The Number of Independent Sets in Hexagonal Graphs

TL;DR

This paper introduces a new method to tightly bound the growth rate of independent sets in hexagonal lattices, significantly improving previous bounds and matching Baxter's numerical estimate to high precision.

Contribution

The authors develop a novel approach to derive rigorous bounds on the growth rate of independent sets in hexagonal graphs, refining prior estimates.

Findings

Established bounds: 1.546440708536001 <= η <= 1.5513

Improved previous bounds from 1.5463 <= η <= 1.5527

Lower bound matches Baxter's estimate to 9 decimal places

Abstract

A new method is proposed to derive rigorous bounds on {\eta}, the growth rate of the logarithm of the number of independent sets on a hexagonal lattice. Specifically, we prove that 1.546440708536001 <= {\eta} <= 1.5513, which improves upon the best known 1.5463 <= {\eta} <= 1.5527 due to Nagy and Zeger. Our lower bound matches the numerical estimate of Baxter up to 9 digits after the decimal point.