Multicanonical simulation of the Domb-Joyce model and the Go model: new enumeration methods for self-avoiding walks

TL;DR

This paper introduces new enumeration methods for self-avoiding walks using multicanonical Monte Carlo sampling, estimating walk counts up to 256 steps and comparing two statistical models for efficiency.

Contribution

The paper develops and applies multicanonical Monte Carlo methods to enumerate self-avoiding walks, comparing the Go and Domb-Joyce models for enumeration efficiency.

Findings

Domb-Joyce model outperforms the Go model in enumeration efficiency.

Estimated the number of 2D self-avoiding walks up to 256 steps.

Identified the impact of phase transitions on model performance.

Abstract

We develop statistical enumeration methods for self-avoiding walks using a powerful sampling technique called the multicanonical Monte Carlo method. Using these methods, we estimate the numbers of the two dimensional N-step self-avoiding walks up to N=256 with statistical errors. The developed methods are based on statistical mechanical models of paths which include self-avoiding walks. The criterion for selecting a suitable model for enumerating self-avoiding walks is whether or not the configuration space of the model includes a set for which the number of the elements can be exactly counted. We call this set a scale fixing set. We selected the following two models which satisfy the criterion: the Go model for lattice proteins and the Domb-Joyce model for generalized random walks. There is a contrast between these two models in the structures of the configuration space. The configuration space of the Go model is defined as the universal set of self-avoiding walks, and the set of the ground state conformation provides a scale fixing set. On the other hand, the configuration space of the Domb-Joyce model is defined as the universal set of random walks which can be used as a scale fixing set, and the set of the ground state conformation is the same as the universal set of self-avoiding walks. From the perspective of enumeration performance, we conclude that the Domb-Joyce model is the better of the two. The reason for the performance difference is partly explained by the existence of the first-order phase transition of the Go model.