Polygons, Polyominoes and Polycubes

1. Front Matter

   Pages i-xix

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2. History and Introduction to Polygon Models and Polyominoes

   • Anthony J Guttmann

   Pages 1-21

3. Lattice Polygons and Related Objects

   • Stuart G Whittington

   Pages 23-41

4. Exactly Solved Models

   • Mireille Bousquet-Mélou, Richard Brak

   Pages 43-78

5. Why Are So Many Problems Unsolved?

   • Anthony J Guttmann

   Pages 79-91

6. The Anisotropic Generating Function of Self-Avoiding Polygons is not D-Finite

   • Andrew Rechnitzer

   Pages 93-115

7. Polygons and the Lace Expansion

   • Nathan Clisby, Gordon Slade

   Pages 117-142

8. Exact Enumerations

   • Ian G. Enting, Iwan Jensen

   Pages 143-179

9. Series Analysis

   • Anthony J. Guttmann, Iwan Jensen

   Pages 181-202

10. Monte Carlo Methods for Lattice Polygons

    • E. J. Janse van Rensburg

    Pages 203-233

11. Effect of Confinement: Polygons in Strips, Slabs and Rectangles

    • Anthony J Guttmann, Iwan Jensen

    Pages 235-246

12. Limit Distributions and Scaling Functions

    • Christoph Richard

    Pages 247-299

13. Interacting Lattice Polygons

    • Aleks L Owczarek, Stuart G Whittington

    Pages 301-315

14. Fully Packed Loop Models on Finite Geometries

    • Jan de Gier

    Pages 317-346

15. Conformal Field Theory Applied to Loop Models

    • Jesper Lykke Jacobsen

    Pages 347-424

16. Stochastic Lowner Evolution and the Scaling Limit of Critical Models

    • Bernard Nienhuis, Wouter Kager

    Pages 425-467

17. Appendix: Series Data and Growth Constant, Amplitude and Exponent Estimates

    • Anthony J Guttmann, Iwan Jensen

    Pages 469-482

18. Back Matter

    Pages 483-490

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The problem of counting the number of self-avoiding polygons on a square grid, - therbytheirperimeterortheirenclosedarea,is aproblemthatis soeasytostate that, at ?rst sight, it seems surprising that it hasn’t been solved. It is however perhaps the simplest member of a large class of such problems that have resisted all attempts at their exact solution. These are all problems that are easy to state and look as if they should be solvable. They include percolation, in its various forms, the Ising model of ferromagnetism, polyomino enumeration, Potts models and many others. These models are of intrinsic interest to mathematicians and mathematical physicists, but can also be applied to many other areas, including economics, the social sciences, the biological sciences and even to traf?c models. It is the widespread applicab- ity of these models to interesting phenomena that makes them so deserving of our attention. Here however we restrict our attention to the mathematical aspects. Here we are concerned with collecting together most of what is known about polygons, and the closely related problems of polyominoes. We describe what is known, taking care to distinguish between what has been proved, and what is c- tainlytrue,but has notbeenproved. Theearlierchaptersfocusonwhatis knownand on why the problems have not been solved, culminating in a proof of unsolvability, in a certain sense. The next chapters describe a range of numerical and theoretical methods and tools for extracting as much information about the problem as possible, in some cases permittingexactconjecturesto be made.

• Counting
• Lattice
• Mathematica
• Monte Carlo method
• Theoretical physics
• algorithms
• biology
• calculus
• computer
• computer science
• distribution
• geometry
• graphs
• mathematical physics
• statistics
• combinatorics

• Department of Mathematics and Statistics, The University of Melbourne, Victoria, Australia

  Anthony J. Guttman

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