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Markov Chains and Mixing Times 2nd Edition by David A Levin, Yuval Peres ISBN 1470429624 978-1470429621

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ISBN 10: 1470429624

ISBN 13: 978-1470429621

Author: David A Levin, Yuval Peres 

This book is an introduction to the modern theory of Markov chains, whose goal is to determine the rate of convergence to the stationary distribution, as a function of state space size and geometry. This topic has important connections to combinatorics, statistical physics, and theoretical computer science. Many of the techniques presented originate in these disciplines.The central tools for estimating convergence times, including coupling, strong stationary times, and spectral methods, are developed. The authors discuss many examples, including card shuffling and the Ising model, from statistical mechanics, and present the connection of random walks to electrical networks and apply it to estimate hitting and cover times.The first edition has been used in courses in mathematics and computer science departments of numerous universities. The second edition features three new chapters (on monotone chains, the exclusion process, and stationary times) and also includes smaller additions and corrections throughout. Updated notes at the end of each chapter inform the reader of recent research developments.

Markov Chains and Mixing Times 2nd Table of contents: 

 

Part 1: Introduction to Markov Chains

  1. Introduction to Markov Chains

    • What are Markov Chains?
    • Basic Terminology and Definitions
    • Examples of Markov Chains
    • The Importance of Mixing Times

  2. Discrete-Time Markov Chains

    • State Space and Transitions
    • Transition Matrices and Probability Vectors
    • Long-term Behavior and Stationary Distributions
    • Examples of Discrete-Time Chains

Part 2: The Theory of Mixing Times 3. Convergence to Stationarity

  • Total Variation Distance
  • Convergence Rates and Mixing Times
  • Coupling Methods
  • Applications and Examples

  1. Conductance and Rapid Mixing

    • Definition of Conductance
    • Relationship Between Conductance and Mixing Time
    • Expander Graphs and Mixing Times

  2. Cheeger’s Inequality and Eigenvalues

    • The Spectrum of Markov Chains
    • Cheeger’s Inequality and Its Implications for Mixing Times
    • Eigenvalue Gaps and Mixing Rates

Part 3: Advanced Topics in Mixing Times 6. Random Walks and Mixing

  • Random Walks on Graphs
  • Mixing Time Bounds for Specific Graphs
  • The Role of Random Walks in Markov Chains

  1. Geometric and High-dimensional Markov Chains

    • Geometric Chains and Mixing Time Bounds
    • High-dimensional State Spaces and Mixing Rates
    • Applications in Random Sampling

  2. Refined Techniques for Bounding Mixing Times

    • Coupling from the Past
    • The Strong Stationary Time Algorithm
    • Multiple Markov Chains and Parallelization

Part 4: Applications of Markov Chains 9. Applications in Randomized Algorithms

  • Markov Chains in Algorithms for Random Sampling
  • Monte Carlo Methods and Markov Chains
  • Markov Chains in Approximation Algorithms
  1. Applications in Statistical Physics and Probability
    • Markov Chains in the Study of Physical Systems
    • Glauber Dynamics and Ising Models
    • The Metropolis-Hastings Algorithm and Its Variants

Part 5: Further Topics and Open Problems 11. Further Research and Open Problems – Current Research Directions in Markov Chains and Mixing Times – Open Problems in the Study of Mixing Times – Future Applications of Markov Chains in Computational Science

Appendices

  • Mathematical Background: Probability Theory and Linear Algebra
  • Additional Resources for Further Study

References

Index

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