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　　《美國數學會經典影印係列：Markov鏈與混閤時間（影印版）》是對Markov鏈理論的現代處理方法的導引，該方法的主要目標是確定一個Markov鏈收斂到作為態空間體積和幾何之函數的平穩分布的收斂速率。作者發展瞭估計收斂時間的關鍵工具，包括耦閤、強平穩時間以及譜方法；一有可能，便強調概率論式的方法。
　　《美國數學會經典影印係列：Markov鏈與混閤時間（影印版）》包括瞭許多例題並對統計力學的中心模型給齣瞭簡短介紹；還講述瞭網絡上的隨機遊動，包括擊中和掩蓋時間，以及對洗牌的各種方法的分析。至於預備知識，作者假定瞭對概率論的適度瞭解以及大學水平的綫性代數。
　　《美國數學會經典影印係列：Markov鏈與混閤時間（影印版）》打算將這個活躍的研究領域的激情帶給大範圍的受眾。

內頁插圖

目錄

Preface
Overview
For the Reader
For the Instructor
For the Expert
Acknowledgements

Part Ⅰ： Basic Methods and Examples
Chapter 1． Introduction to Finite Markov Chains
1．1． Finite Markov Chains
1．2． Random Mapping Representation
1．3． Irreducibility and Aperiodicity
1．4． Random Walks on Graphs
1．5． Stationary Distributions
1．6． Reversibility and Time Reversals
1．7． Classifying the States of a Markov Chain
Exercises
Notes
Chapter 2． Classical （and Useful） Markov Chains
2．1． Gambler's Ruin
2．2． Coupon Collecting
2．3． The Hypercube and the Ehrenfest Urn Model
2．4． The Polya Urn Model
2．5． Birth-and-Death Chains
2．6． Random Walks on Groups
2．7． Random Walks on Z and Reflection Principles
Exercises
Notes
Chapter 3． Markov Chain Monte Carlo： Metropolis and Glauber Chains
3．1． Introduction
3．2． Metropolis Chains
3．3． Glauber Dynamics
Exercises
Notes
Chapter 4． Introduction to Markov Chain Mixing
4．1． Total Variation Distance
4．2． Coupling and Total Variation Distance
4．3． The Convergence Theorem
4．4． Standardizing Distance from Stationarity
4．5． Mixing Time
4．6． Mixing and Time Reversal
4．7． Ergodic Theorem*
Exercises
Notes
Chapter 5． Coupling
5．1． Definition
5．2． Bounding Total．Variation Distance
5．3． Examples
5．4． Grand Couplings
Exercises
Notes
Chapter 6． Strong Stationary Times
6．1． Top-to-Random Shuffle
6．2． Definitions
6．3． Achieving Equilibrium
6．4． Strong Stationary Times and Bounding Distance
6．5． Examples
6．6． Stationary Times and Cesaro Mixing Time
Exercises
Notes
Chapter 7． Lower Bounds on Mixing Times
7．1． Counting and Diameter Bounds
7．2． Bottleneck Ratio
7．3． Distinguishing Statistics
7．4． Examples
Exercises
Notes
Chapter 8． The Symmetric Group and Shuffling Cards
8．1． The Symmetric Group
8．2． Random Transpositions
8．3． Riffle Shuffles
Exercises
Notes
Chapter 9． Random Walks on Networks
9．1． Networks and Reversible Markov Chains
9．2． Harmonic Functions
9．3． Voltages and Current Flows
9．4． Effective Resistance
9．5． Escape Probabilities on a Square
Exercises
Notes
Chapter 10． Hitting Times
10．1． Definition
10．2． Random Target Times
10．3． Commute Time
10．4． Hitting Times for the Torus
10．5． Bounding Mixing Times via Hitting Times
10．6． Mixing for the Walk on Two Glued Graphs
Exercises
Notes
Chapter 11． Cover Times
11．1． Cover Times
11．2． The Matthews Method
11．3． Applications of the Matthews Method
Exercises
Notes
Chapter 12． Eigenvalues
12．1． The Spectral Representation of a Reversible Transition Matrix
12．2． The Relaxation Time
12．3． Eigenvalues and Eigenfunctions of Some Simple Random Walks
12．4． Product Chains
12．5． An l2 Bound
12．6． Time Averages
Exercises
Notes

