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(2+1)Dimensional Quantum Gravity as the Continuum Limit of Causal Dynamical Triangulations
, 2007
"... We perform a nonperturbative sum over geometries in a (2+1)dimensional quantum gravity model given in terms of Causal Dynamical Triangulations. Inspired by the concept of triangulations of product type introduced previously, we impose an additional notion of order on the discrete, causal geometrie ..."
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Cited by 16 (6 self)
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We perform a nonperturbative sum over geometries in a (2+1)dimensional quantum gravity model given in terms of Causal Dynamical Triangulations. Inspired by the concept of triangulations of product type introduced previously, we impose an additional notion of order on the discrete, causal geometries. This simplifies the combinatorial problem of counting geometries just enough to enable us to calculate the transfer matrix between boundary states labelled by the area of the spatial universe, as well as the corresponding quantum Hamiltonian of the continuum theory. This is the first time in dimension larger than two that a Hamiltonian has been derived from such a model by mainly analytical means, and opens the way for a better understanding of scaling and renormalization issues.
Quantum gravity and matter: counting graphs on causal dynamical triangulations
"... An outstanding challenge for models of nonperturbative quantum gravity is the consistent formulation and quantitative evaluation of physical phenomena in a regime where geometry and matter are strongly coupled. After developing appropriate technical tools, one is interested in measuring and classif ..."
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Cited by 6 (2 self)
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An outstanding challenge for models of nonperturbative quantum gravity is the consistent formulation and quantitative evaluation of physical phenomena in a regime where geometry and matter are strongly coupled. After developing appropriate technical tools, one is interested in measuring and classifying how the quantum fluctuations of geometry alter the behaviour of matter, compared with that on a fixed background geometry. In the simplified context of two dimensions, we show how a method invented to analyze the critical behaviour of spin systems on flat lattices can be adapted to the fluctuating ensemble of curved spacetimes underlying the Causal Dynamical Triangulations (CDT) approach to quantum gravity. We develop a systematic counting of embedded graphs to evaluate the thermodynamic functions of the gravitymatter models in a high and lowtemperature expansion. For the case of the Ising model, we compute the series expansions for the magnetic susceptibility on CDT lattices and their duals up to orders 6 and 12, and analyze them by ratio method, Dlog Padé and differential approximants. Apart from providing evidence for a simplification of the model’s analytic structure due to the dynamical nature of the geometry, the technique introduced can shed further light on criteria à la Harris and Luck for the influence of random geometry on the critical properties of matter systems.
Integrability of graph combinatorics via random walks and heaps of dimers
, 2005
"... We investigate the integrability of the discrete nonlinear equation governing the dependence on geodesic distance of planar graphs with inner vertices of even valences. This equation follows from a bijection between graphs and blossom trees and is expressed in terms of generating functions for rand ..."
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Cited by 4 (3 self)
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We investigate the integrability of the discrete nonlinear equation governing the dependence on geodesic distance of planar graphs with inner vertices of even valences. This equation follows from a bijection between graphs and blossom trees and is expressed in terms of generating functions for random walks. We construct explicitly an infinite set of conserved quantities for this equation, also involving suitable combinations of random walk generating functions. The proof of their conservation, i.e. their eventual independence on the geodesic distance, relies on the connection between random walks and heaps of dimers. The values of the conserved quantities are identified with generating functions for graphs with fixed numbers of external legs. Alternative equivalent choices for the set of conserved quantities are also discussed and some applications are presented.
Counting a black hole in Lorentzian product triangulations
, 2005
"... We take a step toward a nonperturbative gravitational path integral for blackhole geometries by deriving an expression for the expansion rate of null geodesic congruences in the approach of causal dynamical triangulations. We propose to use the integrated expansion rate in building a quantum horizon ..."
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We take a step toward a nonperturbative gravitational path integral for blackhole geometries by deriving an expression for the expansion rate of null geodesic congruences in the approach of causal dynamical triangulations. We propose to use the integrated expansion rate in building a quantum horizon finder in the sum over spacetime geometries. It takes the form of a counting formula for various types of discrete building blocks which differ in how they focus and defocus light rays. In the course of the derivation, we introduce the concept of a Lorentzian dynamical triangulation of product type, whose applicability goes beyond that of describing blackhole configurations. 1
Journal of Statistical Mechanics: An IOP and SISSA journal Theory and Experiment
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A statistical formalism of Causal Dynamical
, 2008
"... We rewrite the 1+1 Causal Dynamical Triangulations model as a spin system ..."
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We rewrite the 1+1 Causal Dynamical Triangulations model as a spin system