Contents -- Introduction by C. N. Yang -- Preface -- COMMENTARIES -- 1. Dimer Statistics -- Introduction -- The pfaffian approach -- The rectangular lattice -- Rectangular lattice with a boundary vacancy -- The honeycomb and kagome lattices -- Solution of a three-dimensional dimer model -- Remarks -- References for Chapter 1 -- 2. The Vertex Model -- Introduction -- The eight-vertex model (rectangular lattice) -- The five-vertex model (rectangular lattice) -- The eight-vertex model (honeycomb lattice) -- Vertex models in higher dimensions -- Remarks -- References for Chapter 2 -- 3. Duality and Gauge Transformations -- Introduction -- Duality relation for the Potts and the chiral Potts models -- Gauge transformation for the vertex model and syzygies -- Duality relation for frustrated spin systems -- Duality relation for Potts correlation functions -- References for Chapter 3 -- 4. The Ising Model -- Introduction -- Ising representation of the eight-vertex model -- The Baxter-Wu model -- Density of Fisher zeroes -- Solution of the Ising model on nonorientable surfaces -- Remarks -- References for Chapter 4 -- 5. The Potts Model -- Introduction -- Graphical formulation of the Potts model -- Rigorous determination of the Potts critical point -- Potts partition function zeroes -- Remarks -- References for Chapter 5 -- 6. Critical Frontiers -- Introduction -- The Potts model -- The antiferromagnetic Ising model in a magnetic field -- The Blume-Emery-Griffiths model -- The Ashkin-Teller model -- The O(n) model -- Remarks -- References for Chapter 6 -- 7. Percolation -- Introduction -- The random-cluster model and the Potts model -- Percolation thresholds -- An infinite-range bond percolation -- Directed percolation -- Remarks -- References for Chapter 7 -- 8. Graph Theory -- Introduction -- Graphical analyses in many-body theory.

The Potts model and the Tutte polynomial -- Random graphs -- Spanning trees -- Graphical analysis of lattice models -- References for Chapter 8 -- 9. Knot Invariants -- Introduction -- Lattice models and knot invariants -- Two new knot invariants -- Knot invariants from spin models -- Bracket polynomial and the non intersecting string model -- Remarks -- References for Chapter 9 -- 10. Other Topics -- Theory of electrical circuits -- Quantization of the orbital angular momentum -- The vicious neighbor problem -- Restricted partitions of an integer -- The Hubbard model -- Review of a book by Ta-You Wu -- Eulogy on Shang-Keng Ma -- Professor C. N. Yang and statistical mechanics -- References for Chapter 10 -- Photographs -- Reprinted Papers -- REPRINTED PAPERS -- 1. Dimer Statistics -- PI Exactly Soluble Model of the Ferroelectric Phase Transition in Two Dimensions F. Y. Wu -- P2 Dimers on Two-Dimensional Lattices F. Y. Wu -- 1. Introduction -- 2. The Simple-Quartic Lattice -- 3. The Honeycomb Lattice -- 4. The Checkerboard Square Lattice -- 5. The Triangular Lattice -- 6. The Kagome Lattice -- 7. The 3-12 Lattice -- 8. The 4-8 Lattice -- 9. The Union Jack Lattice -- 10. The Dual of the 3-12 Lattice -- 11. Summary -- Acknowledgments -- References -- P3 Close-Packed Dimers on Nonorientable Surfaces W. T. Lu and F. Y. Wu -- 1. Introduction -- 2. Summary of results -- 3. The Mobius strip -- 3.1. M = even Mobius strip -- 3.2. M = odd Mobius strip -- 4. The Klein bottle -- 4.1. M = even Klein bottle -- 4.2. M = odd Klein bottle -- 5. A reciprocity theorem -- 6. Summary and discussion -- Acknowledgement -- References -- P4 Pfaffian Solution of a Dimer-Monomer Problem: Single Monomer on the Boundary F. Y. Wu -- I. INTRODUCTION -- II. THE SINGLE-MONOMER PROBLEM -- III. EIGENVALUES OF THE DETERMINANT A -- IV. EVALUATION OF THE COFACTOR -- V. DISCUSSIONS.