Part Ⅱ： The Plot Thickens
Chapter 13． Eigenfunctions and Comparison of Chains
13．1． Bounds on Spectral Gap via Contractions
13．2． Wilson's Method for Lower Bounds
13．3． The Dirichlet Form and the Bottleneck Ratio
13．4． Simple Comparison of Markov Chains
13．5． The Path Method
13．6． Expander Graphs
Exercises
Notes
Chapter 14． The Transportation Metric and Path Coupling
14．1． The Transportation Metric
14．2． Path Coupling
14．3． Fast Mixing for Colorings
14．4． Approximate Counting
Exercises
Notes
Chapter 15． The Ising Model
15．1． Fast Mixing at High Temperature
15．2． The Complete Graph
15．3． The Cycle
15．4． The Tree
15．5． Block Dynamics
15．6． Lower Bound for Ising on Square
Exercises
Notes
Chapter 16． From Shuffling Cards to Shuffling Genes
16．1． Random Adjacent Transpositions
16．2． Shuffling Genes
Exercise
Notes
Chapter 17． Martingales and Evolving Sets
17．1． Definition and Examples
17．2． Optional Stopping Theorem
17．3． Applications
17．4． Evolving Sets
17．5． A General Bound on Return Probabilities
17．6． Harmonic Fhnctions and the Doob h-Transform
17．7． Strong Stationary Times from Evolving Sets
Exercises
Notes
Chapter 18． The Cutoff Phenomenon
18．1． Definition
18．2． Examples of Cutoff
18．3． A Necessary Condition for Cutoff
18．4． Separation Cutoff
Exercise
Notes
Chapter 19． Lamplighter Walks
19．1． Introduction
19．2． Relaxation Time Bounds
19．3． Mixing Time Bounds
19．4． Examples
Notes
Chapter 20． Continuous-Time Chains
20．1． Definitions
20．2． Continuous-Time Mixing
20．3． Spectral Gap
20．4． Product Chains
Exercises
Notes
Chapter 21． Countable State Space Chains
21．1． Recurrence and Transience
21．2． Infinite Networks
21．3． Positive Recurrence and Convergence
21．4． Null Recurrence and Convergence
21．5． Bounds on Return Probabilities
Exercises
Notes
Chapter 22． Coupling from the Past
22．1． Introduction
22．2． Monotone CFTP
22．3． Perfect Sampling via Coupling from the Past
22．4． The Hardcore Model
22．5． Random State of an Unknown Markov Chain
Exercise
Notes
Chapter 23． Open Problems
23．1． The Ising Model
23．2． Cutoff
23．3． Other Problems

Appendix A． Background Material
A．1． Probability Spaces and Random Variables
A．2． Metric Spaces
A．3． Linear Algebra
A．4． Miscellaneous

Appendix B． Introduction to Simulation
B．1． What Is Simulation?
B．2． Von Neumann Unbiasing
B．3． Simulating Discrete Distributions and Sampling
B．4． Inverse Distribution Function Method
B．5． Acceptance-Rejection Sampling
B．6． Simulating Normal Random Variables
B．7． Sampling from the Simplex
B．8． About Random Numbers
B．9． Sampling from Large Sets
Exercises
Notes

Appendix C． Solutions to Selected Exercises
Bibliography
Notation Index
Index

前言/序言

　　近年來，我國的科學技術取得瞭長足進步，特彆是在數學等自然科學基礎領域不斷湧現齣一流的研究成果。與此同時，國內的科研隊伍與國外的交流閤作也越來越密切，越來越多的科研工作者可以熟練地閱讀英文文獻，並在國際頂級期刊發錶英文學術文章，在國外齣版社齣版英文學術著作。
　　然而，在國內閱讀海外原版英文圖書仍不是非常便捷。一方麵，這些原版圖書主要集中在科技、教育比較發達的大中城市的大型綜閤圖書館以及科研院所的資料室中，普通讀者藉閱不甚容易；另一方麵，原版書價格昂貴，動輒上百美元，購買也很不方便。這極大地限製瞭科技工作者對於國外先進科學技術知識的獲取，間接阻礙瞭我國科技的發展。
　　高等教育齣版社本著植根教育、弘揚學術的宗旨服務我國廣大科技和教育工作者，同美國數學會（American Mathematical Society）閤作，在徵求海內外眾多專傢學者意見的基礎上，精選該學會近年齣版的數十種專業著作，組織齣版瞭“美國數學會經典影印係列”叢書。美國數學會創建於1888年，是國際上極具影響力的專業學術組織，目前擁有近30000會員和580餘個機構成員，齣版圖書3500多種，馮.諾依曼、萊夫謝茨、陶哲軒等世界級數學大傢都是其作者。本影印係列涵蓋瞭代數、幾何、分析、方程、拓撲、概率、動力係統等所有主要數學分支以及新近發展的數學主題。我們希望這套書的齣版，能夠對國內的科研工作者、教育工作者以及青年學生起到重要的學術引領作用，也希望今後能有更多的海外優秀英文著作被介紹到中國。
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