Weak Multiplex Percolation

In many systems consisting of interacting subsystems, the complex interactions between elements can be represented using multilayer networks. However percolation, key to understanding connectivity and robustness, is not trivially generalised to multiple layers. This Element describes a generalisation of percolation to multilayer networks: weak multiplex percolation. A node belongs to a connected component if at least one of its neighbours in each layer is in this component. The authors fully describe the critical phenomena of this process. In two layers with finite second moments of the degree distributions the authors observe an unusual continuous transition with quadratic growth above the threshold. When the second moments diverge, the singularity is determined by the asymptotics of the degree distributions, creating a rich set of critical behaviours. In three or more layers the authors find a discontinuous hybrid transition which persists even in highly heterogeneous degree distributions, becoming continuous only when the powerlaw exponent reaches $1+1/(M-1)$ for $M$ layers.

Spontaneous emission in inertial and dissipative nematic liquid crystals: the role of critical phenomena

We develop a rigorous, field-theoretical approach to the study of spontaneous emission in inertialand dissipative nematic liquid crystals, disclosing an alternative application of the massive Stueck-elberg gauge theory to describe critical phenomena in these systems. This approach allows one notonly to unveil the role of phase transitions in the spontaneous emission in liquid crystals but also to make quantitative predictions for quantum emission in realistic nematics of current scientific andtechnological interest in the field of metamaterials. Specifically, we predict that one can switchon and off quantum emission in liquid crystals by varying the temperature in the vicinities of thecrystalline-to-nematic phase transition, for both the inertial and dissipative cases. We also predictfrom first principles the value of the critical exponent that characterizes such a transition, whichwe show not only to be independent of the inertial or dissipative dynamics, but also to be in goodagreement with experiments. We determine the orientation of the dipole moment of the emitterrelative to the nematic director that inhibits spontaneous emission, paving the way to achieve direc-tionality of the emitted radiation, a result that could be applied in tuneable photonic devices suchas metasurfaces and tuneable light sources.