Introduction To Percolation Theory Second Edition

Percolation theory deals with clustering, criticality, diffusion, fractals, phase transitions and disordered systems. It provides a quantitative model for understanding these phenomena, and therefore a theoretical and statistical background to many physical and natural sciences. This book explains the basic theory for the graduate while also reaching into the specialized fields of disordered systems and renormalization groups. Much of the book deals with systems lying close to the critical point phase transition point, where the subject is at its most interesting and sensitive. This text is ideal for those who deal with systems which exhibit critical points and phase transition behavior.

This work dealing with percolation theory clustering, criticallity, diffusion, fractals and phase transitions takes a broad approach to the subject, covering basic theory and also specialized fields like disordered systems and renormalization groups.

Preface to the Second EditionPreface to the First EditionIntroduction: Forest Fires, Fractal Oil Fields, and DiffusionWhat is percolation?Forest firesOil fields and fractalsDiffusion in disordered mediaComing attractionsFurther readingCluster NumbersThe truth about percolationExact solution in one dimensionSmall clusters and animals in d dimensionsExact solution for the Bethe latticeTowards a scaling solution for cluster numbersScaling assumptions for cluster numbersNumerical testsCluster numbers away from PcFurther readingCluster StructureIs the cluster perimeter a real perimeter?Cluster radius and fractal dimensionAnother view on scalingThe infinite cluster at the thresholdFurther readingFinite-size Scaling and the Renormalization GroupFinite-size scalingSmall cell renormalizationScaling revisitedLarge cell and Monte Carlo renormalizationConnection to geometryFurther readingConductivity and Related PropertiesConductivity of random resistor networksInternal structure of the infinite clusterMultitude of fractal dimensions on the incipient infinite clusterMultifractalsFractal models Renormalization group for internal cluster structure Continuum percolation, Swiss-cheese models and broad distributions Elastic networks Further reading Walks, Dynamics and Quantum EffectsAnts in the labyrinthProbability distributionsFractons and superlocalizationHulls and external accessible perimetersDiffusion frontsInvasion percolationFurther readingApplication to Thermal Phase TransitionsStatistical 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