An exactly solvable self-avoiding walks model: II. Triangular and honeycomb lattices

Abstract

A modified self-avoiding walks model previously proposed for the square lattice is applied to the triangular and honeycomb lattices. In the model, the walker is restricted not to take any turn which will put the walker in a direction rotated by more than a certain angle, Phi max, from any of the directions previously taken. The generating functions are obtained, and various quantities are evaluated exactly. For all three (square, triangular and honeycomb) lattices, it is found that the model exhibits the characteristics reminiscent of one-dimensional self-avoiding walks in all directions, while strongly retaining the anisotropic effect of the direction of the first step. For the honeycomb lattice, the authors have studied the model for Phi max= pi (A-model) and Phi max=3/2 pi (B-model). Unlike other walks, the B-model exhibits a special property: the oscillations between even and odd steps in various quantities such as the number of N-step walks do not decay as the number of steps increases.