TY - JOUR

T1 - Effect of confinement

T2 - Polygons in strips, slabs and rectangles

AU - Guttmann, Anthony J.

AU - Jensen, Iwan

PY - 2009

Y1 - 2009

N2 - 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 ℤ2, 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 ℤ3 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 ℤ2, 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 ℤ2 of width up to six, based on a transfer matrix formulation.

AB - 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 ℤ2, 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 ℤ3 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 ℤ2, 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 ℤ2 of width up to six, based on a transfer matrix formulation.

UR - http://www.scopus.com/inward/record.url?scp=64249145647&partnerID=8YFLogxK

U2 - 10.1007/978-1-4020-9927-4_10

DO - 10.1007/978-1-4020-9927-4_10

M3 - Article

AN - SCOPUS:64249145647

SP - 235

EP - 246

JO - Lecture Notes in Physics

JF - Lecture Notes in Physics

SN - 0075-8450

ER -