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Table of Contents
Intro; Editorial Board; Preface; Acknowledgements; Contents; Editors, Associate Editors and Contributors; 1 Overview: PCA Models and Issues; 1.1 Introduction; 1.2 Phenomena Addressed by PCA Modeling; 1.2.1 Metastability and Traps; 1.2.2 Epidemics and Extinction; 1.2.3 Wildfire and Percolation Phenomena; 1.3 The Multiple Faces of the PCA Paradigm; 1.3.1 Mathematical Issues; 1.3.2 Computational Issues; 1.3.3 Applications of PCA; 1.4 Future Perspectives; 1.5 Structure of the Book; References; 2 Probabilistic Cellular Automata in the Visual Arts; References
Part I Probability and Statistical Mechanics3 Basic Ideas to Approach Metastability in Probabilistic Cellular Automata; 3.1 Introduction; 3.2 Reversible Probabilistic Cellular Automata; 3.3 The Tuned Cross PCA; 3.4 Main Ingredients for Metastability; 3.5 Metastable Behavior of the Tuned Cross PCA; References; 4 Strategic Interaction in Interacting Particle Systems; 4.1 Introduction; 4.2 Strategic and Non-strategic Interaction; 4.2.1 An Equivalent Optimization Problem; 4.2.2 Strategic Interaction; 4.2.3 Trend-Driven Dynamics; 4.3 A Linear, Mean-Field Model; 4.3.1 The Non-strategic Case
4.3.2 The Strategic Case: The Utility with No Trend4.3.3 The Strategic Case: The Utility with Trend; 4.4 Conclusion; References; 5 Scaling and Inverse Scaling in Anisotropic Bootstrap Percolation; 5.1 Bootstrap Percolation Models; 5.2 A Tractable Example: The (1,2)-Model; 5.3 Inversion; 5.4 Generalisations: Related Models, Higher Dimensions and Other Graphs; References; 6 The Sandpile Cellular Automaton; 6.1 Introduction; 6.2 Definition, Examples and Some Remarks; 6.3 Key Properties; 6.3.1 The Least Action Principle; 6.3.2 Recurrent Sandpiles, Group Structure, Dhar's Formula
6.3.3 Bijection with Spanning Trees6.3.4 Some Computable Quantities; 6.4 Challenges; References; 7 Ising Model on the Torus and PCA Dynamics: Reversibility, Irreversibility, and Fast Tunneling; 7.1 Introduction; 7.2 Reversible and Irreversible PCA on 2D Torus; 7.2.1 Parallel Implementation of the Dynamics; 7.2.2 Stationary Measure of the Dynamics; 7.3 Gibbsian Regime of the Stationary Measure and Fast Mixing; 7.4 Numerical Experiments in the Non-Gibbsian Regime; References; 8 Synchronization in Interacting Reinforced Stochastic Processes; 8.1 A Natural Generalization of PCA Dynamics
8.2 Introduction to the Main Concepts8.2.1 What Is Synchronization?; 8.2.2 A Basic Model of Self-reinforcement: Classical Pólya urn; 8.2.3 Two Examples of Generalized Pólya urns; 8.3 Interacting Urns and Synchronization; 8.3.1 Mean Field Interacting Pólya urns; 8.3.2 First Remarks; 8.3.3 Simulations; 8.4 Main Results; 8.5 Sketch of Synchronization's Proof; 8.6 Related Models of Interacting Urns; 8.6.1 Forced Pólya urn Model; 8.6.2 Urns with a ``Preferred'' Color; 8.6.3 A Simple Finite Range Interacting Case; 8.7 Open Questions; References
Part I Probability and Statistical Mechanics3 Basic Ideas to Approach Metastability in Probabilistic Cellular Automata; 3.1 Introduction; 3.2 Reversible Probabilistic Cellular Automata; 3.3 The Tuned Cross PCA; 3.4 Main Ingredients for Metastability; 3.5 Metastable Behavior of the Tuned Cross PCA; References; 4 Strategic Interaction in Interacting Particle Systems; 4.1 Introduction; 4.2 Strategic and Non-strategic Interaction; 4.2.1 An Equivalent Optimization Problem; 4.2.2 Strategic Interaction; 4.2.3 Trend-Driven Dynamics; 4.3 A Linear, Mean-Field Model; 4.3.1 The Non-strategic Case
4.3.2 The Strategic Case: The Utility with No Trend4.3.3 The Strategic Case: The Utility with Trend; 4.4 Conclusion; References; 5 Scaling and Inverse Scaling in Anisotropic Bootstrap Percolation; 5.1 Bootstrap Percolation Models; 5.2 A Tractable Example: The (1,2)-Model; 5.3 Inversion; 5.4 Generalisations: Related Models, Higher Dimensions and Other Graphs; References; 6 The Sandpile Cellular Automaton; 6.1 Introduction; 6.2 Definition, Examples and Some Remarks; 6.3 Key Properties; 6.3.1 The Least Action Principle; 6.3.2 Recurrent Sandpiles, Group Structure, Dhar's Formula
6.3.3 Bijection with Spanning Trees6.3.4 Some Computable Quantities; 6.4 Challenges; References; 7 Ising Model on the Torus and PCA Dynamics: Reversibility, Irreversibility, and Fast Tunneling; 7.1 Introduction; 7.2 Reversible and Irreversible PCA on 2D Torus; 7.2.1 Parallel Implementation of the Dynamics; 7.2.2 Stationary Measure of the Dynamics; 7.3 Gibbsian Regime of the Stationary Measure and Fast Mixing; 7.4 Numerical Experiments in the Non-Gibbsian Regime; References; 8 Synchronization in Interacting Reinforced Stochastic Processes; 8.1 A Natural Generalization of PCA Dynamics
8.2 Introduction to the Main Concepts8.2.1 What Is Synchronization?; 8.2.2 A Basic Model of Self-reinforcement: Classical Pólya urn; 8.2.3 Two Examples of Generalized Pólya urns; 8.3 Interacting Urns and Synchronization; 8.3.1 Mean Field Interacting Pólya urns; 8.3.2 First Remarks; 8.3.3 Simulations; 8.4 Main Results; 8.5 Sketch of Synchronization's Proof; 8.6 Related Models of Interacting Urns; 8.6.1 Forced Pólya urn Model; 8.6.2 Urns with a ``Preferred'' Color; 8.6.3 A Simple Finite Range Interacting Case; 8.7 Open Questions; References