Percolation

Abstract: Preface to the Second Edition Preface to the First Edition Introduction: Forest Fires, Fractal Oil Fields, and Diffusion What is percolation? Forest fires Oil fields and fractals Diffusion in disordered media Coming attractions Further reading Cluster Numbers The truth about percolation Exact solution in one dimension Small clusters and animals in d dimensions Exact solution for the Bethe lattice Towards a scaling solution for cluster numbers Scaling assumptions for cluster numbers Numerical tests Cluster numbers away from Pc Further reading Cluster Structure Is the cluster perimeter a real perimeter? Cluster radius and fractal dimension Another view on scaling The infinite cluster at the threshold Further reading Finite-size Scaling and the Renormalization Group Finite-size scaling Small cell renormalization Scaling revisited Large cell and Monte Carlo renormalization Connection to geometry Further reading Conductivity and Related Properties Conductivity of random resistor networks Internal structure of the infinite cluster Multitude of fractal dimensions on the incipient infinite cluster Multifractals Fractal models Renormalization group for internal cluster structure Continuum percolation, Swiss-cheese models and broad distributions Elastic networks Further reading Walks, Dynamics and Quantum Effects Ants in the labyrinth Probability distributions Fractons and superlocalization Hulls and external accessible perimeters Diffusion fronts Invasion percolation Further reading Application to Thermal Phase Transitions Statistical physics and the Ising model Dilute magnets at low temperatures History of droplet descriptions for fluids Droplet definition for the Ising model in zero field The trouble with Kertesz Applications Dilute magnets at finite temperatures Spin glasses Further reading Summary Numerical Techniques

Abstract: Mechanisms of transverse conductivity and generation of self-consistent electric fields in strongly ionized magnetized plasma V. Rozhansky. 1. Introduction.- 2. Conductivity tensor in partially ionized plasma.- 3. Main mechanisms of perpendicular conductivity in fully ionized plasma.- 4. Acceleration of plasma clouds in an inhomogeneous magnetic field.- 5. Alfven conductivity.- 6. Perpendicular viscosity, radial current, and radial electric field in an infinite cylinder.- 7. Current systems in front of a biased electrode (flush-mounted probe) and spot of emission.- 8. Currents in the vicinity of a biased electrode that is smaller than the ion gyroradius.- 9. Neoclassical perpendicular conductivity in a tokamak.- 10. Transverse conductivity in a reversed field pinch.- 11. Modeling of electric field and currents in the tokamak edge plasma.- 12. Mechanisms of anomalous perpendicular viscosity and viscosity-driven currents.- 13. Transverse conductivity in a stochastic magnetic field.- 14. Electric fields generated in the shielding layer between hot plasma and a solid state.-- Correlations and anomalous transport models O.G. Bakunin. 1. Introduction.- 2. Turbulent diffusion and transport.- 3. Non-local effects and diffusion equations.- 4. The Corrsin conjecture.- 5. Effects of seed diffusivity.- 6. The diffusive tracer equation and averaging.- 7. The quasi-linear approximation.- 8. The diffusive renormalization.- 9. Anomalous transport and convective cells.- 10. Stochastic instability and transport.- 11. Fractal conceptions and turbulence.- 12. Percolation and scalings.- 13. Percolation and turbulent transport scalings.- 14. The temporal hierarchy of scales and correlations.- 15. The stochastic magnetic field and percolation transport.- 16. Percolation in drift flows.- 17. Multiscale flows.- 18. Subdiffusion and traps.- 19. Continuous time random walks.- 20. Fractional differential equations and scalings.- 21. Correlation and phase-space.- 22. Conclusion.

Abstract: The combination of the compactness of networks, featuring small diameters, and their complex architectures results in a variety of critical effects dramatically different from those in cooperative systems on lattices. In the last few years, important steps have been made toward understanding the qualitatively new critical phenomena in complex networks. The results, concepts, and methods of this rapidly developing field are reviewed. Two closely related classes of these critical phenomena are considered, namely, structural phase transitions in the network architectures and transitions in cooperative models on networks as substrates. Systems where a network and interacting agents on it influence each other are also discussed. A wide range of critical phenomena in equilibrium and growing networks including the birth of the giant connected component, percolation, $k$-core percolation, phenomena near epidemic thresholds, condensation transitions, critical phenomena in spin models placed on networks, synchronization, and self-organized criticality effects in interacting systems on networks are mentioned. Strong finite-size effects in these systems and open problems and perspectives are also discussed.

Abstract: Percolation theory, the theory of the properties of classical particles interacting with a random medium, is of wide applicability and provides a simple picture exhibiting critical behaviour, the features of which are well understood and amenable to detailed calculation In this review the concepts of percolation theory and the general features associated with the critical region about the onset of percolation are developed in detail In particular, several dimensional invariants are examined which make it possible to unify much of the available information, and to extend the insights of percolation theory to processes which have not yet received numerical study The compilation of the results of percolation theory, both exact and numerical, is believed to be complete through 1970 A selective bibliography is given In a concluding chapter several recent applications of percolation theory to classical and to quantum mechanical problems are discussed