How to Estimate the Number of Self-Avoiding Walks over 10¹⁰⁰? 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 c₂₅₆ is 5.6(1)× 10¹⁰⁸ (the number in the parentheses is the statistical error of the last digit) and this is larger than one googol (10¹⁰⁰).