Self-Assembly of Geometric Space from Random Graphs

We present a Euclidean quantum gravity model in which random graphs dynamically self-assemble into discrete manifold structures. Concretely, we consider a statistical model driven by a discretisation of the Euclidean Einstein-Hilbert action; contrary to previous approaches based on simplicial complexes and Regge calculus our discretisation is based on the Ollivier curvature, a coarse analogue of the manifold Ricci curvature defined for generic graphs. The Ollivier curvature is generally difficult to evaluate due to its definition in terms of optimal transport theory, but we present a new exact expression for the Ollivier curvature in a wide class of relevant graphs purely in terms of the numbers of short cycles at an edge. This result should be of independent intrinsic interest to network theorists. Action minimising configurations prove to be cubic complexes up to defects; there are indications that such defects are dynamically suppressed in the macroscopic limit. Closer examination of a defect free model shows that certain classical configurations have a geometric interpretation and discretely approximate vacuum solutions to the Euclidean Einstein-Hilbert action. Working in a configuration space where the geometric configurations are stable vacua of the theory, we obtain direct numerical evidence for the existence of a continuous phase transition; this makes the model a UV completion of Euclidean Einstein gravity. Notably, this phase transition implies an area-law for the entropy of emerging geometric space. Certain vacua of the theory can be interpreted as baby universes; we find that these configurations appear as stable vacua in a mean field approximation of our model, but are excluded dynamically whenever the action is exact indicating the dynamical stability of geometric space. The model is intended as a setting for subsequent studies of emergent time mechanisms.Comment: 26 pages, 9 figures, 2 appendice

Extraordinary behavioral entrainment following circadian rhythm bifurcation in mice.

The mammalian circadian timing system uses light to synchronize endogenously generated rhythms with the environmental day. Entrainment to schedules that deviate significantly from 24 h (T24) has been viewed as unlikely because the circadian pacemaker appears capable only of small, incremental responses to brief light exposures. Challenging this view, we demonstrate that simple manipulations of light alone induce extreme plasticity in the circadian system of mice. Firstly, exposure to dim nocturnal illumination (<0.1 lux), rather than completely dark nights, permits expression of an altered circadian waveform wherein mice in light/dark/light/dark (LDLD) cycles "bifurcate" their rhythms into two rest and activity intervals per 24 h. Secondly, this bifurcated state enables mice to adopt stable activity rhythms under 15 or 30 h days (LDLD T15/T30), well beyond conventional limits of entrainment. Continuation of dim light is unnecessary for T15/30 behavioral entrainment following bifurcation. Finally, neither dim light alone nor a shortened night is sufficient for the extraordinary entrainment observed under bifurcation. Thus, we demonstrate in a non-pharmacological, non-genetic manipulation that the circadian system is far more flexible than previously thought. These findings challenge the current conception of entrainment and its underlying principles, and reveal new potential targets for circadian interventions

Graph-theoretic approaches to injectivity and multiple equilibria in systems of interacting elements

We extend previous work on injectivity in chemical reaction networks to general interaction networks. Matrix- and graph-theoretic conditions for injectivity of these systems are presented. A particular signed, directed, labelled, bipartite multigraph, termed the ``DSR graph'', is shown to be a useful representation of an interaction network when discussing questions of injectivity. A graph-theoretic condition, developed previously in the context of chemical reaction networks, is shown to be sufficient to guarantee injectivity for a large class of systems. The graph-theoretic condition is simple to state and often easy to check. Examples are presented to illustrate the wide applicability of the theory developed.Comment: 34 pages, minor corrections and clarifications on previous versio