Self-avoiding walks in wedges

Abstract

The authors consider the number of self-avoiding walks confined to a subset Zd(f) of the d-dimensional hypercubic lattice Zd, such that the coordinates (x1,x2, . . .,xd) of each vertex in the walk satisfy x1>or=0 and 0<or=xk<or=fk(x1) for k=2,3, . . .,d. They show that if fk(x) to infinity as x to infinity , the connective constant of walks in Zd(f) is identical to the convective constant of walks in Zd. They also explore conditions on fk which lead to a smaller connective constant for walks in Zd(f) and, in particular, consider walks between two parallel (d-1)-dimensional hyperplanes. Finally they contrast some of these results with recent work by Grimmett on percolation on subsets of the square lattice.