Self-avoiding walks subject to boundary constraints

Abstract

A statistical study is carried out for self‐avoiding walks on infintely long square lattice strips, two and three lines wide. The two‐layer problem is solved completely and the three‐layer problem asymptotically. In each instance, the mean square end‐to‐end separation of a walk is found, as expected, to be asymptotically proportional to the square of the number of steps for long walks; in other words, 〈x²〉=an², where 〈x²〉 is the mean square x component of net distance traversed and n is the number of steps (assumed to be large). For the two‐layer problem, a=0.5236, and for the three‐layer problem, a=0.3899.