How to estimate the number of self-avoiding walks over 10^100? Use random walks
Abstract
Counting the number of N-step self-avoiding walks (SAWs) on a lattice is one of the most difficult problems of enumerative combinatorics. Once we give up calculating the exact number of them, however, we have a chance to apply powerful computational methods of statistical mechanics to this problem. In this paper, we develop a statistical enumeration method for SAWs using the multicanonical Monte Carlo method. A key part of this method is to expand the configuration space of SAWs to random walks, the exact number of which is known. Using this method, we estimate a number of N-step SAWs on a square lattice, c_N, up to N=256. The value of c_256 is 5.6(1)*10^108 (the number in the parentheses is the statistical error of the last digit) and this is larger than one googol (10^100).
Keywords
Cite
@article{arxiv.1304.7352,
title = {How to estimate the number of self-avoiding walks over 10^100? Use random walks},
author = {Nobu C. Shirai and Macoto Kikuchi},
journal= {arXiv preprint arXiv:1304.7352},
year = {2013}
}
Comments
5 pages, 3 figures, 1 table, to appear in proceedings of YSMSPIP in Sendai