P5 Remarks on the Modified Potassium Dihydrogen Phosphate Model of a Ferroelectric F. Y. Wu -- I. INTRODUCTION -- II. GENERALIZED MODEL -- III. MODlFmD KDP MODEL IN AN ARBITRARY FIELD -- ACKNOWLEDGMENT -- P6 Exact Solution of Close-Packed Dimers on the Kagome Lattice F. Wang and F. Y. Wu -- I. INTRODUCTION -- II. PFAFFIAN APPROACH -- III. VERTEX·MODEL APPROACH -- IV. DIMER.DIMER CORRELATION -- ACKNOWLEDGMENT -- P7 Exact Solution of a Three-Dimensional Dimer System H. Y. Huang, V. Popkov and F. Y. Wu -- 2. The Vertex Model -- P8 General Lattice Model of Phase Transitions C. Fan and F. Y. Wu -- I. INTRODUCTION -- II. DEFINITION OF PROBLEM -- III. GENERAL CONSIDERATIONS -- IV. FREE·FERMION MODEL -- V. NUMERICAL EXAMPLE -- VI. CONCLUSIONS -- ACKNOWLEDGMENTS -- APPENDIX -- P9 Staggered Ice-Rule Model: The Pfaffian Solution F. Y. Wu and K. Y. Lin -- 1. INTRODUCTION -- II. DEFINITION OF THE MODEL -- III. RELATIONSHIP WITH OTHER LATTICE STATISTICAL PROBLEMS -- IV. PFAFF IAN SOLUTION -- V. APPLICATIONS -- ACKNOWLEDGMENT -- APPENDIX B-PROPERTIES OF s( ) -- P10 Staggered Eight-Vertex Model C. S. Hsue, K. Y. Lin and F. Y. Wu -- I. INTRODUCTION -- 11. SYMMETRY RELATIONS -- III. EQUIVALENCE WITH AN ISING MODEL -- IV. STAGGERED FREE-FERMION MODEL -- V. ANALYTIC PROPERTIES OF -- ACKNOWLEDGMENT -- APPENDIX A: PFAFFIAN SOLUTION -- P11 The Odd Eight-Vertex Model F. Y. Wu and H. Kunz -- 1. INTRODUCTION -- 2. EQUIVALENCE WITH A STAGGERED VERTEX MODEL -- 3. THE FREE-FERMION SOLUTION -- 4. EQUIVALENCE WITH THE FREE-FERMION MODEL OF FAN AND WU -- 5. ISING REPRESENTATIONS OF THE FREE-FERMION MODEL -- 6. SUMMARY -- ACKNOWLEDGMENTS -- REFERENCES -- P12 Eight-Vertex Madelon the Honeycomb Lattice F. Y. Wu -- 1. INTRODUCTION -- 2. DEFINITION OF THE MODEL -- 3. SYMMETRY RELATIONS -- 4. SPECIAL SOLUTIONS -- 5. GENERAL CASE -- 6. SUMMARY -- ACKNOWLEDGMENTS.

