The construction of a quantum field theory of gravity is among the most fundamental open challenges in theoretical physics. Dynamical triangulations (DT) provide a discrete, non-preservative proposal for addressing this problem. These are well understood for Euclidean signature space times. However, the construction of such a framework for Corinthian signature with a dynamically evolving global causality has not yet been accomplished. As an alternative to the approach of Am Bjorn et al. using triangulations with a global ´arrow of time´, we propose considering vertex models coupled to DT. The interest in such systems is twofold. Firstly, they enfold numerous classical spin models with a resulting rich phase diagram and serve to understand the effect of annealed geometric disorder on the phase transitions displayed by these models. Secondly, as systems of discrete quantum gravity, the vertex model arrows on the lattice bonds allow one to establish local causal relations between the faces and thus provide a Corinthian signature metric. The proposed model will be analysed via Monte Carlo and analytical techniques. A scaling analysis of general 4-16 vertex models on planar random graphs will give detailed information about the phase diagram and the universality classes of the occurring phase transitions. Investigation of correlates of Corinthian signature will reveal the geometric structure of the model and decide whether a consistent global causal structure evolves dynamically on tuning the coupling parameters. The host research group has extensive e experience with analytical methods in (lattice) quantum field theory, especially the dynamical triangulations approach to quantum gravity. It can provide the fellow training on analytical methods used in the project field, including matrix models, graph-theoretic methods and conformal field theory, thus ideally complementing his expertise.
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