suited for numerical simulations. In the second part of this review we describe the relevant Monte Carlo techniques, as well as some of the physical results that have been obtained from the simulations of Euclidean gravity. We also explain why the Lorentzian version of dynamical triangulations is a promising candidate for a non-perturbative theory of quantum gravity.Comment: 69 pages, 16 figures, references adde

Quantum Gravity, or The Art of Building Spacetime

The method of four-dimensional Causal Dynamical Triangulations provides a background-independent definition of the sum over geometries in quantum gravity, in the presence of a positive cosmological constant. We present the evidence accumulated to date that a macroscopic four-dimensional world can emerge from this theory dynamically. Using computer simulations we observe in the Euclidean sector a universe whose scale factor exhibits the same dynamics as that of the simplest mini-superspace models in quantum cosmology, with the distinction that in the case of causal dynamical triangulations the effective action for the scale factor is not put in by hand but obtained by integrating out {\it in the quantum theory} the full set of dynamical degrees of freedom except for the scale factor itself.Comment: 22 pages, 6 figures. Contribution to the book "Approaches to Quantum Gravity", ed. D. Oriti, Cambridge University Pres

On directed interacting animals and directed percolation

We study the phase diagram of fully directed lattice animals with nearest-neighbour interactions on the square lattice. This model comprises several interesting ensembles (directed site and bond trees, bond animals, strongly embeddable animals) as special cases and its collapse transition is equivalent to a directed bond percolation threshold. Precise estimates for the animal size exponents in the different phases and for the critical fugacities of these special ensembles are obtained from a phenomenological renormalization group analysis of the correlation lengths for strips of width up to n=17. The crossover region in the vicinity of the collapse transition is analyzed in detail and the crossover exponent $\phi$ is determined directly from the singular part of the free energy. We show using scaling arguments and an exact relation due to Dhar that $\phi$ is equal to the Fisher exponent $\sigma$ governing the size distribution of large directed percolation clusters.Comment: 23 pages, 3 figures; J. Phys. A 35 (2002) 272

(D+1)(D+1)-Colored Graphs - a Review of Sundry Properties

We review the combinatorial, topological, algebraic and metric properties supported by $(D+1)$-colored graphs, with a focus on those that are pertinent to the study of tensor model theories. We show how to extract a limiting continuum metric space from this set of graphs and detail properties of this limit through the calculation of exponents at criticality

Geodesics on Flat Surfaces

This short survey illustrates the ideas of Teichmuller dynamics. As a model application we consider the asymptotic topology of generic geodesics on a "flat" surface and count closed geodesics and saddle connections. This survey is based on the joint papers with A.Eskin and H.Masur and with M.Kontsevich.Comment: (25 pages, 5 figures) Based on the talk at ICM 2006 at Madrid; see Proceedings of the ICM, Madrid, Spain, 2006, EMS, 121-146 for the final version. For a more detailed survey see the paper "Flat Surfaces", arXiv.math.DS/060939

Improved chemistry restraints for crystallographic refinement by integrating the Amber force field into Phenix.

The refinement of biomolecular crystallographic models relies on geometric restraints to help to address the paucity of experimental data typical in these experiments. Limitations in these restraints can degrade the quality of the resulting atomic models. Here, an integration of the full all-atom Amber molecular-dynamics force field into Phenix crystallographic refinement is presented, which enables more complete modeling of biomolecular chemistry. The advantages of the force field include a carefully derived set of torsion-angle potentials, an extensive and flexible set of atom types, Lennard-Jones treatment of nonbonded interactions and a full treatment of crystalline electrostatics. The new combined method was tested against conventional geometry restraints for over 22 000 protein structures. Structures refined with the new method show substantially improved model quality. On average, Ramachandran and rotamer scores are somewhat better, clashscores and MolProbity scores are significantly improved, and the modeling of electrostatics leads to structures that exhibit more, and more correct, hydrogen bonds than those refined using traditional geometry restraints. In general it is found that model improvements are greatest at lower resolutions, prompting plans to add the Amber target function to real-space refinement for use in electron cryo-microscopy. This work opens the door to the future development of more advanced applications such as Amber-based ensemble refinement, quantum-mechanical representation of active sites and improved geometric restraints for simulated annealing