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Table of Contents
Cover
Title page
Chapter 1. Introduction
1.1. Background and motivation
1.2. Summary of results
1.3. Proof ideas
Chapter 2. The setup
2.1. Nearest-neighbour models
2.2. Long-range models
Chapter 3. Definitions and preliminaries
3.1. Graph theoretic definitions
3.2. Exponential tail of the subcritical cluster size distribution: The Aizenman-Newman-Barsky property
3.3. The BK inequality
3.4. Partitions of integers
Chapter 4. The basic technique
4.1. Nearest-neighbour models
4.2. Long-range models
4.3. Analyticity of susceptibility in the subcritical regime
Chapter 5. Analyticity for non-amenable graphs
Chapter 6. Trees
6.1. Regular trees
6.2. Galton-Watson trees
6.3. A tree with percolation density nowhere analytic
Chapter 7. Analyticity above the threshold for planar lattices
7.1. Preliminaries on planar quasi-transitive lattices
7.2. Main result
7.3. Site percolation
Chapter 8. Analyticity of percolation density in all dimensions
8.1. Setting up the renormalisation
8.2. Separating components
8.3. Expressing the percolation density using separating components
8.4. Expanding the percolation density as an infinite sum of polynomials
8.5. Exponential tail of a certain cutset
8.6. Analyticity of k-point function
Chapter 9. Continuum percolation
Chapter 10. Finitely presented groups
10.1. The setup and notation
10.2. A connectedness concept
10.3. Interfaces
10.4. Properties of interfaces
10.5. Using interfaces to prove analyticity
10.6. Extending to site percolation
Chapter 11. Triangulations
11.1. Overview
11.2. Proofs
11.3. Site percolation
Chapter 12. Alternating signs of Taylor coefficients
Chapter 13. The negative percolation threshold
Appendix A. On the number of lattice animals of a given size
Appendix B. Complex analysis basics
Bibliography
Back Cover
Title page
Chapter 1. Introduction
1.1. Background and motivation
1.2. Summary of results
1.3. Proof ideas
Chapter 2. The setup
2.1. Nearest-neighbour models
2.2. Long-range models
Chapter 3. Definitions and preliminaries
3.1. Graph theoretic definitions
3.2. Exponential tail of the subcritical cluster size distribution: The Aizenman-Newman-Barsky property
3.3. The BK inequality
3.4. Partitions of integers
Chapter 4. The basic technique
4.1. Nearest-neighbour models
4.2. Long-range models
4.3. Analyticity of susceptibility in the subcritical regime
Chapter 5. Analyticity for non-amenable graphs
Chapter 6. Trees
6.1. Regular trees
6.2. Galton-Watson trees
6.3. A tree with percolation density nowhere analytic
Chapter 7. Analyticity above the threshold for planar lattices
7.1. Preliminaries on planar quasi-transitive lattices
7.2. Main result
7.3. Site percolation
Chapter 8. Analyticity of percolation density in all dimensions
8.1. Setting up the renormalisation
8.2. Separating components
8.3. Expressing the percolation density using separating components
8.4. Expanding the percolation density as an infinite sum of polynomials
8.5. Exponential tail of a certain cutset
8.6. Analyticity of k-point function
Chapter 9. Continuum percolation
Chapter 10. Finitely presented groups
10.1. The setup and notation
10.2. A connectedness concept
10.3. Interfaces
10.4. Properties of interfaces
10.5. Using interfaces to prove analyticity
10.6. Extending to site percolation
Chapter 11. Triangulations
11.1. Overview
11.2. Proofs
11.3. Site percolation
Chapter 12. Alternating signs of Taylor coefficients
Chapter 13. The negative percolation threshold
Appendix A. On the number of lattice animals of a given size
Appendix B. Complex analysis basics
Bibliography
Back Cover