We study the localization transitions which arise in both one and two dimensions when quantum mechanical particles described by a random Schrödinger equation are subjected to a constant imaginary vector potential. A path-integral formulation relates the transition to flux lines depinned from columnar defects by a transverse magnetic field in superconductors. The theory predicts that, close to the depinning transition, the transverse Meissner effect is accompanied by stretched exponential relaxation of the field into the bulk and a diverging penetration depth.

A delocalization phenomenon is studied in a class of non-Hermitian random quantum-mechanical problems. Delocalization arises in response to a sufficiently large constant imaginary vector potential. The transition is related to depinning of flux lines from extended defects in type-II superconductors subject to a tilted external magnetic field. The physical meaning of the complex eigenvalues and currents of the non-Hermitian system is elucidated in terms of properties of tilted vortex lines. The singular behavior of the penetration length describing stretched exponential screening of a perpendicular magnetic field (transverse Meissner effect), the surface transverse magnetization, and the trapping length is determined near the flux-line depinning point.

We propose a new measure of the communicability of a complex network, which is a broad generalization of the concept of the shortest path. According to the new measure, most of the real-world networks display the largest communicability between the most connected (popular) nodes of the network (assortative communicability). There are also several networks with the disassortative communicability, where the most “popular” nodes communicate very poorly to each other. Using this information we classify a diverse set of real-world complex systems into a small number of universality classes based on their structure-dynamic correlation. In addition, the new communicability measure is able to distinguish finer structures of networks, such as communities into which a network is divided. A community is unambiguously defined here as a set of nodes displaying larger communicability among them than to the rest of the nodes in the network.

Recent literature on delocalization in non-Hermitian systems has stressed criteria based on sensitivity of eigenvalues to boundary conditions and the existence of a nonzero current. We emphasize here that delocalization also shows up clearly in eigenfunctions, provided one studies the product of left and right eigenfunctions, as required on physical grounds, and not simply the squared modulii of the eigenfunctions themselves. We also discuss the right and left eigenfunctions of the ground state in the delocalized regime and suggest that the behavior of these functions, when considered separately, may be viewed as “intermediate” between localized and delocalized.

We present a novel microscopic model of sorption and convection of ions in heterogeneous media. Our model is based on an analogy to electron transport in a semiconductor. A new feature of our model is a power law random distribution of the adsorption time of ions. Diverging standard deviation of the distribution function yields anomalous ion transport. We show that this anomalous transport explains a concentration profile with a long tail that has been observed in column experiments. We successfully fit recent experimental data. Finally, we propose new experiments by which we can check the validity of our model.