Transfer matrix in counting problems made easy

TL;DR

This paper constructs the transfer matrix for the 3-color problem, enabling efficient calculation of the number of configurations in finite systems with various boundary conditions, thus simplifying counting problems in statistical mechanics.

Contribution

It explicitly builds the transfer matrix for the 3-color problem, providing a new tool for counting configurations in related statistical mechanics models.

Findings

Transfer matrix constructed for the 3-color problem.

Calculations for systems with different boundary conditions.

Facilitates counting configurations in statistical models.

Abstract

The transfer matrix is a powerful technique that can be applied to statistical mechanics systems as, for example, in the calculus of the entropy of the ice model. One interesting way to study such systems is to map it onto a 3-color problem. In this paper, we explicitly build the transfer matrix for the 3-color problem in order to calculate the number of possible configurations for finite systems with free, periodic in one direction and toroidal boundary conditions (periodic in both directions)