Polygons, Polyominoes and Polycubes pp 235-246 | Cite as

# Effect of Confinement: Polygons in Strips, Slabs and Rectangles

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.

### Keywords

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