Book contents
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Notation
- Part I Tools and Theory
- 1 Background
- 2 Martingales
- 3 Markov Chains
- 4 Networks and Discrete Analysis
- Part II Results and Applications
- Appendices
- Appendix A Hilbert Space Background
- Appendix B Entropy
- Appendix C Coupling and Total Variation
- References
- Index
4 - Networks and Discrete Analysis
from Part I - Tools and Theory
Published online by Cambridge University Press: 16 May 2024
- Frontmatter
- Contents
- Preface
- Acknowledgments
- Notation
- Part I Tools and Theory
- 1 Background
- 2 Martingales
- 3 Markov Chains
- 4 Networks and Discrete Analysis
- Part II Results and Applications
- Appendices
- Appendix A Hilbert Space Background
- Appendix B Entropy
- Appendix C Coupling and Total Variation
- References
- Index
Summary
Here, we dive deeper into the realm of reversible Markov chains, via the perspective of network theory. The notions of conductance and resistance are defined, as well as voltage and current, and the corresponding mathematical theory.The Laplacian and Green function are defined and their relation to harmonic functions explained. The chapter culminates with a proof (using network theory) that recurrence and transience are essentially group properties: these properties remain invariant when changing between different reasonable random walks on the same group (specifically, symmetric and adapted with finite second moment).
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- Harmonic Functions and Random Walks on Groups , pp. 117 - 158Publisher: Cambridge University PressPrint publication year: 2024