Upper bounds on the growth rates of hard squares and related models via corner transfer matrices

TL;DR

This paper introduces a novel, efficient method combining combinatorics, statistical mechanics, and linear algebra to significantly tighten upper bounds on the growth rates of hard squares and related models, reducing the gap with known lower bounds.

Contribution

It develops a new upper bound algorithm using corner transfer matrix techniques and the Collatz-Wielandt formula, improving computational efficiency and accuracy over previous methods.

Findings

Reduced the gap between upper and lower bounds by 4-6 orders of magnitude.

Developed a faster, parallelizable algorithm for upper bounds calculation.

Achieved dramatic improvements over previous best known bounds.

Abstract

We study the growth rate of the hard squares lattice gas, equivalent to the number of independent sets on the square lattice, and two related models - non-attacking kings and read-write isolated memory. We use an assortment of techniques from combinatorics, statistical mechanics and linear algebra to prove upper bounds on these growth rates. We start from Calkin and Wilf's transfer matrix eigenvalue bound, then bound that with the Collatz-Wielandt formula from linear algebra. To obtain an approximate eigenvector, we use an ansatz from Baxter's corner transfer matrix formalism, optimised with Nishino and Okunishi's corner transfer matrix renormalisation group method. This results in an upper bound algorithm which no longer requires exponential memory and so is much faster to calculate than a direct evaluation of the Calkin-Wilf bound. Furthermore, it is extremely parallelisable and so allows us to make dramatic improvements to the previous best known upper bounds. In all cases we reduce the gap between upper and lower bounds by 4-6 orders of magnitude.