Towards background independent quantum gravity with tensor models

We explore whether the phase diagram of tensor models could feature a pregeometric, discrete and a geometric, continuum phase for the building blocks of space. The latter are associated to rank $d$ tensors of size $N$. We search for a universal large $N$ scaling limit in a rank-3 model with real tensors that could be linked to a transition between the two phases. We extend the conceptual development and practical implementation of the flow equation for the pregeometric setting. This provides a pregeometric "coarse-graining" by going from many microscopic to few effective degrees of freedom by lowering $N$. We discover several candidates for fixed points of this coarse graining procedure, and specifically explore the impact of a novel class of interactions allowed in the real rank-3 model. In particular, we explain how most universality classes feature dimensional reduction, while one candidate, involving a tetrahedral interaction, might potentially be of relevance for three-dimensional quantum gravity.Comment: 23 pages plus appendix and reference

Simulational study of anomalous tracer diffusion in hydrogels

In this article, we analyze different factors that affect the diffusion behavior of small tracer particles (as they are used e.g.in fluorescence correlation spectroscopy (FCS)) in the polymer network of a hydrogel and perform simulations of various simplified models. We observe, that under certain circumstances the attraction of a tracer particle to the polymer network strands might cause subdiffusive behavior on intermediate time scales. In theory, this behavior could be employed to examine the network structure and swelling behavior of weakly crosslinked hydrogels with the help of FCS.Comment: 11 pages, 11 figure

Universal critical behavior in tensor models for four-dimensional quantum gravity

Four-dimensional random geometries can be generated by statistical models with rank-4 tensors as random variables. These are dual to discrete building blocks of random geometries. We discover a potential candidate for a continuum limit in such a model by employing background-independent coarse-graining techniques where the tensor size serves as a pre-geometric notion of scale. A fixed point candidate which features two relevant directions is found. The possible relevance of this result in view of universal results for quantum gravity and a potential connection to the asymptotic-safety program is discussed.Comment: 10 page

Diffusion in Model Networks as Studied by NMR and Fluorescence Correlation Spectroscopy

We have studied the diffusion of small solvent molecules (octane) and larger hydrophobic dye probes in octane-swollen poly(dimethyl siloxane) linear-chain solutions and end-linked model networks, using pulsed-gradient nuclear magnetic resonance (NMR) and fluorescence correlation spectroscopy (FCS), respectively, focusing on diffusion in the bulk polymer up to the equilibrium degree of swelling of the networks, that is, 4.8 at most. The combination of these results allows for new conclusions on the feasibility of different theories describing probe diffusion in concentrated polymer systems. While octane diffusion shows no cross-link dependence, the larger dyes are increasingly restricted by fixed chemical meshes. The simple Fujita free-volume theory proved most feasible to describe probe diffusion in linear long-chain solutions with realistic parameters, while better fits were obtained assuming a stretched exponential dependence on concentration. Importantly, we have analyzed the cross-link specific effect on probe diffusion independently of any specific model by comparing the best-fit interpolation of the solution data with the diffusion in the networks. The most reasonable description is obtained by assuming that the cross-link effect is additive in the effective friction coefficient of the probes. The concentration dependences as well as the data compared at the equilibrium degrees of swelling indicate that swelling heterogeneities and diffusant shape have a substantial influence on small-molecule diffusion in networks.