A connection between the Ice-type model of Linus Pauling and the three-color problem

TL;DR

This paper explores the connection between Pauling's ice-type model and the three-color problem, using theoretical estimates, numerical simulations, and transfer-matrix methods to analyze entropy and configurations in two-dimensional lattices.

Contribution

It establishes a detailed mapping between the ice-type model and the three-color problem, providing numerical and analytical insights into their relationship.

Findings

Estimated low-temperature entropy matches numerical results.

Transfer-matrix calculations for lattices up to 225 sites.

Linear regression aligns with Lieb's exact solution.

Abstract

The ice-type model proposed by Linus Pauling to explain its entropy at low temperatures is here approached in a didactic way. We first present a theoretically estimated low-temperature entropy and compare it with numerical results. Then, we consider the mapping between this model and the three-colour problem, i.e.,colouring a regular graph with coordination equal to 4 (a two-dimensional lattice) with three colours, for which we apply the transfer-matrix method to calculate all allowed configurations for two-dimensional square lattices of $N$ oxygen atoms ranging from 4 to 225. Finally, from a linear regression of the transfer matrix results, we obtain an estimate for the case $N\rightarrow \infty $ which is compared with the exact solution by Lieb.