A Monte Carlo approach for the solution of the simple quadratic and cubic self-avoiding random walk

A procedure for virtually memory-free Monte-Carlo simulation of the self-avoiding random walk (SAW) in a two-dimensional network and in a simple cubic lattice has been developed. It is highly efficient in yielding values of the intrachain meansquare end-to-end distance n > for chains up to n = 1000 and n = 400 steps, for the two and three-dimensional cases respectively. The number N of walks per each chain of n steps involved n = 500 to N = 3000.Accessing a medium-sized computer via a portable terminal, it was possible to obtain values for n up to 21, with N up to 1000, in minutes, and to obtain values for n up to 400 and 1000 with N up to 300 in approximately an hour of computing time.The principal finding is that for n > = anγ, the value of the exponent remains stable at approximately 6/5, consistent with the results of prior work involving complete enumeration for n up to 16 and n up to 10 in the two-dimensional and three-dimensional cases. No comparative data are available for higher levels of n because of the time constraints of memory-dependent computer programs.