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Introduction to Percolation Theory


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Table of Contents

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
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
Dilute magnets at finite temperatures
Spin glasses
Further reading
Numerical Techniques

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