A causal perspective on random geometry

In this thesis we investigate the importance of causality in non-perturbative approaches to quantum gravity. Firstly, causal sets are introduced as a simple kinematical model for causal geometry. It is shown how causal sets could account for the microscopic origin of the Bekenstein entropy bound. Holography and finite entropy emerge naturally from the interplay between causality and discreteness. Going beyond causal set kinematics is problematic however. It is a hard problem to find the right amplitude to attach to each causal set that one needs to define the non-perturbative quantum dynamics of gravity. One approach which is ideally suited to define the non-perturbative gravitational path integral is dynamical triangulation. Without causality this method leads to unappealing features of the quantum geometry though. It is shown how causality is instrumental in regulating this pathological behavior. In two dimensions this approach of causal dynamical triangulations has been analytically solved by transfer matrix methods. In this thesis considerable progress has been made in the development of more powerful techniques for this approach. The formulation through matrix models and a string field theory allow us to study interesting generalizations. Particularly, it has become possible to define the topological expansion. A surprising twist of the new matrix model is that it partially disentangles the large-N and continuum limit. This makes our causal model much closer in spirit to the original idea by 't Hooft than the conventional matrix models of non-critical string theory.Comment: 230 pages, reduced images, PhD thesis in theoretical physics, Imperial College London (2008

Dynamical dimensional reduction in toy models of 4D causal quantum gravity

In recent years several approaches to quantum gravity have found evidence for a scale dependent spectral dimension of space-time varying from four at large scales to two at small scales of order of the Planck length. The first evidence came from numerical results on four-dimensional causal dynamical triangulations (CDT) [Ambjorn et al., Phys. Rev. Lett. 95 (2005) 171]. Since then little progress has been made in analytically understanding the numerical results coming from the CDT approach and showing that they remain valid when taking the continuum limit. Here we argue that the spectral dimension can be determined from a model with fewer degrees of freedom obtained from the CDTs by "radial reduction". In the resulting "toy" model we can take the continuum limit analytically and obtain a scale dependent spectral dimension varying from four to two with scale and having functional behaviour exactly of the form which was conjectured on the basis of the numerical results.Comment: 12 pages, 2 figures, v3: improved discussion, results unchanged, as publishe

Aspects of dynamical dimensional reduction in multigraph ensembles of CDT

We study the continuum limit of a "radially reduced" approximation of Causal Dynamical Triangulations (CDT), so-called multigraph ensembles, and explain why they serve as realistic toy models to study the dimensional reduction observed in numerical simulations of four-dimensional CDT. We present properties of this approximation in two, three and four dimensions comparing them with the numerical simulations and pointing out some common features with 2+1 dimensional Horava-Lifshitz gravity.Comment: 4 pages, 1 figure, Presented at "Gravity, Quantum, and Black Holes" session of IC-MSQUARE 2012, Budapest, to appear in the proceedings, IOP Conference Serie

DeepLOB: Deep Convolutional Neural Networks for Limit Order Books

We develop a large-scale deep learning model to predict price movements from limit order book (LOB) data of cash equities. The architecture utilises convolutional filters to capture the spatial structure of the limit order books as well as LSTM modules to capture longer time dependencies. The proposed network outperforms all existing state-of-the-art algorithms on the benchmark LOB dataset [1]. In a more realistic setting, we test our model by using one year market quotes from the London Stock Exchange and the model delivers a remarkably stable out-of-sample prediction accuracy for a variety of instruments. Importantly, our model translates well to instruments which were not part of the training set, indicating the model's ability to extract universal features. In order to better understand these features and to go beyond a "black box" model, we perform a sensitivity analysis to understand the rationale behind the model predictions and reveal the components of LOBs that are most relevant. The ability to extract robust features which translate well to other instruments is an important property of our model which has many other applications.Comment: 12 pages, 9 figure