Accurate lower bounds on two-dimensional constraint capacities from corner transfer matrices

TL;DR

This paper uses advanced mathematical techniques to compute tight lower bounds on the capacity of various two-dimensional constraint models, significantly improving previous results and suggesting a surprising equivalence between certain models.

Contribution

It introduces a novel application of corner transfer matrix methods to accurately estimate capacities of 2D constraints, providing the tightest bounds to date.

Findings

Significantly improved lower bounds on 2D constraint capacities

Conjecture that the capacities of even and charge(3) constraints are equal

Demonstrated the effectiveness of corner transfer matrix methods in this context

Abstract

We analyse the capacity of several two-dimensional constraint families - the exclusion, colouring, parity and charge model families. Using Baxter's corner transfer matrix formalism combined with the corner transfer matrix renormalisation group method of Nishino and Okunishi, we calculate very tight lower bounds and estimates on the growth rate of these models. Our results strongly improve previous known lower bounds, and lead to the surprising conjecture that the capacity of the even and charge(3) constraints are identical.