Clustering comparison of point processes with applications to random geometric models

In this chapter we review some examples, methods, and recent results involving comparison of clustering properties of point processes. Our approach is founded on some basic observations allowing us to consider void probabilities and moment measures as two complementary tools for capturing clustering phenomena in point processes. As might be expected, smaller values of these characteristics indicate less clustering. Also, various global and local functionals of random geometric models driven by point processes admit more or less explicit bounds involving void probabilities and moment measures, thus aiding the study of impact of clustering of the underlying point process. When stronger tools are needed, directional convex ordering of point processes happens to be an appropriate choice, as well as the notion of (positive or negative) association, when comparison to the Poisson point process is considered. We explain the relations between these tools and provide examples of point processes admitting them. Furthermore, we sketch some recent results obtained using the aforementioned comparison tools, regarding percolation and coverage properties of the Boolean model, the SINR model, subgraph counts in random geometric graphs, and more generally, U-statistics of point processes. We also mention some results on Betti numbers for \v{C}ech and Vietoris-Rips random complexes generated by stationary point processes. A general observation is that many of the results derived previously for the Poisson point process generalise to some "sub-Poisson" processes, defined as those clustering less than the Poisson process in the sense of void probabilities and moment measures, negative association or dcx-ordering.Comment: 44 pages, 4 figure

Spin foam models with finite groups

Spin foam models, loop quantum gravity and group field theory are discussed as quantum gravity candidate theories and usually involve a continuous Lie group. We advocate here to consider quantum gravity inspired models with finite groups, firstly as a test bed for the full theory and secondly as a class of new lattice theories possibly featuring an analogue diffeomorphism symmetry. To make these notes accessible to readers outside the quantum gravity community we provide an introduction to some essential concepts in the loop quantum gravity, spin foam and group field theory approach and point out the many connections to lattice field theory and condensed matter systems.Comment: 47 pages, 6 figure

Equilibration of deep neural networks and carrier chirality in Rashba systems

This thesis reports results of studies conducted on the equilibration of two systems and consists of two parts: the first part deals with the optimisation of deep neural networks, whereas the second part with the decay of non-equilibrium states in strongly Rashba-coupled systems at low temperature. Deep learning is a conceptually simple, highly effective, and widely used tool, yet there remains insufficient understanding for why it works. The optimisation of deep neural networks with common algorithms such as stochastic gradient descent performs unexpectedly well given the complexity of the underlying high-dimensional non-convex minimisation problem. The first part of this thesis therefore looks at the optimisation procedure from the perspective of statistical physics. This allows us to interpret the loss function landscape of deep neural networks as the counterpart of the potential energy landscape in molecular systems and the optimisation of the network as its equilibration dynamics. Using landscape exploration tools developed in theoretical chemistry, we resolve the structure of the loss function landscape, from which we can draw conclusions for the relaxational dynamics of typical optimisers and, consequently, for deep learning. The second part investigates how a non-equilibrium charge-carrier chirality distribution in a clean, strongly Rashba-coupled system at low temperatures decays over time. We first motivate this analysis based on experimental studies of transport properties in Rashba materials at low temperatures and subject to external magnetic fields. We investigate whether chirality imbalances could serve as the source for those experimental observations and develop a framework that models the behaviour of such a system. We then proceed with a more general theoretical study of the equilibration mechanisms of chirality in low-temperature strongly Rashba-coupled systems and compute the relaxation timescales of those mechanisms.This thesis is the outcome of doctoral studies conducted at the University of Cambridge with the financial support of the Engineering and Physical Sciences Research Council of the UK