### Abstract

In this chapter we will be considering the effect of confining polygons to lie in a bounded geometry. This has already been briefly discussed in Chapters 2 and 3, but here we give many more results. The simplest, non-trivial case is that of SAP on the two-dimensional square lattice ℤ^{2}, confined between two parallel lines, say x = 0 and x = w. This problem is essentially 1-dimensional, and as such is in principle solvable. As we shall show, the solution becomes increasingly unwieldy as the distance w between the parallel lines increases. Stepping up a dimension to the situation in which polygons in the simple-cubic lattice ℤ^{3} are confined between two parallel planes, that is essentially a two-dimensional problem, and as such is not amenable to exact solution. Self-avoiding walks in slits were first treated theoretically by Daoud and de Gennes [4] in 1977, and numerically by Wall et al. [14] the same year. Wall et al. studied SAW on ℤ^{2}, in particular the mean-square end-to-end distance. For a slit of width one they obtained exact results, and also obtained asymptotic results for a slit of width two. Around the same time, Wall and co-workers [13, 15] used Monte Carlo methods to study the width dependence of the growth constant for walks confined to strips of width w. In 1980 Klein [9] calculated the behaviour of SAW and SAP confined to strips in ℤ^{2} of width up to six, based on a transfer matrix formulation.

Original language | English |
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Pages (from-to) | 235-246 |

Number of pages | 12 |

Journal | Lecture Notes in Physics |

DOIs | |

Publication status | Published - 2009 |

Externally published | Yes |

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## Cite this

*Lecture Notes in Physics*, 235-246. https://doi.org/10.1007/978-1-4020-9927-4_10