APPENDIX: ISING PARTITION FUNCTION -- P13 Eight-Vertex Model and Ising Model in a Non-zero Magnetic Field: Honeycomb Lattice F. Y. Wu -- References -- P14 Phase Transition in a Vertex Model in Three Dimensions F. Y. Wu -- P15 Exact Solution of a Vertex Model in d Dimensions F. Y. Wu and H. Y. Huang -- 1. Introduction -- 2. An Equivalent Dimer Problem -- 3. The Exact Solution -- 4. The Critical Behavior -- 5. Discussions -- Acknowledgement -- References -- 3. Duality and Gauge Transformations -- P16 Duality Transformation in a Many-Component Spin Model F. Y. Wu and Y. K. Wang -- P17 Duality Properties of a General Vertex Model X. N. Wu and F. Y. Wu -- References -- PI8 Algebraic Invariants of the 0(2) Gauge Transformation J. H. H. Perk, F. Y. Wu and X. N. Wu -- References -- PI9 The 0(3) Gauge Transformation and 3-state Vertex Models L. H. Gwa and F. Y. Wu -- References -- P20 Duality Relation for Frustrated Spin Models D. H. Lee and F. Y. Wu -- I. THE FRUSTRATED ISING MODEL -- II. POTTS SPIN GLASS -- III. SUMMARY -- ACKNOWLEDGMENTS -- P2I Duality Relations for Potts Correlation Functions F.Y. Wu -- 1. Introduction -- 2. The two-point correlation function -- 3. The three-point correlation -- 4. Summary -- Acknowledgement -- References -- P22 Sum Rule Identities and the Duality Relation for the Potts n-Point Boundary Correlation Function F. Y. Wu and H. Y. Huang -- 4. The Ising Model -- P23 Ising Model with Four-Spin Interactions F. Y. Wu -- P24 Exact Solution of an Ising Model with Three-spin Interactions on a Triangular Lattice R. J. Baxter and F. Y. Wu -- P25 Density of the Fisher Zeroes for the Ising Model W. T. Lu and F. Y. WU -- 1. INTRODUCTION -- 2. THE SIMPLE-QUARTIC LATTICE -- 3. THE TRIANGULAR LATTICE -- 4. SIMPLE-QUARTIC ISING MODEL IN A FIELD irr/2 -- 5. TRIANGULAR ISING MODEL IN A FIELD in/2 -- 6. THE HONEYCOMB AND KAGOME LATTICES.

APPENDIX. TWO INTEGRATION IDENTITIES -- ACKNOWLEDGMENTS -- REFERENCES -- P26 Ising Model on Nonorientable Surfaces: Exact Solution for the Mobius Strip and the Klein Bottle W. T. Lu and F. Y. Wu -- I. INTRODUCTION -- II. 2MXN MOBIUS STRIP -- III. EVALUATION OF THE PFAFFlAN -- IV. (2M-l)XN MOBIUS STRIP -- V. KLEIN BOTTLE -- VI. BULK LIMIT AND FINITE-SIZE CORRECTIONS -- VII. SUMMARY -- ACKNOWLEDGMENT -- 5. The Potts Model -- P27 Equivalence of the Potts Model or Whitney Polynomial with an Ice-Type Model R. J. Baxter, S. B. Kelland and F. Y. Wu -- 1. Introduction -- 2. Potts model and Whitney polynomial -- 3. Planar lattices: the surrounding lattice -- 4. Polygon decompositions of -- 5. Equivalent ice-type model on -- 5.1. Proof of equivalence -- 5.2. Four-colouring problem -- 6. The regular lattices -- 6.1. Duality -- 6.2. Cyclic boundary conditions -- References -- P28 The Potts Model F. Y. Wu -- I. INTRODUCTION -- II. DUALITY RELATIONS -- III. SERIES EXPANSIONS -- IV. RELATION WITH OTHER PROBLEMS -- V. CRITICAL PROPERTIES -- VI. RANDOM·BOND MODEL -- VII. UNSOLVED PROBLEMS -- ACKNOWLEDGMENTS -- REFERENCES -- P29 Potts Model of Magnetism (Invited) F. Y. Wu -- I. INTRODUCTION -- II. CRITICAL DIMENSIONS -- III. CRITICAL PROPERTIES -- IV. ANTI FERROMAGNETIC MODEL -- ACKNOWLEDGMENTS -- P30 Exact Results for the Potts Model in Two Dimensions A. Hintermann, H. Kunz and F. Y. Wu -- 1. INTRODUCTION -- 2. POTTS CRITICAL POINT -- 3. CORRELATION FUNCTIONS -- APPENDIX. PROOF OF PROPERTY (ii) -- ACKNOWLEDGMENT -- REFERENCES -- P31 Partition Function Zeros of the Square Lattice Potts Model C. N. Chen, C. K. Hu and F. Y. Wu -- 6. Critical Frontiers -- P32 Critical Point of Planar Potts Models F. Y. Wu -- References -- P33 New Critical Frontiers for the Potts and Percolation Models F. Y. Wu.

P34 Critical Frontier of the Antiferromagnetic Ising Model in a Magnetic Field: The Honeycomb Lattice F. Y. Wu, X. N. Wu and H. W. J. Blote.