Contents
Preface
1. Percolation
1.1 Introduction
1.1.1 Definition of site percolation
1.1.2 Quantities of interest
1.2 Percolation in d=1
1.2.1 Cluster number density
1.2.2 Average cluster size
1.2.3 Transition to percolation
1.2.4 Correlation function
1.2.5 Critical occupation probability
1.3 Percolation on the Bethe Lattice
1.3.1 Definition of the Bethe lattice
1.3.2 Critical occupation probability
1.3.3 Average cluster size
1.3.4 Transition to percolation
1.3.5 Cluster number density
1.3.6 Correlation function
1.4 Percolation in d=2
1.4.1 Transition to percolation
1.4.2 Average cluster size
1.4.3 Cluster number density- exact
1.4.4 Cluster number density - numerical
1.5 Cluster Number Density- Scaling Ansatz
1.5.1 Scaling function and data collapse
1.5.2 Scaling function and data collapse in d = 1
1.5.3 Scaling function and data collapse on the Bethe lattice
1.5.4 Scaling function and data collapse in d = 2
1.6 Scaling Relations
1.7 Geometric Properties of Clusters
1.7.1 Self-similarity and fractal dimension
1.7.2 Mass of a large but finite cluster at p = pc
1.7.3 Correlation length
1.7.4 Mass of the percolating cluster for p > pc
1.8 Finite-Size Scaling
1.8.1 Order parameter
1.8.2 Average cluster size and higher moments
1.8.3 Cluster number density
1.9 Non-Universal Critical Occupation Probabilities
1.10 Universal Critical Exponents
1.11 Real-Space Renormalisation
1.11.1 Self-similarity and the correlation length
1.11.2 Self-similarity and fixed points
1.11.3 Coarse graining and rescaling
1.11.4 Real-space renormalisation group procedure
1.11.5 Renormalisation in d = 1
1.11.6 Renormalisation in d = 2 on a triangular lattice
1.11.7 Renormalisation in d = 2 on a square lattice
1.11.8 Approximation via the truncation of parameter space
1.12 Summary
Exercises

2. Ising Model
2.1 Introduction
2.1.1 Definition of the Ising model
2.1.2 Review of equilibrium statistical mechanics
2.1.3 Thermodynamic limit
2.2 System of Non-Interacting Spins
2.2.1 Partition function and free energy
2.2.2 Magnetisation and susceptibility
2.2.3 Energy and specific heat
2.3 Quantities of Interest
2.3.1 Magnetisation
2.3.2 Response functions
2.3.3 Correlation length and spin-spin correlation function
2.3.4 Critical temperature and external field
2.3.5 Symmetry breaking
2.4 Ising Model in d = 1
2.4.1 Partition function
2.4.2 Free energy
2.4.3 Magnetisation and susceptibility
2.4.4 Energy and specific heat
2.4.5 Correlation function
2.4.6 Critical temperature
2.5 Mean-Field Theory of the Ising Model
2.5.1 Partition function and free energy
2.5.2 Magnetisation and susceptibility
2.5.3 Energy and specific heat
2.6 Landau Theory of the Ising Model
2.6.1 Free energy
2.6.2 Magnetisation and susceptibility
2.6.3 Specific heat
2.7 Landau Theory of Continuous Phase Transitions
2.8 Ising Model in d = 2
2.8.1 Partition function
2.8.2 Magnetisation and susceptibility
2.8.3 Energy and specific heat
2.8.4 Critical temperature
2.9 Widom Scaling Ansatz
2.9.1 Scaling ansatz for the free energy
2.9.2 Scaling ansatz for the specific heat
2.9.3 Scaling ansatz for the magnetisation
2.9.4 Scaling ansatz for the susceptibility
2.9.5 Scaling ansatz for the spin-spin correlation function
2.10 Scaling Relations
2.11Widom Scaling Form and Critical Exponents in d = 1
2.12 Non-Universal Critical Temperatures
2.13 Universal Critical Exponents
2.14 Ginzburg Criterion
2.15 Real-Space Renormalisation
2.15.1 Kadanoffs block spin transformation
2.15.2 Kadanoffs block spin and the free energy
2.15.3 Kadanoffs block spin and the correlation function
2.15.4 Renormalisation in d = 1
2.15.5 Renormalisation in d = 2 on a square lattice
2.16 Wilsons Renormalisation Group Theory
2.16.1 Coupling space and renormalisation group flow
2.16.2 Self-similarity and fixed points
2.16.3 Basin of attraction of fixed points
2.16.4 RG flow in coupling and configurational space
2.16.5 Universality and RG flow near fixed point
2.16.6 Widom scaling form
2.17 Summary
Exercises

3. Self-Organised Criticality
3.1 Introduction
3.1.1 Sandpile metaphor
3.2 BTW Model in d = 1
3.2.1 Algorithm of the BTW model in d = 1
3.2.2 Transient and recurrent configurations
3.2.3 Avalanche time series
3.2.4 Avalanche-size probability
3.3 Mean-Field Theory of the BTW Model
3.3.1 Random neighbour BTW model
3.3.2 Algorithm of the random neighbour BTW model
3.3.3 Steady state and the average avalanche size
3.4 Branching Process
3.4.1 Branching ratio
3.4.2 Avalanche-size probability - exact
3.4.3 Avalanche-size probability - scaling form
3.5 Avalanche-Size Probability- Scaling Ansatz
3.6 Scaling Relations
3.7 Moment Analysis of Avalanche-Size Probability
3.8 BTW Model in d = 2
3.8.1 Algorithm of the BTW model in d = 2
3.8.2 Steady state and the average avalanche size
3.8.3 Avalanche time series
3.8.4 Avalanche-size probability
3.9 Ricepile Experiment and the Oslo Model
3.9.1 Ricepile experiment
3.9.2 Ricepile avalanche time series
3.9.3 Ricepile avalanche-size probability density
3.9.4 Ricepile modelling
3.9.5 Algorithm of the Oslo model
3.9.6 Transient and recurrent configurations
3.9.7 Avalanche time series
3.9.8 Avalanche-size probability
3.10 Earthquakes and the OFC Model
3.10.1 Earthquake mechanism
3.10.2 Earthquake time series
3.10.3 Earthquake-size frequency
3.10.4 Earthquake modelling
3.10.5 Algorithm of the OFC model
3.10.6 Steady state and the average avalanche size
3.10.7 Avalanche time series
3.10.8 Avalanche-size probability
3.11 Rainfall
3.11.1 Rainfall mechanism
3.11.2 Rainfall time series
3.11.3 Rainfall-size number density
3.12 Summary
Exercises

Appendix A Taylor Expansion
Appendix B Hyperbolic Functions
Appendix C Homogeneous and Scaling Functions
Appendix D Fractals
Appendix E Data Binning
Appendix F Boltzmann Distribution
Appendix G Free Energy
Appendix H Metropolis Algorithm
Bibliography
List of Symbols