How to estimate the number of self-avoiding walks over 10^100? Use random walks

TL;DR

This paper introduces a statistical enumeration method using multicanonical Monte Carlo to estimate the number of N-step self-avoiding walks on a lattice, reaching sizes over 10^100, which was previously computationally infeasible.

Contribution

It develops a novel approach that expands the configuration space to random walks, enabling estimation of large self-avoiding walk counts beyond previous limits.

Findings

Estimated c_256 as 5.6(1)*10^108

Method extends enumeration to walks larger than 10^100

Demonstrates effectiveness of statistical mechanics in combinatorics

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).