Monte Carlo of Chains with Excluded Volume: a Way to Evade Sample Attrition

TLDR: In this paper, a new method for the generation of long self-avoiding walks on lattice is described, where short self avoiding walks are generated by direct Monte Carlo method, then linked in pairs to form "dimers".

Abstract: A new method for the generation of long self‐avoiding walks on lattice is described: Short self‐avoiding walks—say of length N = 50—are generated by direct Monte Carlo method, then linked in pairs to form “dimers.” Each dimer is tested for intersections between its two halves—those passing the test giving a sample of self‐avoiding walks N = 100, which is dimerized in turn, etc. In this manner the repeated checking for intersections formed with only a few steps (short loops) is substantially avoided, while precisely such intersections are responsible for the heavy attrition with the direct Monte Carlo method. Thus the attrition accompanying the dimerization is quite insignificant even for very large N. With the help of this method walks N = 50 × 27 = 6400 were generated on the 4‐choice cubic lattice, for which the expansion coefficient of the end to end distance is α = 2.2. (The limit reached by others is N ≃ 2000 on the tetrahedral lattice, corresponding to only α ≃ 1.6, while α = 2.2 would require a leng...